NUMERICAL STUDIES OF THE KLEIN-GORDON-SCHRÖDINGER EQUATIONS LI YANG. (M.Sc., Sichuan University)

Transcription

1 NUMERICAL STUDIES OF THE KLEIN-GORDON-SCHRÖDINGER EQUATIONS LI YANG (M.Sc., Sichuan University) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 6

2 Acknowledgements I would like to thank my advisor, Associate Professor Bao Weizhu, who gave me the opportunity to work on such an interesting research project, paid patient guidance to me, reviewed my thesis and gave me much invaluable help and constructive suggestions on it. It is also my pleasure to epress my appreciation and gratitude to Zhang Yanzhi, Wang Hanquan, and Lim Fongyin, from whom I got valuable suggestions and great help on my research project. I would also wish to thank the National University of Singapore for her financial support by awarding me the Research Scholarship during the period of my MSc candidature. My sincere thanks go to the Mathematics Department of NUS for its kind help during my two-year study here. Li Yang June 6 ii

3 Contents Acknowledgements ii Summary vi List of Tables i List of Figures Introduction. Physical background The problem Contemporary studies Overview of our work Numerical studies of the Klein-Gordon equation 8. Derivation of the Klein-Gordon equation Conservation laws of the Klein-Gordon equation Numerical methods for the Klein-Gordon equation iii

4 Contents iv.3. Eisting numerical methods Our new numerical method Numerical results of the Klein-Gordon equation Comparison of different methods Applications of CN-LF-SP The Klein-Gordon-Schrödinger equations 6 3. Derivation of the Klein-Gordon-Schrödinger equations Conservation laws of the Klein-Gordon-Schrödinger equations Dynamics of mean value of the meson field Plane wave and soliton wave solutions of KGS Reduction to the Schrödinger-Yukawa equations (S-Y) Numerical studies of the Klein-Gordon-Schrödinger equations Numerical methods for the Klein-Gordon-Schrödinger equations Time-splitting for the nonlinear Schrödinger equation Phase space analytical solver+time-splitting spectral discretizations (PSAS-TSSP) Crank-Nicolson leap-frog time-splitting spectral discretizations (CN-LF-TSSP) Properties of numerical methods For plane wave solution Conservation and decay rate Dynamics of mean value of meson field Stability analysis Numerical results of the Klein-Gordon-Schrödinger equation Comparisons of different methods

5 Contents v 4.3. Application of our numerical methods Application to the Schrödinger-Yukawa equations 7 5. Introduction to the Schrödinger-Yukawa equations Numerical method for the Schrödinger-Yukawa equations Numerical results of the Schrödinger-Yukawa equations Convergence of KGS to S-Y in nonrelativistic limit regime Applications Conclusion 78

6 Summary In this thesis, we present a numerical method for the nonlinear Klein-Gordon equation and two numerical methods for studying solutions of the Klein-Gordon-Schrödinger equations.we begin with the derivation of the Klein-Gordon equation (KG) which describes scalar (or pseudoscalar) spinless particles, analyze its properties and present Crank-Nicolson leap-frog spectral method (CN-LF-SP) for numerical discretization of the nonlinear Klein-Gordon equation. Numerical results for the Klein-Gordon equation demonstrat that the method is of spectral-order accuracy in space and second-order accuracy in time and it is much better than the other numerical methods proposed in the literature. It also preserves the system energy, linear momentum and angular momentum very well in the discretized level. We continue with the derivation of the Klein-Gordon-Schrödinger equations (KGS) which describes a system of conserved scalar nucleons interacting with neutral scalar mesons coupled through the Yukawa interaction and analyze its properties. Two efficient and accurate numerical methods are proposed for numerical discretization of the Klein-Gordon-Schrödinger equations. They are phase space analytical solver+timesplitting spectral method (PSAS-TSSP) and Crank-Nicolson leap-frog time-splitting spectral method (CN-LF-TSSP). These methods are eplicit, unconditionally stable, of spectral accuracy in space and second order accuracy in time, easy to etend vi

7 Summary vii to high dimensions, easy to program, less memory-demanding, and time reversible and time transverse invariant. Furthermore, they conserve (or keep the same decay rate of) the wave energy in KGS when there is no damping (or a linear damping) term, give eact results for plane-wave solutions of KGS, and keep the same dynamics of the mean value of the meson field in discretized level. We also apply our new numerical methods to study numerically soliton-soliton interaction of KGS in D and dynamics of KGS in D. We numerically find that, when a large damping term is added to the Klein-Gordon equation, bound state of KGS can be obtained from the dynamics of KGS when time goes to infinity. Finally, we etend our numerical method, time-splitting spectral method (TSSP) to the Schrödinger-Yukawa equations and present the numerical results of the Schrödinger-Yukawa equations in D and D cases. The thesis is organized as follows: Chapter introduces the physical background of the Klein-Gordon equation and the Klein-Gordon-Schrödinger equations. We also review some eisting results of them and report our main results. In Chapter, the Klein-Gordon equation, which describes scalar (or pseudoscalar) spinless particles, is derived and its analytical properties are analyzed. The Crank-Nicolson leap-frog spectral method for the nonlinear Klein-Gordon equation is presented and other eisting numerical methods are introduced. We also report the numerical results of the nonlinear Klein-Gordon equation, i.e., the breather solution of KG, solitonsoliton collision in D and D problems. In Chapter 3, the Klein-Gordon-Schrödinger equations, describing a system of conserved scalar nucleons interacting with neutral scalar mesons coupled through the Yukawa interaction, is derived and its analytical properties are analyzed. In Chapter 4, two new efficient and accurate numerical methods are proposed to discretize KGS and the properties of these two numerical methods are studied. We test the accuracy and stability of our methods for KGS with a solitary wave solution, and apply them to study numerically dynamics of a plane wave, soliton-soliton collision in D with/without damping terms and a D

8 Summary viii problem of KGS. In Chapter 5, we etend our methods to the Schrödinger-Yukawa equations and report some numerical results of them. Finally, some conclusions based on our findings and numerical results are drawn in Chapter 6.

9 List of Tables. Spatial discretization errors e(t) at time t = for different mesh sizes h under k = Temporal discretization errors e(t) at time t = for different time steps k under h = / Conserved quantities analysis: k =. and h = / Spatial discretization errors e (t) and e (t) at time t = for different mesh sizes h under k =.. I: For γ = (cont d): II: For γ = Temporal discretization errors e (t) and e (t) at time t = for different time steps k. I: For γ = (cont d): II. For γ =.5 and h = / Conserved quantities analysis: k =. and h = Error analysis between KGS and its reduction S-Y: Errors are computed at time t = under h = 5/8 and k = i

10 List of Figures. Time evolution of soliton-soliton collision in Eample.. a): surface plot; b): contour plot Time evolution of a stationary Klein-Gordon s breather solution in Eample.. a): surface plot; b): contour plot Circular and elliptic ring solitons in Eample.3 (from top to bottom: t =, 4, 8,.5 and 5) Collision of two ring solitons in Eample.4 (from top to bottom : t =,, 4, 6 and 8) Collision of four ring solitons in Eample.5 (from top to bottom: t =,.5, 5, 7.5 and ) Numerical solutions of the meson field φ (left column) and the nucleon density ψ (right column) at t = for Eample with Type initial data in the nonrelativistic limit regime by PSAS-TSSP. - : eact solution given in (4.96), : numerical solution. I. With the meshing strategy h = O(ε) and k = O(ε): (a) Γ = (ε, h, k ) = (.5,.5,.4), (b) Γ /4, and (c) Γ /

11 List of Figures i 4. (cont d): II. With the meshing strategy h = O(ε) and k =.4- independent of ε: (d) Γ = (ε, h ) = (.5,.5), (e) Γ /4, and (f) Γ / Numerical solutions of the density ψ(, t) and the meson field φ(, t) at t = in the nonrelativistic limit regime by PSAS-TSSP with the same mesh (h = / and k =.5). - : eact solution, : numerical solution. The left column corresponds to the meson field φ(, t): (a) ε = /, (c) ε = /6. (e) ε = /8. The right column corresponds to the density ψ(, t) : (b) ε = /, (d) ε = /6. (f) ε = / Numerical solutions for plane wave of KGS in Eample 4. at time t = (left coumn) and t = 4 (right column). : eact solution given in (4.96), : numerical solution. (a): Real part of nucleon filed Re(ψ(, t)); (b): imaginary part of nucleon field Im(ψ(, t)); (c): meson filed φ Numerical solutions of soliton-soliton collison in standard KGS in Eample 4.3 I: Nucleon density ψ(, t) (cont d): II. Meson field φ(, t) Time evolution of nucleon density ψ(, t) (left column) and meson field φ(, t) (right column) for soliton-soliton collision of KGS in Eample 4.4 for different values of γ Time evolution of the Hamiltonian H(t) ( left ) and mean value of the meson field N(t) ( right ) in Eample 4.4 for different values of γ Time evolution of the Hamiltonian H(t) ( left ) and mean value of the meson field N(t) ( right ) in Eample 4.6 for different values of γ Numerical solutions of the nucleon density ψ(, y, t) (right column) and meson field φ(, y, t) (left column) in Eample 4.5 at t =. th row: ε = /; nd row: ε = /8; 3rd row : ε = /

12 List of Figures ii 4. Numerical solutions of the nucleon density ψ(, y, t) (right column) and meson field φ(, y, t) (left column) in Eample 4.5 at t =. th row: ε = /; nd row: ε = /8; 3rd row : ε = / Surface plots of the nucleon density ψ(, y, t) (left column) and meson field φ(, y, t) (right column) in Eample 4.6 with γ = at different times Numerical results for different scales of the Xα term in Eample 5., i.e., α =, ε, ε,. a) and b) : small time t =.5, pre-break, a) for ε =.5, b) for ε =.5. c)-f): large time, t = 4., post-break. c) for ε =., d) for ε =.5, e) for ε =.375, f) ε = Time evolution of the position density for Xα term at O() in Eample 5., i.e., α =.5, with ε =.5, h = /5 and k =.5. a) surface plot; b) pseudocolor plot Time evolution of the position density for attractive Hartree interaction in Eample 5.. C =, α =.5, ε =.5, k =.5. a) surface plot; b) pseudocolor plot

13 Chapter Introduction In this chapter, we introduce the physical background of the nonlinear Klein-Gordon equation (KG) and the Klein-Gordon-Schröding equations (KGS) and review some eisting analytical and numerical results of them and report our main results of these two problems.. Physical background The Klein-Gordon equation (or Klein-Fock-Gordon equation) is a relativistic version of the Schrödinger equation, which describes scalar (or pseudoscalar) spinless particles. The Klein-Gordon equation was actually first found by Schödinger, before he made the discovery of the equation that now bears his name. He rejected it because he couldn t make it fit the data (the equation doesn t take into account the spin of the electron); the way he found his equation was by making simplification in the Klein-Gordon equation. Later, it was revived and it has become commonly accepted that Klein-Gordon equation is the appropriate model to describe the wave function of the particle that is charge-neutral, spinless and relativistic effects can t be ignored. It has important applications in plasma physics, together with Zakharov equation describing the interaction of Langmuir wave and the ion acoustic wave in a plasma

14 . The problem [57], in astrophysics together with Mawell equation describing a minimally coupled charged boson field to a spherically symmetric space time [], in biophysics together with another Klein-Gordon equation describing the long wave limit of a lattice model for one-dimensional nonlinear wave processes in a bi-layer [47] and so on. Furthermore, Klein-Gordon equation coupled with Schrödinger equation (Klein- Gordon-Schrödinger equations or KGS) is introduced in [54, 9] and it describes a system of conserved scalar nucleons interacting with neutral scalar mesons coupled through the Yukawa interaction. As is well known, KGS is not eactly integrable, so the numerical study on it is very important.. The problem One of the problems we will study numerically is the general nonlinear Klein-Gordon equation (KG) tt φ φ + F (φ) =, R d, t >, (.) φ(, ) = φ () (), t φ(, ) = φ () (), R d, (.) with the requirements t φ, φ, as, (.3) where t is time, is the spatial coordinate, the real-valued function φ(, t) is the wave function in relativistic regime, G (φ) = F (φ). The general form of (.) covers many different generalized Klein-Gordon equations arising in various physical applications. For eample: a) when F (φ) = ±(φ φ 3 ), (.) is referred as the φ 4 equation, which describes the motion of the system in field theory [3]; b) when F (φ) = sin(φ), (.) becomes the well-known sine-gordon equation, which is widely used in physical world. It can be found in the motion of a rigid pendulum attached to an etendible string [6], in rapidly rotating fluids [3], in the physics of Josephson junctions and other applications [4, 49].

