A UNIFORMLY CONVERGENT NUMERICAL SCHEME FOR SINGULARLY PERTURBED NON-LINEAR EIGENVALUE PROBLEM UNDER CONSTRAINTS CHAI MING HUANG

Transcription

1 A UNIFORMLY CONVERGENT NUMERICAL SCHEME FOR SINGULARLY PERTURBED NON-LINEAR EIGENVALUE PROBLEM UNDER CONSTRAINTS CHAI MING HUANG (B.Sc.(Hons.), NUS) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 6

2 Acknowledgments First and foremost, I would like to thank my supervisor, Associate Professor Bao Weizhu for his patience, guidance and invaluable advice. He had been etremely patient with me throughout my studies and was always ready to guide me in my research. It is also my pleasure to epress my appreciation and thanks to my fellow postgraduates Yang Li, Lim Fong Yin and especially my seniors Dr. Wang Hanquan and Dr. Zhang Yanzhi. They provided immense help especially with the Mathematical derivation and programming. This dissertation would not be completed smoothly without their kind assistance and heartfelt encouragement. Lastly, I would like to dedicate this work with love to my wife, my parents and my family for being always there for me. Chai Ming Huang Dec 6 ii

3 Contents Acknowledgments ii Summary vi Introduction. Brief history of Bose-Einstein condensation Review of eisting numerical methods The problem The organization of the thesis The Gross-Pitaevskii equation 7. The time-dependent GPE Non-dimensionalization of GPE Reduction of the GPE to lower dimensions Stationary states of GPE Ground state Ecited states iii

4 Contents iv 3 The singularly perturbed nonlinear eigenvalue problem 4 3. The singularly perturbed nonlinear eigenvalue problem For bounded domain Ω = [, ] d For the whole space Ω = R d General formulation Approimations in D bo potential Thomas-Fermi approimation for ground state Matched asymptotic approimations for ground state Matched asymptotic approimations for ecited states Approimations for D harmonic potential Thomas-Fermi approimation for ground state Thomas-Fermi approimation for the first ecited state Matched asymptotic approimations for the first ecited state 5 4 Numerical Methods for Singularly Perturbed Eigenvalue Problems 8 4. Gradient flow with discrete normalization Discretization with uniform mesh in D Discretization with piecewise uniform mesh in D The full discretization with piecewise uniform mesh Piecewise uniform mesh for ground state with bo potential Piecewise uniform mesh for first ecited state with bo potential Piecewise uniform mesh for first ecited state with harmonic potential Choice of initial data analysis of uniform mesh analysis of piecewise uniform mesh Numerical comparisons

5 Contents v 5 Numerical Applications Numerical results in D Ground state and ecited states with bo potential Ground state and ecited states with harmonic potential in D Numerical results in D for bo potential Choice of mesh Choice of initial data Results Numerical results in D for harmonic potential Choice of mesh Choice of initial data Results Numerical results in D for harmonic plus optical lattice potential Results Conclusions 89

6 Summary The time-independent Gross-Pitaevskii equation (GPE) in the semiclassical regime is used to describe the equilibrium properties of Bose-Einstein Condensate at etremely low temperature. In this regime, the GPE is a singular perturbed nonlinear eigenvalue problem. The aim of this thesis is to present a uniformly convergent numerical scheme to solve the singularly perturbed nonlinear eigenvalue problem. The adaptive numerical scheme proposed is based on a piecewise uniform mesh. The scheme is found to be able to treat the interior layers or boundary layers inherent in solutions of singularly perturbed nonlinear eigenvalue problems. A comparison of the new proposed scheme based on piecewise uniform mesh is made against the classical numerical scheme based on uniform mesh. We found that the numerical accuracy of the new numerical scheme proposed is greatly improved over the classical numerical scheme. An etension of the new numerical scheme is made to two dimensions. The scheme is then applied to solve the singular perturbed nonlinear eigenvalue problem in two dimensions. vi