15 .3 Contemporary studies 3 Another specific problem we study numerically is the Klein-Gordon-Schrödinger (KGS) equations describing a system of conserved scalar nucleons interacting with neutral scalar mesons coupled through the Yukawa interaction [54, 9]: i t ψ + ψ + φψ + iνψ =, R d, t >, (.4) ε tt φ + γε t φ φ + φ ψ =, R d, t >, (.5) ψ(, ) = ψ () (), φ(, ) = φ () (), t φ(, ) = φ () (), R d ; (.6) where t is time, is the spatial coordinate, the comple-valued function ψ = ψ(, t) represents a scalar nucleon field, the real-valued function φ = φ(, t) represents a scalar meson field, ε > is a parameter inversely proportional to the speed of light, and γ and ν are two nonnegative parameters. The general form of (.4) and (.5) covers many different generalized Klein-Gordon- Schrödinger equations arising in many various physical applications. In fact, when ε =, γ = and ν =, it reduces to the standard KGS [9]. When ν >, a linear damping term is added to the nonlinear Schrödinger equation (.4) for arresting blowup. When γ >, a damping mechanism is added to the Klein-Gordon equation (.5). When ε (corresponding to infinite speed of light or nonrelativistic limit regime) in (.5), formally, we get the well-known Schrödinger-Yukawa (S-Y) equations without (ν = ) or with (ν > ) a linear damping term: i t ψ + ψ + φψ + iνψ =, R d, t >, (.7) φ + φ = ψ, R d, t >. (.8).3 Contemporary studies There was a series of mathematical study from partial differential equations for the KG (.). J. Ginibre et al. [3] studied the Cauchy problem for a class of nonlinear Klein-Gordon equations by a contraction method and proved the eistence and uniqueness of strongly continuous global solutions in the energy space

16 .3 Contemporary studies 4 H (R n ) L (R n ) for arbitrary space dimension n. In [68], Weder developed the scattering theory for the Klein-Gordon equation and proved the eistence and completeness of the wave operators, and invariance principle as well. On the other hand, numerical methods for the nonlinear Klein-Gordon equation were studied in the last fifty years. Strauss et al. [6] proposed a finite difference scheme for the one-dimensional (D) nonlinear Klein-Gordon equation, which is based on radial coordinate and second-order central difference for the terms φ tt and φ rr. In [4], Jiménez presented four eplicit finite difference methods to integrate the nonlinear Klein-Gordon equation and compared the properties of these four numerical methods. Numerical treatment for damped nonlinear Klein-Gordon equation, based on variational method and finite element approach, is studied in [45, 65]. In [45], Khalifa et al. established the eistence and uniqueness of the solution and a numerical scheme was developed based on finite element method. In [36], Guo et al. proposed a Legendre spectral scheme for solving the initial boundary value problem of the nonlinear Klein-Gordon equation, which also kept the conservation. There are also some other numerical methods for solving it [44, 66]. In particular, the Sine-Gordon equation is a typical eample of the nonlinear Klein-Gordon equation. There has been a considerable amount of recent discussions on computations of sine-gordon type solitons, in particular via finite difference and predictor-corrector scheme [, 8, 9], finite element approaches [, 4], perturbation methods [48] and symplectic integrators [5]. There was also a series of mathematical study from partial differential equations for the KGS (.4)-(.5) in the last two decades. For the standard KGS, i.e. ε =, γ = and ν =, Fukuda and Tsutsumi [8, 9, 3] established the eistence and uniqueness of global smooth solutions, Biler [7] studied attractors of the system, Guo [33] established global solutions, Hayashi and Von Wahl [37] proved the eistence of global strong solution, Guo and Miao [34] studied asymptotic behavior of

17 .4 Overview of our work 5 the solution, Ohta [56] studied the stability of stationary states for KGS. For plane, solitary and periodic wave solutions of the standard KGS, we refer to [, 38, 5, 67]. For dissipative KGS, i.e. ε =, γ > and ν >, Guo and Li [35, 5], Ozawa and Tsutsumi [58] studied attractor of the system and asymptotic smoothing effect of the solution, Lu and Wang [53] found global attractors. For the nonrelativistic limit of the Klein-Gordon equation, we refer to [5, 6, 64, ]. In order to study effectively the dynamics and wave interaction of the KGS, especially in D & 3D, an efficient and accurate numerical method is one of the key issues. However, numerical methods and simulation for the KGS in the literature remain very limited. Xiang [69] proposed a conservative spectral method for discretizating the standard KGS and established error estimate for the method. Zhang [7] studied a conservative finite difference method for the standard KGS in D. Due to that both methods are implicit, it is a little complicated to apply the methods for simulating wave interactions in KGS, especially in D & 3D. Usually very tedious iterative method must be adopted at every time step for solving nonlinear system in the above discretizations for KGS and thus they are not very efficient. In fact, there was no numerical result for KGS based on their numerical methods in [69, 7]. To our knowledge, there is no numerical simulation results for the KGS reported in the literature. Thus it is of great interests to develop an efficient, accurate and unconditionally stable numerical method for the KGS..4 Overview of our work In this thesis, we propose a Crank-Nicolson leap-frog spectral discretization (CN- LF-SP) for the nonlinear Klein-Gordon equation and we also present two different numerical methods, i.e., phase space analytical solver+time-splitting spectral discretization (PSAS-TSSP) and Crank-Nicolson leap-frog time-splitting spectral discretization (CN-LF-TSSP) for the damped Klein-Gordon-Schrödinger equations.

18 .4 Overview of our work 6 Our numerical method for the KG is based on discretizing spatial derivatives in the Klein-Gordon equation (.) by Fourier pseudospectral method and then applying Crank-Nicolson/leap-frog for linear/nonlinear terms for time derivatives. The key points in designing our new numerical methods for the KGS are based on: (i) discretizing spatial derivatives in the Klein-Gordon equation (.5) by Fourier pseudospectral method, and then solving the ordinary differential equations (ODEs) in phase space analytically under appropriate chosen transmission conditions between different time intervals or applying Crank-Nicolson/leap-frog for linear/nonlinear terms for time derivatives [, ]; and (ii) solving the nonlinear Schrödinger equation (.4) in KGS by a time-splitting spectral method [63, 6, 5, 8, 9], which was demonstrated to be very efficient and accurate and applied to simulate dynamics of Bose-Einstein condensation in D & 3D [6, 7]. Our etensive numerical results demonstrate that the methods are very efficient and accurate for the KGS. In fact, similar techniques were already used for discretizing the Zakharov system [,, 4] and the Mawell-Dirac system [, 39]. This thesis consists of si chapters arranged as following. Chapter introduces the physical background of the Klein-Gordon equation and the Klein-Gordon-Schrödinger equations. We also review some eisting results of them and report our main results. In Chapter, the Klein-Gordon equation, which describes scalar (or pseudoscalar) spinless particles, is derived and its analytical properties are analyzed. The Crank-Nicolson leap-frog spectral method for the nonlinear Klein-Gordon equation is presented and other eisting numerical methods are introduced. We also report the numerical results of the nonlinear Klein-Gordon equation, i.e., the breather solution of KG, soliton-soliton collision in D and D problems. In Chapter 3, the Klein-Gordon-Schrödinger equations, describing a system of conserved scalar nucleons interacting with neutral scalar mesons coupled through the Yukawa interaction, is derived and its analytical properties are analyzed. In Chapter 4, two new efficient and accurate numerical methods are proposed to discretize KGS and the properties

19 .4 Overview of our work 7 of these two numerical methods are studied. We test the accuracy and stability of our methods for KGS with a solitary wave solution, and apply them to study numerically the dynamics of a plane wave, soliton-soliton collision in D with/without damping terms and a D problem of KGS. In Chapter 5, we etend our methods to the Schrödinger-Yukawa equations and report some numerical results of it. Finally, some conclusions based on our findings and numerical results are drawn in Chapter 6.

20 Chapter Numerical studies of the Klein-Gordon equation In this chapter, the Klein-Gordon equation, which is the relativistic quantum mechanical equation for a free particle, is derived and its properties are analyzed. We present the Crank-Nicolson leap-frog spectral discretization (CN-LF-SP) for the nonlinear Klein-Gordon equation (.) with the periodic boundary conditions, show the numerical simulations of (.) in D and D eamples, and compare our method with other eisting numerical methods.. Derivation of the Klein-Gordon equation This section is devoted to derive the Klein-Gordon equation. From elementary quantum mechanics [6], we know that the Schrödinger equation for free particle is i t φ = P φ, (.) m where φ is the wave function, m is the mass of the particle, is Planck s constant, and P = i is the momentum operator. The Schrödinger equation suffers from not being relativistically covariant, meaning 8

21 . Derivation of the Klein-Gordon equation 9 that it does not take into account Einstein s special relativity. It is natural to try to use the identity from special relativity E = P c + m c 4 φ, (.) for the energy (c is the speed of light); then, plugging into the quantum mechanical momentum operator, yields the equation i t φ = ( i ) c + m c 4 φ. (.3) This, however, is a cumbersome epression to work with because of the square root. In addition, this equation, as it stands, is nonlocal. Klein and Gordon instead worked with more general square of this equation (the Klein-Gordon equation for a free particle), which in covariant notation reads ( + µ )φ =, (.4) where µ = mc and = c t. This operator ( ) is called as the d Alember operator. This wave equation (.4) is called as the Klein-Gordon equation. It was in the middle 9 s by E. Schrödinger, as well as by O. Klein and W. Gordon, as a candidate for the relativistic analog of the nonrelativistic Schrödinger equation for a free particle. In order to obtain a dimensionless form of the Klein-Gordon equation (.4), we define the normalized variables t = µc t, = µ. (.5) Then plugging (.5) into (.4) and omitting all, we get the following dimensionless standard Klein-Gordon equation tt φ φ + φ =. (.6) For more general case, we consider the nonlinear Klein-Gordon equation where G(φ) = φ F (φ) dφ. tt φ φ + F (φ) =, (.7)

22 . Conservation laws of the Klein-Gordon equation. Conservation laws of the Klein-Gordon equation There are at least three invariants in the nonlinear Klein-Gordon equation (.). Theorem.. The nonlinear Klein-Gordon equation (.) preserves the conserved quantities. They are the energy [ H(t) := H(φ(, t)) = R ( tφ(, t)) + ] φ(, t) + G(φ(, t)) d [ d (φ() ()) + ] φ() () + G(φ () ()) d R d := H(), t, (.8) the linear momentum P(t) := P(φ(, t)) = ( t φ(, t))( φ(, t))d R d φ () ()( φ () ())d := P(), t, (.9) R d and angular momentum A(t) := A(φ(, t)) = R d [ ( R d [ ( ( tφ(, t)) + ) ( φ(, t)) + G(φ(, t)) t φ(, t) ] d ) +t t φ(, t) φ(, t) (φ() ()) + ( φ() ()) + G(φ () ()φ () ()) ] +t φ () () φ () () d := A(), t. (.) Proof. Multiplying (.) by φ t, and integrating over R d, we can get [ t (φ t) + ] φ + G(φ) d ( φφ t ) d =. (.) R d R d From (.), noting (.3), we can have the conservation of energy [ d dt H = t (φ t) + ] φ + G(φ) d =. (.) R d

23 . Conservation laws of the Klein-Gordon equation Multiplying (.) by φ, and integrating over R d, we can get [ ( φφ t ) t d R d R (φ t) + ] φ G(φ) d =. (.3) d From (.3), noting (.3), we can obtain the conservation of linear momentum Multiplying (.) by φ t, we get d dt P = ( φφ t ) t d =. (.4) R d φ t (φ tt φ + G(φ)) =. (.5) Multiplying (.) by t φ, we get t φ(φ tt φ + G(φ)) =. (.6) Subtracting (.5) by (.6) and integrating over R d, we can obtain [ ( R (φ t) + ) ] φ + G(φ)φ t + tφ t φ d d t [ ( ( φ)φ t + t (φ t) + )] φ G(φ) d =. (.7) R d From (.7), noting (.3), we can obtain the conservation of angular momentum d dt A = R d [ ( t (φ t) + ) ] ( φ) + G(φ)φ t + tφ t φ d =. (.8) In one dimension case, the above conserved quantities become [ H = (φ t(, t)) + ] (φ (, t)) + G(φ(, t)) d, (.9) P = A = [φ t (, t)φ (, t)] d, (.) [ ( (φ t(, t)) + ) (φ (, t)) + G(φ)φ t (, t) ] +tφ t (, t)φ (, t) d. (.)

24 .3 Numerical methods for the Klein-Gordon equation.3 Numerical methods for the Klein-Gordon equation In this section, we review some eisting numerical methods for the nonlinear Klein- Gordon equation and present a new method for it. For simplicity of notation, we shall introduce the methods in one spatial dimension (d = ). Generalization to d > is straightforward by tensor product grids and the results remain valid without modification. For d =, the problem becomes tt φ φ + F lin (φ) + F non (φ) =, a < < b, t >, (.) φ(a, t) = φ(b, t), φ(a, t) = φ(b, t), t, (.3) φ(, ) = φ () (), t φ(, ) = φ () (), a b, t, (.4) where F lin (φ) represents the linear part of F (φ) and F non (φ) represents the nonlinear part of it. As it is known in Section., the KG equation has the properties b [ H(t) = φ t(, t) + ] φ (, t) + G(φ) d = H(), (.5) P (t) = A(t) = a b a b a [φ t (, t)φ (, t)] d = P (), (.6) [ ( φ t(, t) + ) φ (, t) + G(φ)φ t (, t) ] +tφ t (, t)φ (, t) d = A(). (.7) In some cases, the boundary condition (.3) may be replaced by φ(a, t) = φ(b, t) =, t. (.8) We choose the spatial mesh size h = > with h = (b a)/m for M being an even positive integer, the time step being k = t > and let the grid points and the time step be j := a + jh, j =,,, M; t m := mk, m =,,. (.9) Let φ m j be the approimation of φ( j, t m ).

25 .3 Numerical methods for the Klein-Gordon equation 3.3. Eisting numerical methods There are several numerical methods proposed in the literature [3, 7, 4] for discretizing the nonlinear Klein-Gordon equation. We will review these numerical schemes for it. The schemes are the following A). This is the simplest scheme for the nonlinear Klein-Gordon equation and has had wide use [7]: φ n+ j φ n j + φ n j k φn j+ φ n j + φ n j h + F (φ n j ) =, j =,, M, (.3) φ n+ M The initial conditions are discretized as φ j = φ () ( j ), = φn+, φ n+ = φ n+ M. (.3) φ j φ j k = φ () ( j ), j M. (.3) B). This scheme was proposed by Ablowitz, Kruskal, and Ladik [3]: φ n+ j φ n j + φ n j k φn j+ φ n j + φ n j + F ( φn j+ + φ n j ) =, h j =,, M, (.33) φ n+ M = φn+, φ n+ = φ n+ M. (.34) The initial conditions are discretized as φ j = φ () ( j ), φ j φ j k C). This scheme has been studied by Jiménez [4]: = φ () ( j ), j M. (.35) φ n+ j φ n j + φ n j k φn j+ φ n j + φ n j + G(φn j+) G(φ n j ) h φ n j+ =. φn j j =,, M, (.36) φ n+ M = φn+, φ n+ = φ n+ M. (.37) The initial conditions are discretized as φ j = φ () ( j ), φ j φ j k = φ () ( j ), j M. (.38)

26 .3 Numerical methods for the Klein-Gordon equation 4 The eisting numerical methods are of second-order accuracy in space and secondorder accuracy in time. Our new method shown in the net section is of spectralorder accuracy in space, which is much more accurate than them..3. Our new numerical method We discretize the Klein-Gordon equation (.) by using a pseudospectral method for spatial derivatives, followed by application of a Crank-Nicolson/leap-frog method for linear/nonlinear terms for time derivative. φ m+ j φ m j + φ m j k D f ( +F lin βφ m+ j + ( β)φ m j + βφ m j [ ] βφ m+ j + ( β)φ m j + βφ m j ) + Fnon (φ m j ) = j =,, M, m =,, (.39) where β / is a constant; D f, a spectral differential operator approimation of, is defined as D f U =j = M/ l= M/ µ l (Ũ) l e iµ l( j a), (.4) where (Ũ) l, the Fourier coefficient of a vector U = (U, U, U,, U M ) T with U = U M, is defined as (Ũ) l = M M j= U j e iµ l( j a), µ l = πl b a, l = M,, M. (.4) The initial condition (.4) are discretized as φ j = φ ( j ), φ j φ j k = φ (), j =,,,, M. (.4) Remark.. If the periodic boundary condition (.3) is replaced by (.8), then the Fourier basis used in the above algorithm can be replaced by the sine basis. In fact, the generalized nonlinear Klein Gordon equation (.) with the homogeneous Dirichlet boundary condition (.8) and initial condition (.4) can be discretized

27 .4 Numerical results of the Klein-Gordon equation 5 by φ m+ j φ m j + φ m j k D s ( +F lin βφ m+ j + ( β)φ m j + βφ m j [ ] βφ m+ j + ( β)φ m j + βφ m j ) + Fnon (φ m j ) = j =,, M, m =,, (.43) where D f, a spectral differential operator approimation of based on sine-basis, is defined as D s U =j = M/ l= M/ η l (Ũ) l sin(η l ( j a)), (.44) where (Ũ) l, the sine coefficient of a vector U = (U, U, U,, U M ) T with U = U M =, is defined as (Ũ) l = M M j= U j sin(η l ( j a)), η l = πl, l =,,, M. (.45) b a.4 Numerical results of the Klein-Gordon equation In this section, we report numerical results of the nonlinear Klein-Gordon equation with interaction of two solitary wave solutions in D to compare the accuracy and stability of different methods described in the previous section. We also present numerical eamples including breather solution, soliton-soliton collisions in D, as well as ring solitary solutions and soliton-soliton collisions in D to demonstrate the efficiency and spectral accuracy of the Crank-Nicolson leap-frog spectral method (CN-LF-SP) for the nonlinear Klein-Gordon equation..4. Comparison of different methods Eample. The nonlinear Klein-Gordon equation with the interaction between two solitary solutions in D, i.e., d =, F (φ) = sin(φ) in (.)-(.). The well-known

28 .4 Numerical results of the Klein-Gordon equation 6 kink solitary solution for the Klein-Gordon equation is ( φ(, t) = 4tan [ep ± θt )], (.46) θ where θ <. We consider the collision of two solitons, one with the + sign (kink) and the other with the sign (antikink). The two solitons have equal amplitude but opposite velocities. The initial condition in (.4) is chosen as ( + φ () () = 4tan [ep θ )] + 4tan [ ep ( )] +, (.47) θ φ () () =. (.48) We solve the problem on the interval [ 3, 3], i.e., a = 3 and b = 3 with mesh size h = /3 and time step k =. by using our method CN-LF-SP method (β =.5). For the numerical results shown in Figure., we choose = 5., θ =.3. The above numerical results can be compared with the previous ones for this problem in [3, 5]. There is no analytical solution in this case and we let φ be the eact solution which is obtained numerically by using our numerical method with a very fine mesh and small time step size, e.g. h = 3 and k =.. Let φ h,k be the numerical solution obtained by using a numerical method with mesh size h and time step k. To quantify the numerical methods, we define the error functions as e(t) = φ(., t) φ h,k (t) l. First we test the discretization error in space. In order to do this, we choose a very small time step, e.g., k =., such that the error from time discretization is negligible compared with to spatial discretization error, and solve the nonlinear Klein-Gordon equation with different methods under different mesh size h. Table. lists the numerical errors of e(t) at t = with different mesh size h for different numerical methods. Then we test the discretization error in time. Table. shows the numerical errors e(t) under different time steps k for different numerical methods. Finally we test the conservation of conserved quantities. Table.3 presents the

29 .4 Numerical results of the Klein-Gordon equation 7 a) 5 t b) 3 3 Figure.: Time evolution of soliton-soliton collision in Eample.. a): surface plot; b): contour plot. quantities at different times with mesh size h = /6 and time step k =. for different numerical methods. From Tables.-.3, we can draw the following observations: our numerical method CN-LF-SP is of spectral-order accuracy in space discretization and second-order accuracy in time. Finite difference methods (A, B and C) are of second-order accuracy in space discretization. Moreover, CN-LF-SP with β = / or β = /4 are unconditionally stable, where CN-LF-SP with β = is conditionally stable. However finite

30 .4 Numerical results of the Klein-Gordon equation 8 Mesh h =. h = / h = /4 FDM A E-.3E- FDM B E-.986E- FDM C E-.44E- CN-LF-SP (β = ).98E- 5.37E-5.9E-7 CN-LF-SP (β = /4).98E- 5.33E-5.89E-7 CN-LF-SP (β = /).98E- 5.3E-5.E-7 Table.: Spatial discretization errors e(t) at time t = for different mesh sizes h under k =.. Time Step k = 3 k = 64 k = 8 k = 56 k = 5 CN-LF-SP(β = ) E-6.73E E-7.868E-7 CN-LF-SP(β = /4) 6.673E-5.668E-5 4.7E-6.44E-6.8E-7 CN-LF-SP(β = /).669E E-5.43E-5.66E E-7 Table.: Temporal discretization errors e(t) at time t = for different time steps k under h = /6. difference methods (A, B and C) are all conditionally stable in time. All of these numerical methods conserve the energy H, the linear momentum P and angular momentum A very well..4. Applications of CN-LF-SP Breather solution of the Klein-Gordon equation Eample. The nonlinear Klein-Gordon equation with a breather solution, i.e., we choose d =, F (φ) = sin(φ) in (.)-(.) and consider the problem on the interval [a, b] with a = 4 and b = 4 with mesh size h = 5/56, time step k =. and β =.5 for CN-LF-SP. The well-known breather solution is of the

31 .4 Numerical results of the Klein-Gordon equation 9 Time H P A FDM A FDM B FDM C CN-LF-SP(β = ) CN-LF-SP(β = /4) CN-LF-SP(β = /) Table.3: Conserved quantities analysis: k =. and h = /6. form [49] [ φ(, t) = 4tan m m sin(t ] m ) + c. (.49) cosh(m + c ) The initial condition is taken as φ () () = φ(, ), φ () () = φ t (, ) in (.49). In the present eample, we choose m =.5, c =, c = m. From the results shown in Figure., it is easy to see that they completely agree with those presented in [5]. Ring solitary solution of the D Klein-Gordon equation Eample.3 The Klein-Gordon equation with a circular ring soliton solution in D case, i.e., we choose d =, F (φ) = sin(φ) in (.)-(.). The initial condition is

32 .4 Numerical results of the Klein-Gordon equation a) 3 5 t 5 5 b) 5 5 Figure.: Time evolution of a stationary Klein-Gordon s breather solution in Eample.. a): surface plot; b): contour plot. taken as [ φ(, y, ) = α tan ep(3 ] + y ), (.5) φ t (, y, ) =, (, y) R, (.5) where α = 4. in our numerical simulation. We solve this problem on the rectangle [ 6, 6] with mesh size h = /6 and time step k =. by using our method CN-LF-SP (β =.5). Figure.3 shows the surface plots of sin(φ/) (left column)

33 .4 Numerical results of the Klein-Gordon equation and the contour plots of it (right column) at different times, i.e., t =, 4, 8,.5, 5. The resolution is outstanding as compared with the eisting numerical solutions [4, 4]. It is found that the ring solitons shrink at the initial stage, but oscillations and radiations begin to form and continue slowly as time goes on. clearly viewed in the contour plots. This can be Eample.4 Two circular soliton-soliton collision in D of Klein Gordon equation, i.e., we choose d =, F (φ) = sin(φ) in (.)-(.). The initial condition is taken as { [ φ(, y, ) = α tan ep γ (4 )]} ( + 3) + (y + 7) { [ +α tan ep γ (4 )]} ( + 3) + (y + 7), (.5) [ φ t (, y, ) = θ sech γ (4 )] ( + 3) + (y + 7) [ +θ sech γ (4 )] ( + 3) + (y + 7), (, y) R,(.53) where α = 4., θ = 4.3, γ =.9 in our simulation. We solve this problem on the rectangle [ 3, ] [ 7, 3] with mesh size h = /6 and time step k =. by using our method CN-LF-SP (β =.5). Numerical simulation presented in Figure.4 is for sin(φ/) at time t =,, 4, 6, 8, respectively. The solution shown includes the etension across = and y = 7 by symmetry properties of the problem [4]. Figure.4 demonstrates the collision between two epanding circular ring solitons in which two smaller ring solitons bounding an annular region merge into a large ring soliton. The simulated solution is again precisely consistent to eisting results [4]. Contour plots are given to show more clearly the movement of solitons. Though minor disturbances can be observed in the middle of the numerical solution, probably due to the transactions following the symmetry features in computations, the overall simulation results match well with those described in [9, 4] with satisfaction. Eample.5 The collision of four circular solitons in D of Klein Gordon equation,

34 .4 Numerical results of the Klein-Gordon equation i.e., we choose d =, F (φ) = sin(φ) in (.)-(.). The initial condition is taken as { [ φ(, y, ) = α tan ep γ (4 )]} ( + 3) + (y + 3) { [ +α tan ep γ (4 )]} ( + 3) + (y + 7) { [ +α tan ep γ (4 )]} ( + 7) + (y + 3) { [ +α tan ep γ (4 )]} ( + 7) + (y + 7), (.54) [ φ t (, y, ) = θ sech γ (4 )] ( + 3) + (y + 3) [ +θ sech γ (4 )] ( + 3) + (y + 7) [ +θ sech γ (4 )] ( + 7) + (y + 3) [ +θ sech γ (4 )] ( + 7) + (y + 7), (, y) R, (.55) where α = 4., θ = 4.3, γ =.9 in our simulation. We solve this problem on the rectangle [ 3, ] [ 7, 3] with mesh size h = /6 and time step k =. by using our method CN-LF-SP (β =.5). Numerical simulation is presented in Figure.5 in terms of sin(φ/) at t =,.5, 5, 7.5,. The simulation is based on an etension across = and y = due to the symmetry [4, 4, 46]. The computation can be continued for t without major boundary disturbances. This feature is significant in potential multi-dimensional applications. Figure.5 demonstrates precisely the collision between four epanding circular ring solitons in which the smaller ring solitons bounding an annular region merge into a large ring soliton. This matches known eperimental results [8, 4]. Again, contour plots are given to illustrate more clearly the movement of the solitons.

35 .4 Numerical results of the Klein-Gordon equation y y y y y Figure.3: Circular and elliptic ring solitons in Eample.3 (from top to bottom: t =, 4, 8,.5 and 5).

36 .4 Numerical results of the Klein-Gordon equation y y y y y Figure.4: Collision of two ring solitons in Eample.4 (from top to bottom : t =,, 4, 6 and 8).

37 .4 Numerical results of the Klein-Gordon equation 5 5 y y y y y Figure.5: Collision of four ring solitons in Eample.5 (from top to bottom: t =,.5, 5, 7.5 and ).