7 Chapter Introduction. Brief history of Bose-Einstein condensation In 95, Indian physicist Satyendra Nath Bose published a paper devoted to the statistical description of the quanta of light. Based on Bose s results, Albert Einstein [3] predicted that a phase transition in a gas of noninteracting atoms could occur due to quantum statistical effects. During this phase of transition period, a Bose- Einstein Condensate (BEC) will be formed when a macroscopic number of noninteracting bosons simultaneously occupy the single quantum state of the lowest energy [3]. For many years, there was no practical application of BEC. In 938, after superfluditiy was discovered in liquid helium, F. London theorized that the superfluidity could be a manifestation of BEC. However in 955, eperiments on superfluid helium showed that only a small fraction of condensate is found. In the 97s, eperimental studies on dilute atomic gases were developed. The first of these studies focused on spin-polarized hydrogen. This gas was chosen as it has a very light mass and is thus likely to achieve BEC. After numerous attempts, BEC was almost achieved but it was not pure [44].

8 . Brief history of Bose-Einstein condensation In the 98s, there was remarkable progress made in the application of laserbased cooling techniques and magneto-optical trapping. In 995, a historical milestone was achieved when the eperimental teams of Cornell and Wieman at Boulder of JILA and of Ketterle at MIT succeeded in reaching the ultra low temperature and densities required to observe BEC in vapors of 87 Rb [] and 3 Na []. Later in the same year, occurrence of BEC in vapors of 7 Li was also reported [5]. For their achievement, the Nobel Prize of Physics was awarded to the first three researchers who created this fifth state of matter in the laboratory. After realizing BEC in dilute bosonic atomic gases, BEC was also reached in other atomic matter, including the spin-polarized hydrogen, metastable 4 He and 4 K [8]. Since all the particles occupy the same state in the BEC at ultra low temperature, the condensate is characterized by a comple-valued wave function ψ(, t), whose time evolution is governed by the time-dependent Gross-Pitaevskii equation (GPE) [, 37]. It is impossible to solve the GPE analytically ecept for the simplest cases of GPE. Various numerical methods are used to solve the GPE instead. When the problems involve the static properties of the condensate, the numerical solutions of the time-independent GPE are of interest. Over the last several years, there were etensive progress made towards developing innovative approaches and algorithms in solving both time-dependent and time-independent GPE. We will survey some of the more important recent research papers written in the field, with more emphasis of the numerical methodology in solving the time-independent GPE, which is the main subject of interest in this dissertation.

9 . Review of eisting numerical methods 3. Review of eisting numerical methods The earliest attempts to solve the GPE might be started by Edwards and Burnetts [7]. They developed a Runge-Kutta method based on finite-difference to solve the time-independent GPE for spherical condensates. Edwards [6] also designed a basis set approach to solve GPE. For the solving of time-independent GPE in ground state and the vorte states in anisotropic traps, a finite-difference based imaginary time method was developed by Dalfovo and Stringari []. Adhikari [] used a finitedifference based approach to solve the two-dimensional time-independent GPE. Cerimele, together with his coworkers [7], developed a finite-difference and imaginarytime approach for solving the time-independent GPE. Schneider and Feder [48] used a discrete variable representation that is coupled with a Gaussian quadrature integration scheme, to attain the ground and the ecited states of GPE in three dimensions. Recently, Bao and Tang [] used a different approach for obtaining the ground state of GPE. They did this by directly minimizing the corresponding energy functional with a finite element discretization. Utilizing the harmonic oscillator as the basis set, Dion and Cancés [3] proposed a Gauss-Hermite quadrature integration scheme to solve both the time-dependent and time-independent GPE. More recently, Bao and Du [4] developed a novel method called the gradient flow with discrete normalization to find the ground state of the GPE. This numerical method is perhaps one of the most efficient ways to solve the time-independent GPE [4, 5, 9, 7, 9, ]..3 The problem However, there are numerical difficulties when the time-independent GPE is in a semiclassical regime, i.e. BEC is a strong repulsively interacting condensate. In such a regime, the GPE is reduced into a singularly perturbed non-linear eigenvalue