38 Chapter 3 The Klein-Gordon-Schrödinger equations In this chapter, the Klein-Gordon-Schrödinger equations describing a system of conserved scalar nucleons interacting with neutral scalar mesons, are derived and its analytical properties are analyzed. 3. Derivation of the Klein-Gordon-Schrödinger equations This section is devoted to derive the Klein-Gordon-Schrödinger equations. The Lagrangian density of the dynamic system that describes the interaction between the comple scalar nucleon field and real scalar meson field in one dimension (D) is L = L o + L int = i ψ ψ t + m ψ ψ + (φ t φ µ φ ) + g φ ψ, (3.) where ψ represents the comple scalar nucleon field, φ describes the real scalar meson field, g φ ψ is the Yukawa interaction potential, and m, µ respectively describe the mass of nucleon and mass of meson. 6

39 3. Derivation of the Klein-Gordon-Schrödinger equations 7 Since the Lagrangian of the whole system must satisfy the Euler-Lagrange equation (that is, L + L L ct ( η ct ) ( η ) η the whole system. Let η = ψ, we get the equation Let η = φ,we get another equation = ). Then we can get the equations of motion of i ψ t + m ψ + g ψφ =. (3.) φ + µ φ g ψ =, (3.3) where = c t. In general, we can obtain the equations of motion of the dynamic system in d- dimension (d =,, 3). i ψ t + m ψ + g ψφ =, (3.4) φ + µ φ g ψ =, (3.5) where = c t. In order to obtain a dimensionless form of the KGS (3.4)- (3.5), we define the normalized variables Then defining µ t = t, = µ, (3.6) m ψ = g m mg ψ, φ = φ. (3.7) µ µ ε = µ mc, (3.8) and plugging (3.6)-(3.7) into (3.4)-(3.5), and then removing all, we get the following dimensionless standard Klein-Gordon-Schrödinger equations i t ψ + ψ + φψ =, (3.9) ε tt φ φ + φ ψ =. (3.)

40 3. Conservation laws of the Klein-Gordon-Schrödinger equations 8 Adding dissipative terms to (3.9), we can obtain the generalized Klein-Gordon- Schrödinger equations i t ψ + ψ + φψ + iνψ =, (3.) ε tt φ + γε t φ φ + φ ψ =. (3.) 3. Conservation laws of the Klein-Gordon-Schrödinger equations Define the wave energy D(t) := D(ψ(, t)) = ψ(, t) d, R d t, (3.3) and the Hamiltonian [ ( H(t) = φ (, t) + ε ( t φ(, t)) + φ(, t) ) + ψ(, t) R d ψ(, t) φ(, t) ] d, t. (3.4) Lemma 3.. Suppose (ψ, φ) be the solution of the generalized Klein-Gordon-Schrödinger equations (.4)-(.5), its wave energy and Hamiltonian has the following properties d dt D(t) = d ψ(, t) d = ν ψ(, t) d, t, (3.5) dt R d R d d dt H(t) = d [ ( φ (, t) + ε ( t φ(, t)) + φ(, t) ) + ψ(, t) dt R d ψ(, t) φ(, t) ] d [ = γε( t φ(, t)) + iν ( t ψ (, t)ψ(, t) R d t ψ(, t)ψ (, t))] d, t. (3.6) Proof. Multiplying (.4) by ψ, the conjugate of ψ, we get i ψ t ψ + ψψ + φ ψ + i ν ψ =. (3.7) Then taking the conjugate of (.5) and multiplying it by ψ, we obtain i ψ t ψ + ψ ψ + φ ψ i ν ψ =. (3.8)

41 3. Conservation laws of the Klein-Gordon-Schrödinger equations 9 Subtracting (3.8) from (3.7) and then multiplying both sides by i, we can get ψ t ψ + ψ t ψ + i (ψ ψ ψ ψ) + ν ψ =. (3.9) Integrating over R d, integration by parts, (3.9) leads to d dt D(t) = d ψ(, t) d = ν ψ(, t) d, dt R d R d t, (3.) which implies that D(t) preserves as a constant when ν =. Multiplying (.4) by ψ t, we get i ψ t + ψψ t + φψψ t + i νψψ t =. (3.) Then taking the conjugate of (.5) and multiplying it by ψ t, we obtain i ψ t + ψ t ψ + φψ ψ t i νψ ψ t =. (3.) Adding (3.) to (3.), we can get (ψ ψ t + ψ t ψ ) + φ( ψ ) t + i ν(ψψ t ψ ψ t ) = (3.3) Integrating over R d (3.9) leads to R d [ (ψ ψ t + ψ t ψ ) + φ( ψ ) t + i ν(ψψ t ψ ψ t ) ] d =. (3.4) Multiplying (.5) by φ t, and integrating over R d, we can get R d [ ε φ tt φ t + γε(φ t ) φφ t + φφ t ψ φ t ] d = (3.5) Noting (3.5), (3.4) and (3.), we obtain d dt H(t) = [ ] φφt + ε φ t φ tt + φ φ t + ( ψ ψ φ) t d R d [ ] = γε(φt ) + φφ t + ψ φ t + φ φ t + ( ψ ψ φ) t d R d [ ] = [ φφ t + φ φ t ] d + ( ψ ) t φ + ( ψ ) t d γε(φ t ) d R d R d R d [ = γε(φt ) + i ν (ψ t ψ ψt ψ) ] d, t, (3.6) R d

42 3.3 Dynamics of mean value of the meson field 3 which implies that H(t) is decreasing if ν = as time t is increasing and H(t) preserves as a constant when ν = and γ =. From the above discussion, we find that when γ = and ν = in (.4)-(.5), the KGS has at least two invariants. Theorem 3.. The generalized Klein-Gordon-Schrödinger equations (.4)-(.5) with γ = ν = preserve the conserved quantities. They are the wave energy D(t) := D(ψ(, t)) = ψ(, t) d R d ψ () () d := D(), R d t, and the Hamiltonian [ ( H(t) = φ (, t) + ε ( t φ(, t)) + φ(, t) ) + ψ(, t) R d ψ(, t) φ(, t) ] d [ ( (φ () ()) + ε (φ () ()) + φ () () ) + ψ () () R d ψ () () φ () () ] d (3.7) := H(), t. (3.8) In one dimension case, the conserved quantities become H = A = [ ( φ (, t) + ε (φ t (, t)) + (φ (, t)) ) + ψ(, t) ] ψ(, t) φ(, t) d, (3.9) ψ(, t) d. (3.3) 3.3 Dynamics of mean value of the meson field When ν =, the generalized KGS (.4)-(.5) collapses to i t ψ + ψ + φψ =, R d, t >, (3.3) ε tt ψ + γε t φ φ + φ ψ =, R d, t >. (3.3)

43 3.4 Plane wave and soliton wave solutions of KGS 3 Define the mean value of the meson field as N(t) := N(φ(, t)) = φ(, t) d, t. (3.33) R d Integrating (3.3) over R d, integration by parts and noticing (3.7), we obtain ε N (t) + γεn (t) + N(t) = D(), t, (3.34) with initial condition as N() = N(φ () ) := φ () () d, R d N () = N(φ () ) := φ () () d. R d (3.35) Denote λ = γ + γ 4 ε, λ = γ γ 4, λ = γ ε ε, β = 4 γ. (3.36) ε Solving the ODE (3.34) with the initial data (3.35), we get the dynamics of the mean value of the meson field when ν = : (i) For γ > N(t) = D()+ N(φ() ) λ (N(φ () ) D()) λ λ (ii) for γ = e λ t + N(φ() ) + λ (N(φ () ) D()) e λ λ λ t ; N(t) = D() + ( N(φ () ) D() ) e λ t + ( N(φ () λ (N(φ () ) D()) ) te λ t ; and (iii) for γ < [ N(t) = D()+e λ t (N(φ () ) D() ) ] cos(β t) + N(φ() ) λ (N(φ () ) D()) sin(β t). β These immediately imply that lim t N(t) = D() when γ >, and N(t) is a periodic function with period T = επ when γ =. 3.4 Plane wave and soliton wave solutions of KGS When d =, γ =, ν =, and ε =, the generalized KGS (.4)-(.5) collapses to i t ψ + ψ + φψ =, R, t >, (3.37) tt φ φ + φ ψ =, R, t >, (3.38)

44 3.5 Reduction to the Schrödinger-Yukawa equations (S-Y) 3 which admits plane and soliton wave solutions. Firstly, it is instructive to eamine some eplicit solutions to (3.37) and (3.38). The initial data in (.6) is chosen as φ () () = d >, φ () () =, ψ () () = d ep ( i πl ), R, (3.39) b a with l an integer, and a, b and d constants, the KGS admits the plane wave solution []. φ(, t) = d, where ψ(, t) = [ ( )] πl d ep i b a ωt, R, t, (3.4) ω = ( ) πl d. b a Secondly, as it is well known, the standard KGS is not completely integrable. Therefore the generalized KGS can not be eactly integrable either. one-soliton solutions to (3.37) and (3.38) for d =, γ =, ν = [38] However, it has ψ ± (, t) = 3B sech (B + c ± t) ep [ i ( d ± + ( 4B d ±) t )], (3.4) φ ± (, t) = 6B sech (B + c ± t), R, t, (3.4) where with B / a constant. 4B 4B c ± = ±, d ± = ε 4B = c ± B, Note that the phase in ψ oscillates as O( ). This eact solution is used in Chapter ε 4 to test the accuracy of our new numerical methods. 3.5 Reduction to the Schrödinger-Yukawa equations (S-Y) In the nonrelativistic regime, i.e., ε in (.4)-(.6), which corresponds to infinite speed of light, we can obtain the well-known Schrödinger-Yukawa equations

45 3.5 Reduction to the Schrödinger-Yukawa equations (S-Y) 33 without (ν = ) or with (ν > ) a linear damping term: i t ψ + ψ + φψ + i νψ =, R d, t >, (3.43) φ + φ = ψ, R d, t >. (3.44) When ν =, the S-Y equations (.7)-(.8) is time reversible, time transverse invariant, and preserves the following wave energy and Hamiltonian: D SY = ψ(, t) d, (3.45) R d [ ( H SY = φ (, t) + φ(, t) ) ] + ψ(, t) ψ(, t) φ(, t) d.(3.46) R d Similarly, letting ε in (3.8), we get formally the quadratic convergence rate of the Hamiltonian from KGS with ν = γ = to S-Y in the nonrelativistic limit regime, i.e., < ε : [ ( H(t) = φ (, t) + φ(, t) ) ] + ψ(, t) ψ(, t) φ(, t) d R d +ε R d ( t φ(, t)) d H SY + O(ε ). (3.47) Our numerical results in Chapter 5 confirm this asymptotic result.

46 Chapter 4 Numerical studies of the Klein-Gordon-Schrödinger equations In this Chapter, we present two accurate and efficient numerical schemes, phase space analytical solver+time-splitting spectral method (PSAS-TSSP) and Crank- Nicolson leap-frog time-splitting spectral method (CN-LF-TSSP) for the Klein- Gordon-Schrödinger equations (.4)-(.5) with the periodic boundary conditions and analyze the numerical properties for these two numerical methods. Then we compare the accuracy and stability of different numerical methods for the Klein- Gordon-Schrödinger equations with a solitary wave solution, and also present the numerical results for plane waves, soliton-soliton collisions in D, D problems and the generalized KGS with a damping term. 4. Numerical methods for the Klein-Gordon-Schrödinger equations In this section we present two new accurate and efficient numerical methods, i.e., phase space analytical+time-splitting spectral discretizations (PSAS-TSSP) and 34

47 4. Numerical methods for the Klein-Gordon-Schrödinger equations 35 Crank-Nicolson leap-frog spectral discretization (CN-LF-TSSP) for the Klein-Gordon- Schrödinger equations (.4)-(.6) with the periodic boundary conditions. For simplicity of notation, we shall introduce the method in D of the KGS with periodic boundary conditions. Generalizations to higher dimensions are straightforward for tensor product grids and the results remain valid without modifications. For d =, the problem becomes i t ψ(, t) + ψ + iνψ + φψ =, a < < b, t >, (4.) ε tt φ + εγ t φ φ + φ ψ =, a < < b, t >, (4.) ψ(a, t) = ψ(b, t), ψ(a, t) = ψ(b, t), t, (4.3) φ(a, t) = φ(b, t), φ(a, t) = φ(b, t), t, (4.4) ψ(, ) = ψ () (), φ(, ) = φ () (), t φ(, ) = φ () (), a b. (4.5) As it is well known, the above KGS in D has the following properties: D(t) = b a b ψ(, t) d = e νt ψ () () d = e νt D(), t. (4.6) a So when ν =, D(t) D(), i.e., it is an invariant of the KGS. When ν >, it decays to eponentially. Furthermore, when ν = and γ =, the KGS also conserves the Hamiltonian: a [ ( H(t) = φ(, t) + ε ( t φ(, t)) + ( φ(, t)) ) + ψ(, t) b ψ(, t) φ(, t) ] d H(), t. (4.7) In some cases, the periodic boundary conditions (4.3) and (4.4) may be replaced by the homogeneous Dirichlet boundary conditions ψ(a, t) = ψ(b, t) =, φ(a, t) = φ(b, t) =, t. (4.8) We choose the spatial mesh size h = > with h = (b a)/m for M being an even positive integer, the time step size being k = t > and let the grid points and the time step be j := a + jh, j =,,, M; t m := mk, m =,,,.

48 4. Numerical methods for the Klein-Gordon-Schrödinger equations 36 Let ψj m and φ m j be the approimations of ψ( j, t m ) and φ( j, t m ), respectively. Furthermore, let ψ m and φ m be the solution vector at time t = t m = mk with components ψj m and φ m j, respectively. 4.. Time-splitting for the nonlinear Schrödinger equation From time t = t m and t = t m+, the first NLS-type equation (4.) is solved in two splitting steps [5, 6, 8, 9, 63]. One solves first for the time step of length k, followed by solving i t ψ + ψ =, (4.9) i t ψ + iνψ + φψ =, (4.) for the same time step. Equation (4.9) will be discretized in space by the Fourier spectral method and integrated in time eactly. For each fied [a, b], integrating (4.) from time t = t m to t = t m+ = t m + k, and then approimating the integral on [t m, t m+ ] via the trapezoidal rule [,, ], we obtain [ tm+ ] ψ(, t m+ ) = ep ( ν + iφ(, τ)) dτ ψ(, t m ) t [ m = ep νk + ik φ(, t ] m) + φ(, t m+ ) ψ(, t m ), a b. (4.) 4.. Phase space analytical solver+time-splitting spectral discretizations (PSAS-TSSP) The Klein-Gordon equation (4.) in KGS is discretized by using a pseudospectral method for spatial derivatives and then solving the ODEs in phase space analytically under appropriate chosen transmission conditions between different time intervals. From time t = t m to t = t m+, assume φ(, t) = M/ l= M/ φ m l (t)e iµ l( a), a b, t m t t m+, (4.)

49 4. Numerical methods for the Klein-Gordon-Schrödinger equations 37 where µ l = πl b a and φ m l (t) is the Fourier coefficient of lth mode. Plugging (4.) into (4.) and noticing the orthogonality of the Fourier functions, we get the following ODEs for m and t m t t m+ : ε d φm l (t) + εγ d φ m l (t) dt dt φ m l (t m ) = (φ () ) l, m =, φ m l (t m ), m >, + (µ l + ) φ m l (t) ( ψ(t m ) ) l =, (4.3) l = M,..., M. (4.4) For each fied l ( M/ l M/ ), equation (4.3) is a second-order ODE. It needs two initial conditions such that the solution is unique. When m = in (4.3) and (4.4), we have the initial condition (4.4) and we can pose the other initial condition for (4.3) due to the initial condition (4.5): d dt φ l (t ) = d dt φ l () = (φ () ) l, l = M,..., M. (4.5) Denote λ = γ + γ 4(µ l + ) ε, λ = γ γ 4(µ l + ), β = ε 4(µ l + ) γ. ε (4.6) Then the solution of (4.3), (4.4) with m = and (4.5) for l ( M/ l < M/) and t t is: (i) For γ 4(µ l + ) > φ l (t) = (ii) for γ 4(µ l φ l (t) = [ ( ( ψ () ) l µ l + + ε γ 4(µ l + ) (φ () ) l (φ () ) l [ ε γ 4(µ l + ) (φ () ) l + ) =, [ ( ψ () ) l µ l + + (φ () ) l ( (φ () ) l ( ψ () ) l µ l + ( ( ( ψ () ) l µ l + + (φ () ) l + γ (φ ε () ) l ) ] ( ψ () ) l µ l + λ ) ] λ e λ t e λ t ; (4.7) ( ψ () ) l µ l + )) t ] e γt ε ; (4.8)

50 4. Numerical methods for the Klein-Gordon-Schrödinger equations 38 and (iii) for γ 4(µ l + ) < [ φ l (t) = e γt ε β + ( (φ () ) l ( (φ () ) l + γ ε ( (φ () ) l ( ψ () ) l µ l + ) cos(βt) ( ψ () ) l µ l + ) ] + ) sin(βt) ( ψ () ) l µ l +. (4.9) But when m >, we only have one initial condition (4.4). One can t simply pose the continuity between d dt φ m l (t) and d dt m φ l (t) across the time t = t m, because the last term in (4.3) is usually different in two adjacent time intervals [t m, t m ] and [t m, t m+ ]; i.e., ( ψ(tm ) ) l ( ψ(t m ) ) l. Since our goal is to develop an eplicit scheme and we need to linearize the nonlinear term in (4.) in our discretization (4.3), in general, d dt m φ l (t m) = lim t t m d dt m φ l (t) lim t t + m d dt φ m l (t) = d dt φ m l (t + m), m =,,..., l = M,..., M. (4.) Unfortunately, we don t know the jump d dt φ m l (t + m) d dt m φ l (t m) across the time t = t m. In order to get a unique solution of (4.3) and (4.4) for m >, we pose here an additional condition: φ m l (t m ) = φ m l (t m ), l = M,..., M. (4.) The condition (4.) is equivalent to posing that the solution φ m l (t) on the time interval [t m, t m+ ] of (4.3)-(4.4) is also continuous at the time t = t m. After a simple computation, we get the solution of (4.3), (4.4) and (4.) with m > for l (l = M/,..., M/ ) and t m t t m+ : (i) For γ 4(µ l + ) > [ φ m ( ψm l (t) = ) l e kλ e kλ µ l + (e kλ ) + [ ( ψm + ) l e kλ e kλ µ l + (e kλ ) + m m φ l (t m ) φ l (t m )e kλ m m φ l (t m ) φ l (t m )e kλ ] e λ (t t m) ] e λ (t t m) + ( ψ m ) l µ l + ; (4.)

51 4. Numerical methods for the Klein-Gordon-Schrödinger equations 39 (ii) for γ 4(µ l [ + ) ( = φ m t t m l (t) = k + φ m l φ m ( ψ l (t m ) m ) l µ l + + e ( ψ (t m ) ] m ) l µ l + e γ(t tm) ε kγ ε ( )) ( ψm ) l m φ µ l (t m ) l + ( ψ + m ) l µ l + ; (4.3) and (iii) for γ 4(µ l [( + ) < φ m l (t) = e γ(t tm) ε φ m ( ψ l (t m ) ) m ) l µ l + (cos(β(t t m )) + cot(βk) sin(β(t t m ))) ( ) ] +e γk ( ψm ) ε l m sin(β(t t m )) ( ψ φ µ l (t m ) + m ) l l + sin(βk) µ l +. (4.4) From time t = t m to t = t m+, we combine the splitting steps via the standard Strang splitting [6, 5, 8]: φ m+ j = ψ j = ψ j M/ l= M/ M/ l= M/ (φ m+ ) l e iµ l( j a), (4.5) e ikµ l / (ψm ) l e iµ l( j a), = e νk+ik(φm j +φm+ j )/ ψ j, ψ m+ j = M/ l= M/ e ikµ l / (ψ ) l e iµ l( j a), j M, m ; (4.6) where Ũl, the Fourier coefficients of a vector U = (U, U, U,, U M ) T with U = U M, are defined as (φ m+ ) l = Ũ l = M M j= U j e iµ l( j a), l = M,, M and (i) for γ 4(µ l + ) > ε(e λ k e λ k ) (φ () ) γ 4(µ l +) l + ε(λ e λ k λ e λ k ) (φ () ) γ 4(µ l +) l m = ( ) ε(λ + e λ k λ e λ k ) + γ 4(µ l +) ( ψ () ) l µ l +,, (4.7) (4.8) e (λ +λ )k (φ m ) l + (e λ k + e λ k ) (φ m ) l m + ( e λ k ) ( e λ k ) ( ψ m ) l µ l + ;

52 4. Numerical methods for the Klein-Gordon-Schrödinger equations 4 (ii) for γ 4(µ l + ) = ( + γk γk )e ε ε (φ () ) l + ke γk ε (φ () ) l m = ( ) ( + γk γk ( ψ )e ε () ) l, ε µ l + (φ m+ ) l = e kγ ε (φm ) l e γk ε (φ m ) l m ( ) + e γk ε ( ψ m ) l µ l + ; (4.9) and (iii) for γ 4(µ l + ) < (cos(βk) + γ γk sin(βk))e ε βε (φ () ) l + sin(βk) ( (cos(βk) + γ γk sin(βk))e ε βε (φ m+ ) l = e γk ε (φ m ) l + cos(βk)e γk ( cos(βk)e γk ε e γk ε e γk ε β ) ( ψ () ) l, µ l + (φ () ) l m = ε (φm ) l m ) ( ψ m ) l µ l +. (4.3) The initial conditions (4.5) are discretized as ψ j = ψ () ( j ), φ j = φ () ( j ), φ () j = φ () ( j ), j M. (4.3) Note that the spatial discretization error of the above method is of spectral order accuracy in h, and the time discretization error is demonstrated to be of second order accuracy in k in section 4.3. from our numerical results Crank-Nicolson leap-frog time-splitting spectral discretizations (CN-LF-TSSP) Another way to discretize the Klein-Gordon equation (4.) in the KGS is by using a pseudospectral method for spatial derivatives, followed by application of a Crank- Nicolson/leap-frog method for linear/nonlinear terms for time derivatives: ε φm+ j φ m j + φ m j k + εγ φm+ j φ m j k + ( βφ m+ j + ( β)φ m j + βφ m j ( D f βφ m+ + ( β)φ m + βφ m ) ) = j ) ψ m j =, j M, m, (4.3)

53 4. Numerical methods for the Klein-Gordon-Schrödinger equations 4 where β / is a constant; D f, a spectral differential operator approimation of, is defined as D f U =j = M/ l= M/ µ l Ũ l e iµ l( j a). (4.33) When β = in (4.3), the discretization (4.3) to the equation (4.) is eplicit. When < β /, the discretization is implicit, but can be solved eplicitly. In fact, suppose φ m j = M/ l= M/ (φ m ) l e iµ l( j a), j =,, M, m =,,... (4.34) Plugging (4.34) into (4.3) and using the orthogonality of the Fourier functions, we obtain for m ε (φ m+ ) l (φ m ) l + (φ m ) l (φ + εγ m+ ) l (φ m ) l k k +(µ l + ) ( ψ m ) l ( β (φ m+ ) l + ( β) (φ m ) l + β (φ m ) l ) =, j M.(4.35) Solving the above equation, we get (φ m+ ) l = 4ε + (β )(µ l + ( ) )k εγk k β(µ l + ) + εγk + (φm ) l (φ ε k β(µ l + ) + εγk + m ) l ε k + ( ψm k β(µ l + ) + εγk + ) l, M ε l < M, m. (4.36) From time t = t m to t = t m+, we combine the splitting steps via the standard Strang splitting [6, 5, 8]: φ m+ j = ψ j = ψ j M/ l= M/ M/ l= M/ (φ m+ ) l e iµ l( j a), (4.37) e ikµ l / (ψm ) l e iµ l( j a), = e νk+ik(φm j +φm+ j )/ ψj ψ m+ j = M/ l= M/ e ikµ l / (ψ ) l e iµ l( j a), j M, m. (4.38)

54 4. Properties of numerical methods 4 The initial conditions are discretized as ψ j = ψ () ( j ), φ j = φ () ( j ), This implies that φ j φ j k = φ () ( j ), j M. (4.39) (φ ) l = ε + (β )(µ l + )k (ε + β(µ l + )k ) k (φ () ) l + k (k β(µ l + ) εγk + ε ) (ε + β(µ l + (φ )k ) () ) l + (ε + β(µ l + ( ψ )k ) () ) l. (4.4) Note that the spatial discretization error of the method is of spectral order accuracy in h and the time discretization error is demonstrated to be second order accurate in k in section 4.3. from our numerical results. 4. Properties of numerical methods 4.. For plane wave solution Choose the initial data in (4.3)-(4.4) as φ () = d, φ () =, a < < b, (4.4) ψ () = ( d ep i πr ), a < < b, (4.4) b a then the generalized KGS (3.37)-(3.38) admits the plane wave solution (3.4). In this case, our numerical method TSSP gives eact solution provided M ( r + ). Plugging (4.4)-(4.4) into (4.39), we get φ and note (4.7), we also get j = φ j, (4.43) ψj = ( d ep i πr ) j, j =,,,, M. (4.44) b a ψ l = d e iµra δ lr, l = M,, M, (4.45)

55 4. Properties of numerical methods 43 with, l r, δ lr =, l = r, µ r = πr b a. (4.46) Plugging (4.43), (4.44) and (4.45) into (4.37)-(4.38) with m =, we have and φ j = φ j = d, (4.47) ψ j = d e iµr j e ikµ r/, (4.48) ψ j = d e iµr j e ikµ r / e νk+ikd = d e µr j e i(kµ r / kd), ν =. (4.49) Then we obtain ψ j = d e i(kµ r kd µra), ν =. (4.5) By induction, we can obtain φ m+ j = d, (4.5) ψ m+ j = d e i(tµ r t d µ ra), ν =, (4.5) with t = t m+ = (m + )k, m =,,. (4.53) Here we use the identity M j= e iπ(r l)j/m =, r l nm, M, r l = nm. for n integer (4.54) Thus in this case our numerical method TSSP really gives eact results provided that the number of grid points M ( r + ). 4.. Conservation and decay rate Define the usual l -norm and mean value of a vector U = (U, U,..., U M ) T which is a discretization of a periodic function U() on the interval [a, b] with U j = U( j )

56 4. Properties of numerical methods 44 (j =,,, M) as U l = b a M M j= U j, N(U) = b a M M j= U j. (4.55) For the discretizations (4.5)-(4.6) and (4.37)-(4.38), we have the following result for dynamics of wave energy in discretized level: Theorem 4.. The discretizations PSAS-TSSP(4.5)-(4.6) and CN-LF-TSSP (4.37)- (4.38) for KGS possess the following properties: ψ m l = e νtm ψ l = e νtm ψ () l, m =,,,. (4.56) Proof. From (4.38) in the scheme TSSP, noting (4.7), we have M M b a ψm+ l = ψ m+ j = j= = M = M = M j= M/ l= M/ M/ l= M/ M = e νk = e νk M = e νk M M j= M/ l= M/ M/ e ikµ l / (ψ ) l = M M j= ψj e iµ l( j a) e νk+ik(φm j +φm+ j )/ ψj j= M/ l= M/ M/ l= M/ M/ l= M/ = Me νk b a ψm l = e ikµ l / (ψ ) l e iµ l( j a) l= M/ = M j= (ψ ) l ψ j M = e νk j= e kµ l / (ψ ) l e iµ l( j a) e iµ l / (ψ ) l = e νk M ψ j M/ l= M/ ψ j e iµ l( j a) M = e νk ψ m j j= (ψ ) l = Me νt m+ ψ l, m. (4.57) b a