10 .3 The problem 4 problem under a constraint as shown µφ( ) = ε φ( ) + V ( )φ( ) + φ( ) φ( ), Ω, (.) φ( ) Ω =, (.) under the normalization condition φ = φ( ) d =, (.3) Ω where φ( ) is a real function, Ω R d, V ( ) is an eternal potential, µ > and < ε. When ε goes to zero, the solutions of the problem have boundary layers or interior layers [8]. The classical numerical scheme based on uniform mesh to discretize the gradient flow would be difficult to track these layers [4]. In order to obtain a reliable numerical solution for (.) when ε, it is desirable to use an adaptive mesh that concentrates nodes in the boundary layers or interior layers. Ideally, the mesh should be generated by adapting it to the features of the computed solution. There has been a great deal of research done on the use of adaptive methods for steady and unsteady partial differential equations recently [6, 8, 9, 4, 4, 34, 33, 35, 45, 46]. Among which, Shishkin [49] in 99 proposed an upwind scheme based on a piecewise uniform mesh to solve the two-point boundary layer problems fine in the boundary and coarse in the rest of the domain. This scheme is useful and has been demonstrated to be ε-uniform convergenct by Miller et al. [4, 4]. It has also been shown that the scheme is uniformly convergent near the boundary layer and it has been pointed out that uniform convergence cannot be obtained at all interior mesh points unless the mesh is specially tailored to the solution of the problem. In this thesis, we aim to design a uniformly convergent numerical scheme based on piecewise uniform mesh for discretizing the gradient flow so that we can treat problems with complicated boundary layer or interior layers effectively.

11 .4 The organization of the thesis 5.4 The organization of the thesis This thesis is organized as follows. In Chapter, starting from the time-dependent GPE, we first rescale it to a dimensionless form and then reduce the time-dependent GPE from three dimensions into lower dimensions. We net describe how to obtain the stationary states of BEC and the time-independent GPE in a semiclassical regime, i.e., the ground state and ecited states. In Chapter 3, we arrive at the singularly perturbed nonlinear eigenvalue problem under a constraint to be solved. For the sake of comparison with numerical approimation later, we present some analytical approimations for the ground and ecited states in BEC with bo potential in one dimension (D). We also present some analytical approimations for the first ecited states in BEC with harmonic potential in D. We demonstrate that there are boundary layers or interior layers in these solutions. In Chapter 4, we describe the numerical methods for solving such singularly perturbed nonlinear eigenvalue problem under a constraint. We apply one of the most efficient numerical technique the gradient flow with discrete normalization to solve the singularly perturbed and constrained nonlinear eigenvalue problem. We first show a classical numerical scheme based on uniform mesh to discretize the gradient flow. We then analyze the shortcomings of the scheme and introduce the detailed algorithm of our newly proposed numerical scheme based on piecewise uniform mesh to discretize the gradient flow to treat boundary layers or interior layers. Finally we provide numerical error analysis for both uniform mesh and piecewise uniform mesh. The limitations of uniform mesh are shown and the advantages from using piecewise uniform mesh are presented. Comparisons between solutions obtained by our proposed piecewise uniform mesh and solutions generated with the classical uniform

12 .4 The organization of the thesis 6 mesh are shown in more details. In Chapter 5, we apply our new proposed scheme based on piecewise uniform mesh to calculate the ground state, first, third, and ninth ecited states of BEC with bo potential in D and the first ecited state of BEC with harmonic potential in D. We compare the numerical results with those asymptotic approimation shown in Chapter 3. We then etend our numerical scheme based on piecewise uniform mesh to find numerical solutions of the singularly perturbed and constrained nonlinear eigenvalue problem in two dimensions (D), for eample, ground state and ecited states of BEC in three different potentials, bo potential, harmonic potential and harmonic plus optical potential. This is to illustrate the capability of the proposed piecewise uniform scheme in solving the time-independent GPE under different potentials and conditions, more specifically, to treat the boundary layers or interior layers in two dimensions. Finally in Chapter 6, some conclusions on our results are drawn and possible future works are highlighted.