57 4. Properties of numerical methods 45 Thus, the equality (4.56) is proved. Here, we use the identities M, k l nm, e iπ(k l)j/m = for n integer, (4.58), k l = nm, j= and M/ l= M/, k j nm, e iπ(k j)l/m =, k j = nm, for n integer. (4.59) 4..3 Dynamics of mean value of meson field When ν =, for the discretization PSAS-TSSP (4.5)-(4.6), we have the following results for dynamics of mean value of the meson field in discretized level: Theorem 4.. When ν =, the discretization PSAS-TSSP (4.5)-(4.6) for KGS possesses the following property: (i) when γ > (ii) when γ = N(φ m+ ) = D() + N(φ() ) λ (N(φ () ) D()) e λ λ λ t m+ + N(φ() ) + λ (N(φ () ) D()) e λ λ λ t m+ ; (4.6) N(φ m+ ) = D() + ( N(φ () ) D() ) e λ t m+ and (iii) when γ < + ( N(φ () ) λ ( N(φ () ) D() )) t m+ e λ t m+ ; (4.6) where [ N(φ m+ ) = D() + e λ t m+ (N(φ () ) D() ) cos(β t m+ ) ( + N(φ() ) λ N(φ () ) D() ) ] sin(β t m+ ) ; (4.6) β D() = N( ψ ) = b a M M j= ψ () ( j ). (4.63)

58 4. Properties of numerical methods 46 Proof. From (4.55), (4.5) and (4.7), noticing the orthogonality of the discrete Fourier series, we have N(φ m+ ) = b a M = b a M M j= M/ l= M/ φ m+ j = b a M M (φ m+ ) l Similarly, from (4.56) with ν =, we have ( ψ m+ ) = M ψ m+ j = ψm+ l M b a j= j= M j= M/ l= M/ (φ m+ ) l e iµ l( j a) e iµ l( j a) = (b a) (φ m+ ). (4.64) = ψ l b a = ( ψ () ), m. (i) When γ >, denote p = e λ k and q = e λ k. From (4.8) and (4.6) with l =, (3.36) and (4.65), we obtain (φ m+ ) = pq(φ m ) + (p + q) (φ m ) + (p )(q ) ( ψ m ) (φ ) = = pq (φ m ) + (p + q) (φ m ) + (p )(q ) ( ψ () ), m ;(4.65) p q (φ λ λ () ) + λ q λ p (φ λ λ () ) + Rewrite (4.65), by induction, we obtain for m : (φ m+ ) p (φ m ) + (p ) ( ψ () ) From (4.67), by induction again, we get (φ m+ ) = m r= ( ) + λ p λ q λ λ = q ( (φm ) p (φ m ) + (p ) ( ψ ) () ) ( ψ () ). (4.66) ( = q m (φ ) p (φ () ) + (p ) ( ψ ) () ). (4.67) [ ( p r q m r (φ ) p (φ ) () ) + p r (q m r )(p ) ( ψ ] () ) +p m+ (φ () ) = qm+ p m+ q p (φ ) + qpm+ pq m+ q p (φ () ) + pm+ q m+ + pq m+ qp m+ + q p ( ψ q p () ). (4.68)

59 4. Properties of numerical methods 47 Combining (4.64), (4.68) and (4.66), noticing (4.65) and (4.55), we obtain [ N(φ m+ ) = qm+ p m+ p q N(φ () ) + λ q λ p N(φ () ) q p λ λ λ λ ( ) ] + + λ p λ q N( ψ () ) + qpm+ pq m+ N(φ () ) λ λ q p + pm+ q m+ + pq m+ qp m+ + q p N( ψ () ) q p = pm+ q m+ N(φ () ) + λ q m+ λ p m+ N(φ () ) λ λ λ λ ) + ( λ q m+ λ p m+ N( ψ () ) λ λ = N( ψ () ) + N(φ() ) + λ (N(φ () ) N( ψ () )) e λ λ λ t m+ + N(φ() ) λ (N(φ () ) N( ψ () )) e λ λ λ t m+. (4.69) Thus (4.6) is a combination of (4.69) and (4.63). (ii) When γ =, denote p = e λ k. From (4.9) and (4.6) with l =, (3.36) and (4.65), we obtain (φ m+ ) = p (φ m ) + p (φ m ) + (p ) ( ψm ) = p (φ m ) + p (φ m ) + (p ) ( ψ () ), m ; (4.7) (φ ) = kp (φ () ) + ( λ k)p (φ () ) + ( ( λ k)p) ( ψ () ). (4.7) Rewrite (4.7), by induction, we obtain for m : (φ m+ ) p (φ m ) + (p ) ( ψ () ) = p ( (φm ) p (φ m ) + (p ) ( ψ ) () ) = p m ( (φ ) p (φ () ) + (p ) ( ψ () ) ). (4.7)

60 4. Properties of numerical methods 48 From (4.7), by induction again, we get m [ ( (φ m+ ) = p m (φ ) p (φ ) () ) + p r (p m r )(p ) ( ψ ] () ) r= +p m+ (φ () ) = (m + )p m (φ ) mp m+ (φ () ) + ( p m+ + (m + )(p )p m) ( ψ () ). (4.73) Combining (4.64), (4.73) and (4.7), noticing (4.65) and (4.55), we obtain N(φ m+ ) = (m + )p m [ kpn(φ () ) + ( λ k)pn(φ () ) + ( ( λ k)p) N( ψ () ) ] mp m+ N(φ () ) + [ p m+ + (m + )(p )p m] N( ψ () ) = (m + )kp m+ N(φ () ) + ( (m + )λ k) p m+ N(φ () ) + [ p m+ + (m + )λ kp m+] N( ψ () ) = N( ψ () ) + [ N(φ () ) N( ψ () ) ] e λ t m+ + [ N(φ () ) λ ( N(φ () ) N( ψ () ) )] t m+ e λ t m+. (4.74) Thus (4.6) is a combination of (4.74) and (4.63). (iii) When γ <, denote p = e (λ +iβ )k and q = e (λ iβ )k. From (4.8) and (4.6) with l =, (3.36) and (4.65), we obtain (φ m+ ) = pq(φ m ) + (p + q) (φ m ) + (p )(q ) ( ψ m ) = pq (φ m ) + (p + q) (φ m ) + (p )(q ) ( ψ () ), m ; (4.75) (φ ) = sin(β k)e λ k (φ () ) β + [ + ( cos(β k) λ ( cos(β k) λ β sin(β k) Rewrite (4.75), by induction, we obtain for m : ) ) sin(β k) β ] e λ k (φ m+ ) p (φ m ) + (p ) ( ψ () ) = q ( (φm ) p (φ m ) + (p ) ( ψ ) () ) e λ k (φ () ) ( ψ () ). (4.76) = q m ( (φ ) p (φ () ) + (p ) ( ψ () ) ). (4.77)

61 4. Properties of numerical methods 49 From (4.77), by induction again, we get (φ m+ ) = m r= [ ( p r q m r (φ ) p (φ ) () ) + p r (q m r )(p ) ( ψ ] () ) +p m+ (φ () ) = qm+ p m+ q p (φ ) + qpm+ pq m+ q p + pm+ q m+ + pq m+ qp m+ + q p q p (φ () ) ( ψ () ). (4.78) Combining (4.64), (4.78) and (4.76), noticing (4.65) and (4.55), we obtain [ ( ( N(φ m+ ) = qm+ p m+ cos(β k) λ ) ) sin(β k) e λ k N( ψ () ) q p β ) ] + sin(β k)e λ k β N(φ () ) + ( cos(β k) λ β sin(β k) e λ k N(φ () ) + qpm+ pq m+ N(φ () ) + pm+ q m+ + pq m+ qp m+ + q p N( ψ () ) q p q p ( = e λ (m+)k cos((m + )β k) λ ) sin((m + )β k) N(φ () ) β [ ( + e λ (m+)k cos((m + )β k) λ )] sin((m + )β k) N( ψ () ) β ( ) +e λ (m+)k sin((m + )β k) N(φ () ) β [ = N( ψ () ) + e λ t m+ (N(φ () ) N( ψ () ) ) cos(β t m+ ) ( + N(φ() ) λ N(φ () ) N( ψ () ) ) ] sin(β t m+ ). (4.79) β Thus (4.6) is a combination of (4.79) and (4.63) Stability analysis By using the standard von Neumann analysis for the discretizetion PSAS-TSSP (4.5)-(4.6) and CN-LP-TSSP (4.37)-(4.38), we have Theorem 4.3. The discretization PSAS-TSSP (4.5)-(4.6) is unconditionally stable for any parameter value γ, time step k > and mesh size h >. When

62 4. Properties of numerical methods 5 /4 β / and γ =, the discretization CN-LP-TSSP (4.37)-(4.38) is unconditionally stable; and when β < /4 and γ =, it is conditionally stable under the stability condition k hε ( 4β)(π + h ). (4.8) Proof. For the discretization PSAS-TSSP (4.5)-(4.6), setting ( ψ m ) l = and plugging m+ φ l = µ φ m l = µ φm l into (4.8), (4.9) and (4.3) with µ the amplification factor, we obtain: (i) When γ 4(µ l + ) >, the characteristic equation for µ is µ (e λ k + e λ k )µ + e (λ +λ )k =. (4.8) Solving the above equation, we have µ = e λ k, µ = e λ k. (4.8) Thus the amplification factor satisfies G l = ma{ µ, µ } = e γ+ γ 4(µ l +), l = M,..., M. (4.83) (ii) When γ 4(µ l + ) =, the characteristic equation for µ is µ e γk µ + e γk =. (4.84) Solving the above equation, we have µ = e γk. Thus the amplification factor satisfies G l = µ = e γk, l = M,..., M. (4.85) (iii) When γ 4(µ l + ) <, the characteristic equation for µ is µ cos(βk)e γk µ + e γk =. (4.86)

63 4. Properties of numerical methods 5 Solving the above equation, we have µ = e γk [cos(βk) ± i sin(βk)]. (4.87) Thus the amplification factor satisfies G l = µ = e γk cos (βk) + sin (βk) = e γk, l = M,..., M. (4.88) (4.83), (4.85) and (4.88), together with (4.56), imply that PSAS-TSSP is unconditionally stable for any time step k >, mesh size h > and parameter values γ. Similarly for the discretization CN-LF-TSSP (4.37)-(4.38), noting (4.36), we have the characteristic equation ( µ (µ l + ) )k + εγk µ + k β(µ l + ) + εγk + ε k β(µ l Solving the above equation, we obtain εγk =. + ) + εγk + ε (4.89) (µ l µ = + )k + εγk k β(µ l + ) + εγk + ε ( (µ ± l + ) )k + εγk + k β(µ l + ) + εγk + ε k β(µ l εγk. + ) + εγk + ε When /4 β / and γ =, we have (µ l + )k k β(µ l + ) +, k >. ε Thus (µ l µ = + )k ± i k β(µ l + ) + ε ( (µ l + ) )k. k β(µ l + ) + ε This implies that the amplification factor satisfies ( (µ l G l = µ = + ) ( )k (µ l + + )k k β(µ l + ) + ε k β(µ l + ) + ε =, l = M/,, M/. )

64 4.3 Numerical results of the Klein-Gordon-Schrödinger equation 5 This, together with (4.56), implies that CN-LP-TSSP with /4 β / is unconditionally stable. On the other hand, when β < /4 and γ =, we need the stability condition (µ l + )k k β(µ l + ) +, l = M/,, M/. ε This implies that k min M/ l M/ 4ε ( 4β)( + µ l ) = hε ( 4β)(π + h ). Thus we get the stability condition (4.8). 4.3 Numerical results of the Klein-Gordon-Schrödinger equation In this section, we will compare the accuracy and stability of different numerical methods for the Klein-Gordon-Schrödinger equations with a solitary wave solution, and also present the numerical results for plane waves, soliton-soliton collisions in D, D problems and the generalized KGS with a damping term Comparisons of different methods Eample 4.. The standard KGS with a solitary wave solution in D, i.e., we choose d =, ε =, ν = in (.4)-(.6). The initial data in (.6) is chosen as ψ () () = ψ + (, t = ), φ () () = φ + (, t = ), (4.9) φ () () = t φ + (, t = ), R; (4.9) where ψ + and φ + are given in (3.4)-(3.4). We present computations for two different regimes of acoustic speed, i.e., /ε. C ase I. O()-acoustic speed, i.e., ε =. In our computation, we take B = in (3.4)-(3.4) and solve the problem on the interval [ 3, 3], i.e., a = 3 and

65 4.3 Numerical results of the Klein-Gordon-Schrödinger equation 53 Mesh h =. h = / h = /4 PSAS-TSSP e E-3 9.6E-8 e E E-7 CN-LF-TSSP e E E-7 (β = ) e E E-7 CN-LF-TSSP e E E-8 (β = /4) e E-3 7.5E-7 CN-LF-TSSP e E-3.4E-7 (β = /) e E E-7 Table 4.: Spatial discretization errors e (t) and e (t) at time t = for different mesh sizes h under k =.. I: For γ =. b = 3 with periodic boundary conditions. When γ =, the KGS admits the wellknown solitary wave solution (3.4)-(3.4) as eact solution. When γ >, there is no analytical solution and we let ψ and φ be the eact solutions which are obtained numerically by using our numerical method with a very fine mesh and small time step size, e.g. h = 3 and k =.. Let ψ h,k and φ h,k be the numerical solution obtained by using a method with mesh size h and time step k. To quantify the numerical methods, we define the error functions as e (t) = ψ(., t) ψ h,k (t) l, e (t) = φ(., t) φ h,k (t) l, e(t) = ψ(., t) ψ h,k(t) l ψ(., t) l + φ(., t) φ h,k(t) l φ(., t) l First we test the discretization error in space. = e (t) ψ(., t) l + e (t) φ(., t) l In order to do this, we choose a very small time step, e.g., k =., such that the error from time discretization is negligible compared to the spatial discretization error, and solve the KGS with different methods under different mesh size h and γ. Table 4. lists the numerical errors of e (t) and e (t) at t = with different mesh sizes h and parameter values γ for different numerical methods. Then we test the discretization error in time. Table 4. shows the numerical errors.

66 4.3 Numerical results of the Klein-Gordon-Schrödinger equation 54 Mesh h =. h = / h = /4 PSAS-TSSP e E-3.7E-6 e E-3.79E-6 CN-LF-TSSP e E-3.6E-6 (β = ) e E-3.78E-6 CN-LF-TSSP e E-3.65E-6 (β = /4) e E-3.5E-6 CN-LF-TSSP e E-3.6E-6 (β = /) e E-3.E-6 Table 4.: (cont d): II: For γ =.5. e (t) and e (t) at t =. under different time steps k and mesh sizes h for different numerical methods. Finally we test the conservation of conserved quantities. Table 4.3 presents the quantities and numerical errors at different times with mesh size h = and time 8 step k =. for different numerical methods. From Tables 4.-4., we can draw the following observations: Our new numerical methods PSAS-TSSP and CN-LF-TSSP are of spectral-order accuracy in space discretization and second-order accuracy in time. Moreover, PSAS-TSSP and CN-LP- TSSP with β = / or β = /4 are unconditionally stable, where CN-LP-TSSP with β = is conditionally stable. Both numerical methods conserve the wave energy D eactly and the Hamiltonian H very well (up to 8 significant digits). Furthermore, these two methods are eplicit, easy to program, less memory requirement, easy to etend to D & 3D cases, and keep most properties of KGS in the discretized level. In the following net section, we always use PSAS-TSSP for solving KGS. C ase II. nonrelativistic limit regime, i.e., < ε. We choose B = and γ = in (3.4)-(3.4). Here we test the ε-resolution of different numerical methods when ε. Two types of initial data are chosen:

67 4.3 Numerical results of the Klein-Gordon-Schrödinger equation 55 h Error k = 5 k = k = 4 k = 6 PSAS-TSSP 4 e 7.44E E-4.97E-5.86E-6 e.49e-3.538e E-6 7.4E-7 e E E-4.97E-5.859E-6 e.49e-3.538e E E-7 CN-LF-TSSP 4 e 7.935E E E-5.995E-6 (β = ) e 7.88E E-5 3.4E E-7 e E E E-5.993E-6 e 7.88E E-5 3.E-6.3E-7 CN-LF-TSSP 4 e 6.74E E-4.68E-5.68E-6 (β = /4) e 5.69E-3 3.6E-4.75E-5.486E-6 e E E-4.68E-5.678E-6 e 5.69E-3 3.6E-4.74E-5.43E-6 CN-LF-TSSP 4 e 6.737E E-4.69E-5.685E-6 (β = /) e.6e- 6.75E E-5.686E-6 e E E-4.69E-5.683E-6 e.6e- 6.75E E-5.65E-6 Table 4.: Temporal discretization errors e (t) and e (t) at time t = for different time steps k. I: For γ =. Type. O(ε)-wavelength in the initial data, i.e., we choose the initial data as (4.9)-(4.9) with ε. Type. O()-wavelength in the initial data, i.e., we choose the initial data as where φ () satisfies ψ(, ) = ψ () () = sech( + p)e i(+p) + sech( p)e i( p),(4.9) φ(, ) = φ () (), t ψ(, ) =, < <, (4.93) φ () () + φ () () = ψ () (), < <. (4.94)

68 4.3 Numerical results of the Klein-Gordon-Schrödinger equation 56 Error k = 4 k = 8 k = 6 k = 3 k = 64 PSAS-TSSP e 4.35E E-.885E E-3.73E-3 e 9.93E-.87E- 5.5E-3.364E-3 3.4E-4 CN-LF-TSSP e 3.948E- 7.E-.656E- 4.7E-3.8E-3 (β = ) e 5.475E-.46E-.957E-3 7.3E-4.87e-4 CN-LF-TSSP e 3.848E E-.3E- 3.7E E-4 (β = /4) e.58e- 5.E-.35E- 3.79E E-4 CN-LF-TSSP e 3.966E E-.33E- 3.83E-3 8.6E-4 (β = /) e 3.57E E-.6E- 5.54E-3.385E-3 Table 4.: (cont d): II. For γ =.5 and h = /4. Time e D H PSAS-TSSP..69E E CN-LF-TSSP E β =..E CN-LF-TSSP..63E β = / E CN-LF-TSSP..473E β = /. 4.85E Table 4.3: Conserved quantities analysis: k =. and h = 8. We take p = 8 in (4.9). Figure 4. shows the numerical results of PSAS-TSSP at t = when we choose the meshing strategy h = O(ε) and k = O(ε):Γ = (ε, h, k ) = (.5,.5,.4), Γ /4, Γ /6; and h = O(ε) and k =.4-independent of ε: Γ = (ε, h ) = (.5,.5), Γ /4, Γ /6 for Type initial data; and Figure 4. shows similar results for Type initial data. In addition, CN-LF-TSSP with β = /4 or / gives similar numerical results at the same meshing strategies.

69 4.3 Numerical results of the Klein-Gordon-Schrödinger equation 57 From Tables and Figures 4.-4., we can draw the following observations: (i) In O()-speed of light regime, i.e., ε = O() fied, our new numerical methods PSAS-TSSP and CN-LP-TSSP are of spectral-order accuracy in space discretization and second-order accuracy in time. Moreover, PSAS-TSSP and CN-LF-TSSP with β = / or β = /4 are unconditionally stable, where CN-LF-TSSP with β = is conditionally stable. Both numerical methods conserve the wave energy D eactly and the Hamiltonian H very well (up to 8 significant digits). Furthermore, these two methods are eplicit, easy to program, less memory requirement, easy to etend to D & 3D cases, and keep most properties of KGS in the discretized level. (ii) in the nonrelativistic limit regime, i.e., < ε, the ε-resolution of our numerical methods PSAS-TSSP and CN-LF-TSSP with ε = /4 or / is h = O(ε) and k = O(ε) for O(ε)-wavelength, and resp., h = O() and k = O() for O()-wavelength, initial data. The method CN-LF-TSSP with β = gives correct numerical results only at meshing strategy h = O(ε) and k = O(ε ) for O(ε)- wavelength, and resp., h = O() and k = O(ε) for O()-wavelength, initial data Application of our numerical methods Plane wave solution of the standard KGS Eample 4.. The standard KGS with a plane-wave solution, i.e., we choose d =, ε =, ν = and γ = in (.4)-(.6) and consider the problem on the interval [a, b] with a = and b = π. The initial condition is taken as ψ(, ) = ψ () () = e i7, φ(, ) = φ () () =, φ t (, ) = φ () () =. (4.95) It is easy to see that KGS (4.)-(4.) with periodic boundary conditions (4.3)-(4.4) and initial condition (4.95) admits the plane wave solution ψ(, t) = e i(7 48t), φ(, t) =, a b, t. (4.96) We solve this problem by using the PSAS-TSSP (4.5)-(4.6) with mesh size h = π, 8 time step k =.. Figure 4.3 shows the numerical results at t = and t = 4.

70 4.3 Numerical results of the Klein-Gordon-Schrödinger equation 58 (a) (b) (c) Figure 4.: Numerical solutions of the meson field φ (left column) and the nucleon density ψ (right column) at t = for Eample with Type initial data in the nonrelativistic limit regime by PSAS-TSSP. - : eact solution given in (4.96), : numerical solution. I. With the meshing strategy h = O(ε) and k = O(ε): (a) Γ = (ε, h, k ) = (.5,.5,.4), (b) Γ /4, and (c) Γ /6. From Figure 4.3, we can see that the PSAS-TSSP method for KGS really provides the eact plane-wave solution of the KGS equations (4.)-(4.5).

71 4.3 Numerical results of the Klein-Gordon-Schrödinger equation a) 3 3 b) c) d) e) f) 3 3 Figure 4.: (cont d): II. With the meshing strategy h = O(ε) and k =.4- independent of ε: (d) Γ = (ε, h ) = (.5,.5), (e) Γ /4, and (f) Γ /6. Soliton-Soliton collisions of standard KGS Eample 4.3. Interaction between solitary wave solutions in D for the standard KGS, i.e., we choose d =, ε =, ν = and γ = in (.4)-(.6). The initial condition is chosen as ψ(, ) = ψ + ( + p, t = ) + ψ ( p, t = ), (4.97) φ(, ) = φ + ( + p, t = ) + φ ( p, t = ), R, (4.98) t φ(, ) = t φ + ( + p, t = ) + t φ ( p, t = ), (4.99)

72 4.3 Numerical results of the Klein-Gordon-Schrödinger equation a) c) e) b) d) f) 3 3 Figure 4.: Numerical solutions of the density ψ(, t) and the meson field φ(, t) at t = in the nonrelativistic limit regime by PSAS-TSSP with the same mesh (h = / and k =.5). - : eact solution, : numerical solution. The left column corresponds to the meson field φ(, t): (a) ε = /, (c) ε = /6. (e) ε = /8. The right column corresponds to the density ψ(, t) : (b) ε = /, (d) ε = /6. (f) ε = /8. where ψ ± and φ ± are defined as in (3.4) and (3.4), and = ±p are initial locations of the two solitons.

73 4.3 Numerical results of the Klein-Gordon-Schrödinger equation Re(ψ(,t)) Re(ψ(,t)).5.5 a) Im(ψ(,t)) Im(ψ(,t)).5.5 b) φ(,t) φ(,t).5.5 c) Figure 4.3: Numerical solutions for plane wave of KGS in Eample 4. at time t = (left coumn) and t = 4 (right column). : eact solution given in (4.96), : numerical solution. (a): Real part of nucleon filed Re(ψ(, t)); (b): imaginary part of nucleon field Im(ψ(, t)); (c): meson filed φ. We solve the problem in the interval [ 4, 4], i.e., a = 4 and b = 4 with mesh size h = 5/8 and time step k =. by using our method PSAS-TSSP, and take p = 8 and B =. Figure 4.4 shows the values of ψ(, t) and φ(, t) at different times. From Figure 4.4, the time t = 9. corresponds to the time when the two solitons are at the same location and the time t = 8.5 corresponds to the time when the collision is nearing completion (cf. Figure 4.4). From the figure we can see that during the collision, waves are emitted, and that after the collision the two

74 4.3 Numerical results of the Klein-Gordon-Schrödinger equation 6 t=. 3 t=7.8 3 ψ ψ t=8.8 t= ψ 3 ψ 3 t=. 3 t=. 3 ψ ψ 3 t=.5 t=8.5 3 ψ ψ Figure 4.4: Numerical solutions of soliton-soliton collison in standard KGS in Eample 4.3 I: Nucleon density ψ(, t). solitons have a reduced peak value.

75 4.3 Numerical results of the Klein-Gordon-Schrödinger equation 63 7 t=. 7 t= φ 3 φ 3 5 t=8.8 6 t=9. 4 φ φ φ φ 8 t= φ φ t=. 6 t=.5 6 t= Figure 4.5: (cont d): II. Meson field φ(, t). Soliton-soliton collisions of KGS with damping term Eample 4.4. Soliton-soliton collision in D of KGS with damping term, i.e., we choose d =, ε = and ν = in (.4)-(.6). The initial condition is chosen as (4.97)-(4.98). Again, we solve the problem in the interval [ 4, 4], i.e., a = 4 and b = 4 with mesh size h = 5/8 and time step k =. by using our method

76 4.3 Numerical results of the Klein-Gordon-Schrödinger equation 64 PSAS-TSSP, and take p = 8 and B =. Figure 4.6 displays time evolution of ψ(, t) and φ(, t) for different values of γ. Figure 4.7 shows time evolution of the Hamiltonian H(t) and mean value of the meson field N(t) for different values of γ. From Figures , we can draw the following conclusions: (i) When γ =, the collision between two solitons seems quite elastic (cf. Figure 4.6 top row) although there are some waves are emitted; when γ > but small, damping effect can be observed in the collision and the emission of the sound waves is inconspicuous; when γ > and large, a soliton wave which is a bound state of the KGS is generated after the collision (cf last row). This observation seems new for the KGS. (ii). When γ =, the Hamiltonian is conserved; when γ >, it decreases when time increases and converges to a constant when time goes infty (cf. Figure 4.7 left). (iii) When γ =, the mean value of the meson field changes periodically; where it oscillates and decays when γ > (cf. Figure 4.7 right). These agree very well with the analytical results in section. (iv) The results here also demonstrate the efficiency and high resolution of our numerical method for studying soliton-soliton collision in KGS. Soliton-soliton collisions of KGS in nonrelativistic limit regime Eample 4.5. Soliton-soliton collision in D of KGS in nonrelativistic limit regime, i.e., we choose d =, γ = and ν = in (.4)-(.6). Let ψ () () = sech( + p)e i(+p) + sech( p)e i( p), (4.) φ () () = sech( + p) sech( p), R. (4.) We solve the KGS (.4)-(.6) in one dimension with the initial conditions ψ KGS (, ) = ψ () (), φ KGS (, ) = φ () (), t φ KGS (, ) =, R, (4.) in the interval [ 8, 8]. We take p = 6.. Figures show that there are many oscillations when ε, which are quite different from the numerical results obtained by the different initial conditions (4.9)-(4.94).

77 4.3 Numerical results of the Klein-Gordon-Schrödinger equation 65 Figure 4.6: Time evolution of nucleon density ψ(, t) (left column) and meson field φ(, t) (right column) for soliton-soliton collision of KGS in Eample 4.4 for different values of γ.

78 4.3 Numerical results of the Klein-Gordon-Schrödinger equation 66 3 γ= γ=. γ= γ=4 4. N(φ) γ= γ=. γ= γ=4 H a) t b) t Figure 4.7: Time evolution of the Hamiltonian H(t) ( left ) and mean value of the meson field N(t) ( right ) in Eample 4.4 for different values of γ γ= γ=. γ= γ=4 5 γ= γ=. γ= γ=4 34 H 33.5 N(φ) t t Figure 4.8: Time evolution of the Hamiltonian H(t) ( left ) and mean value of the meson field N(t) ( right ) in Eample 4.6 for different values of γ. Solution of D standard KGS Eample 4.6. Dynamics of KGS in D, i.e., we choose d =, ν = and ε = in (.4)-(.6). The initial condition is taken as ψ(, y, ) = e +y + e ( +y ) ei5/ cosh( 4 +y ), φ(, y, ) = e ( +y ) +y ), φ t (, y, ) = e (, (, y) R.

79 4.3 Numerical results of the Klein-Gordon-Schrödinger equation Figure 4.9: Numerical solutions of the nucleon density ψ(, y, t) (right column) and meson field φ(, y, t) (left column) in Eample 4.5 at t =. th row: ε = /; nd row: ε = /8; 3rd row : ε = /3. We solve this problem on the rectangle [ 64, 64] with mesh size h = /6 and time step k =. by using our method PSAS-TSSP. Figure 4. shows the surface plots of φ and ψ with γ = at different times. Figure 4.8 depicts time evolution of the Hamiltonian H(t) and mean value of the meson field N(t) for different values of γ. From Figures , we can draw the following conclusions: (i). When γ =, the Hamiltonian is conserved; when γ >, it decreases when time increases and

80 4.3 Numerical results of the Klein-Gordon-Schrödinger equation Figure 4.: Numerical solutions of the nucleon density ψ(, y, t) (right column) and meson field φ(, y, t) (left column) in Eample 4.5 at t =. th row: ε = /; nd row: ε = /8; 3rd row : ε = /3. converges to a constant when time goes infinity (cf. Figure 4.8 left). (ii) When γ =, the mean value of the meson field changes periodically; where it oscillates and decays when γ > (cf. Figure 4.8 right). These agree very well with the analytical results in section. (iii) The results here also demonstrate the efficiency and high resolution of our numerical method for studying dynamics of KGS in D & 3D.

81 4.3 Numerical results of the Klein-Gordon-Schrödinger equation 69 t= y Figure 4.: Surface plots of the nucleon density ψ(, y, t) (left column) and meson field φ(, y, t) (right column) in Eample 4.6 with γ = at different times.

82 Chapter 5 Application to the Schrödinger-Yukawa equations In this chapter, we will etend our numerical method, time-splitting spectral method (TSSP) to the Schrödinger-Yukawa equations and also present the numerical results of the Schrödinger-Yukawa equations in D and D cases. 5. Introduction to the Schrödinger-Yukawa equations The generalized Schrödinger-Yukawa equations are i η t ψ = η θ ψ + Cφψ α ψ d ψ i γψ + Vet ψ, R d, t >, (5.) φ = ψ + βφ, R d, t >, (5.) ψ(, ) = ψ () (), φ(, ) = φ () (), R d. (5.3) where η is the scaled Planck constant, C > is a fied constant, α >, θ, β are real parameters and V et is a given eternal potential, for eample a confining potential. 7

83 5. Introduction to the Schrödinger-Yukawa equations 7 When η =, θ =, α =, V et =, β =, we can get the well-known Schrödinger- Yukawa (S-Y) equations i t ψ + ψ + φψ + i νψ =, R d, t >, (5.4) φ + φ = ψ, R d, t > ; (5.5) which can be also derived from (.4) and (.5) when ε (corresponding to infinite speed of light or nonrelativistic limit regime). When θ =, γ =, β =, the system of (5.) and (5.) reduces to the well-known Schrödinger-Poisson-Xα (S-P-Xα) model, which has been firstly derived in [, 3] as a local one particle approimation of the time dependent Hartree-Fock equations. It describes the time evolution of electrons in a quantum model respecting the Pauli principle in an approimate fashion. It reads i η t ψ = η ψ + Cφψ α ψ d ψ + Vet ψ, R d, t >, (5.6) φ = ψ, R d, t >. (5.7) It is an NLS with two nonlinear terms of different nature: the nonlocal Hartree potential and a local power term, with focusing sign and an eponent that is subcritical for finite time blowup. Such time dependent density functional theory models yield approimations of the time-dependent Hartree-Fock (TDHF) equations that are much easier to solve numerically: firstly, for large number of particles N the system of coupled NLS of the TDHF system becomes too large, and secondly, the echange terms are very costly to calculate.

84 5. Numerical method for the Schrödinger-Yukawa equations 7 5. Numerical method for the Schrödinger-Yukawa equations For d =, the Schrödinger-Yukawa equations with periodic boundary conditions can be written as i η t ψ = η θ ψ + Cφψ α ψ ψ i νψ + V et ψ, a < < b, t > (5.8) φ = ψ + βφ, a < < b, t >, (5.9) ψ(a, t) = ψ(b, t), ψ(a, t) = ψ(b, t), t, (5.) φ(a, t) = φ(b, t), φ(a, t) = φ(b, t), t, (5.) ψ(, ) = ψ () (), φ(, ) = φ () (), a b. (5.) The idea to construct the TSSP for the KGS (.4)-(.6) can be easily etended to the Schrödinger-Yukawa equations (5.)-(5.). The details of the numerical scheme are shown below. From time t = t m to t = t m+, we use the Strang splitting formula: M M/ l M/ ( ψ µ l +β m ) l e iµ l( j a), β, φ m+ j = ψj = M ψ = ψ n+ j = M M M/ l M/, l ( ψ m ) l e iµ l( j a), β =, µ l +β (5.3) j =,,,, M. (5.4) M/ l= M/ e iηµ l k/ (ψm ) l e iµ l( j a), j =,,,, M. (5.5) e i[c(φm j +φm+ j )/+V et( j )+α(e νk ) ψ j /ν i ν]k/η ψ j, ν, e i[c(φm j +φm+ j )/+V et( j ) α ψ j i ν]k/η ψ j, ν =, (5.6) j =,,,, M. (5.7) M/ l= M/ e iηµ l k/ (ψ ) l e iµ l( j a), j =,,,, M ; (5.8) where Ũl (l = M/,, M/ ), the Fourier coefficients of a vector U = (U, U,, U M ) T with U = U M, are defined as µ l = πl M b a, Ũ l = U j e iµ l( j a), l = M,, M j=. (5.9)

85 5.3 Numerical results of the Schrödinger-Yukawa equations Numerical results of the Schrödinger-Yukawa equations In this section, we study the convergence of KGS to S-Y in the nonrelativistic limit ( < ε ), where the parameter ε is inversely proportional to the acoustic speed. We also present the numerical solution of the Schrödinger-Yukawa equations in the D case Convergence of KGS to S-Y in nonrelativistic limit regime Eample 5.. Reduction from the Klein-Gordon-Schrödinger equations to the Schrödinger-Yukawa equations, i.e., we choose d =, ν = and γ = in (.4)-(.6). Let and φ () () satisfies ψ () () = sech( + p)e i(+p) + sech( p)e i( p), (5.) φ () () + φ () () = ψ () (). (5.) We solve the KGS (.4)-(.6) in one dimension with the initial conditions ψ KGS (, ) = ψ () (), φ KGS (, ) = φ () (), t φ KGS (, ) =, R, (5.) and the S-Y (5.)-(5.) in one dimension with the initial condition ψ SY (, ) = ψ () (). (5.3) in the interval [ 8, 8] with mesh size h = 5/8 and time step k =.5. We take p = 8. Let ψ KGS and φ KGS be the numerical solutions of the KGS (.4)-(.6), ψ SY and φ SY be of the S-Y (5.)-(5.) by using PSAS-TSSP and TSSP respectively. Table 5. shows the errors between the solutions of the KGS and its reduction S-Y at time t =. under different ε.

86 5.3 Numerical results of the Schrödinger-Yukawa equations 74 ε = /4 ε/ ε/4 ε/8 φ KGS φ SY l E- 7.76E-3.89E-3 ψ KGS ψ SY l 3.6E E-3.399E E-4 H KGS H SY E- 8.7E-3.49E-3 Table 5.: Error analysis between KGS and its reduction S-Y: Errors are computed at time t = under h = 5/8 and k =.5. From Table 5., we can see that the meson field φ KGS, nucleon density ψ KGS, and the Hamiltonian H KGS of the KGS (.4)-(.6) converge to φ SY in l -norm, ψ SY in l -norm, H SY of the Schrödinger-Yukawa equations (5.)-(5.) quadratically when ε Applications Eample 5.. -D S-P-Xα model, i.e., we choose d =, ε =, θ = /, V et and C = in (5.)-(5.). Note that the local interaction term in (5.) is the focusing cubic NLS interaction in the case d =. The initial condition is hence taken the same as in the simulations of [55] ψ(, t = ) = A I ()e i SI()/ε, R, (5.4) where A I () = e, (S I ()) = tanh(). Note that S I is such that the initial phase is compressive. This means that even the linear evolution develops caustics in finite time. We solve this problem either on the interval [ 4, 4] or on [ 8, 8] depending on the time for which the solution is calculated. We present numerical results for four different regimes of α: Case I. α =, i.e., Schrödinger-Poisson regime; Case II. α = ε, i.e., Schrödinger-Poisson equation with O(ε) cubic nonlinearity; Case III. α = ε, i.e., Schrödinger-Poisson equation with O( ε) cubic nonlinearity;

87 5.3 Numerical results of the Schrödinger-Yukawa equations 75 Case IV. α =, i.e., Schrödinger-Poisson equation with O() cubic nonlinearity. Figure 5. displays comparisons of the position density n(, t) = ψ(, t) at fied time for the above different parameter regimes with different ε. Figure 5. plots the evolution of the position density of the wave function for α = and ε =.5. Figure 5.3 shows the analogous results for the attractive Hartree integration, i.e., C =, α =.5 and ε =.5. From Figure 5., we can see that before the break (part a) and b)), the result is essentially independent of ε. After the break the behavior of the position density n(, t) changes substantially with respect to the different regimes of α. For α = the solution stays smooth. For α = ε it also stays smooth, but it concentrates at the origin. For α = ε a pronounced structure of peaks develop, they look like the soliton structure typical for the NLS [43]. The number of peaks is doubled when ε is halved. For α =, the number of peaks increases again and they occur at different locations than that for α = ε. We can see that the scaling α = O( ε) is critical in the sense that the solution has a substantially behavior than for the smaller scales of α. Beyond this scaling, the semiclassical limit can not be obtained by naive numerics. Figure 5.3 is a test to see what happens if the Hartree potential is attractive instead of repulsive, with all other parameters kept the same, i.e., Figure 5. and Figure 5.3 differ only by the sign of C. The resulting effect corresponds to the physical intuition that the pattern of caustics that is typical for focusing NLS would be enhanced and focused in physical space by an additional attractive force.

88 5.3 Numerical results of the Schrödinger-Yukawa equations Position Density at t=.5,ε=.5 α= α=sqrt(ε) α=ε α= Position Density at t=.5,ε=.5 α= α=sqrt(ε) α=ε α=.. n().8 n() a) b) Position Density at t=4.,ε=. α= α=sqrt(ε) α=ε α= Position Density at t=4.,ε=.5 α= α=sqrt(ε) α=ε α= n().5.5 n() c) d) Position Density at t=4.,ε=.375 α= α=sqrt(ε) α=ε α= 6 5 Position Density at t=4.,ε=.5 α= α=sqrt(ε) α=ε α= 4 4 n() 3 n() 3 e) f) Figure 5.: Numerical results for different scales of the Xα term in Eample 5., i.e., α =, ε, ε,. a) and b) : small time t =.5, pre-break, a) for ε =.5, b) for ε =.5. c)-f): large time, t = 4., post-break. c) for ε =., d) for ε =.5, e) for ε =.375, f) ε =.5.

89 5.3 Numerical results of the Schrödinger-Yukawa equations t Figure 5.: Time evolution of the position density for Xα term at O() in Eample 5., i.e., α =.5, with ε =.5, h = /5 and k =.5. a) surface plot; b) pseudocolor plot t a) b) Figure 5.3: Time evolution of the position density for attractive Hartree interaction in Eample 5.. C =, α =.5, ε =.5, k =.5. a) surface plot; b) pseudocolor plot.