13 Chapter The Gross-Pitaevskii equation In this chapter, we derive the time-independent GPE from the well-known timedependent GPE. As preparatory steps, we introduce the time-dependent GPE with two kinds of eternal potentials, i.e., the harmonic oscillator potential and the bo potential. The GPE is then non-dimensionalized, rescaled and reduced into lower-dimensional formulations. Finally the solutions of the time-independent GPE, ground state and ecited states are summarized.. The time-dependent GPE At temperatures T much lower than the critical temperature T c, the BEC is well described by the macroscopic wave function ψ = ψ(, t). The evolution of this wave function is governed by a self-consistent nonlinear Schrödinger equation known as the Gross-Pitaevskii equation [3, 43] ψ(, t) i t = m ψ(, t) + V ( )ψ(, t) + NU ψ(, t) ψ(, t), (.) where = (, y, z) T is the spatial coordinate vector, is the Planck constant, m is the atomic mass, N is number of atoms in the condensate, U = 4π a s /m describes the interactions between atoms in the condensate with a s the atomic scattering 7

14 . Non-dimensionalization of GPE 8 length (positive for repulsive interaction and negative for attractive interaction), V ( ) is an eternal trapping potential. Two important invariants of (.) are the normalization of the wave function ψ(, t) = ψ(, t) d =, (.) R 3 and the energy E(ψ) = R 3 [ m ψ(, t) + V ( ) ψ(, t) + NU ] ψ(, t) 4 d. (.3) There are two typical eternal potentials V ( ) considered in this dissertation:. The bo potential:, <, y, z < L, V bo ( ) =, otherwise. (.4). The harmonic oscillator potential: V ho ( ) = V ho () + V ho (y) + V ho (z), R 3, (.5) V ho (τ) = m ω ττ, τ =, y, z, (.6) where ω τ is the trap frequency in τ direction.. Non-dimensionalization of GPE We introduce the following parameters in order to scale (.) under the normalization (.) [8] t = t t s, = s, ψ(, t) = 3/ s ψ(, t), (.7) where t s and s are the dimensionless time and length units. Substituting (.7) into (.), multiplying throughout by t s m s, then removing all, we obtain a dimensionless GPE under the normalization (.) in three dimensions (3D): ψ(, t) i t = ψ(, t) + V ( )ψ(, t) + β ψ(, t) ψ(, t), (.8)

15 .3 Reduction of the GPE to lower dimensions 9 and the dimensionless energy functional E(ψ) is defined as follows: [ E(ψ) = ψ(, t) + V ( ) ψ(, t) + β ] ψ(, t) 4 d, (.9) R 3 where the interaction parameter β = 4πasN s. The choices used for the scaling parameters, t s and s for the two different dimensionless potential V ( ) are:. The bo potential: t s = ml, s = L, (.), <, y, z <, V ( ) = (.), otherwise.. The harmonic oscillator potential: t s =, ω s =, mω (.) where γ y = ωy ω and γ z = ωz ω. V ( ) = ( + γ yy + γ zz ), (.3).3 Reduction of the GPE to lower dimensions In order to illustrate dimension reduction of the GPE in 3D to two dimensions (D) or one dimension (D), we first consider the dimensionless GPE with the harmonic potential. The dimensionless GPE with its normalization is given by: ψ(, t) i t where β = NU 3 s ω = ψ(, t) + ( + γ yy + γ zz )ψ(, t) + β ψ(, t) ψ(, t),(.4) = 4πasN s. In a disk-shaped condensation with parameters ω ω y and ω z ω ( γ y and γ z ), the three-dimensional GPE (.4) can be reduced to a twodimensional GPE by assuming that the time evolution does not cause ecitations

16 .3 Reduction of the GPE to lower dimensions along the z-ais, since the ecitations along the z-ais have large energy (of order ω z ) compared to that along the - and y-ais with energies of order ω. Thus we may assume that the condensation wave function along the z-ais is always well described by the ground state wave function and set ψ(, y, z, t) = ψ (, y, t)φ 3 (z), (.5) where φ 3 (z) φ ho (z) = (γ z /π) /4 e γzz /. Plugging (.5) into (.4), then multiplying by φ 3 (z) (the conjugate of φ 3 (z)), integrating with respect to z over (, ), we get the two-dimensional GPE with = (, y) T i ψ (, t) t = ψ + ( + γ yy + C ) ψ + β ψ ψ, (.6) where C = γ z β = β z φ 3 (z) dz + φ 3 (z) 4 dz β dφ 3 (z) dz dz, φ ho (z) 4 dz = β γ z /π. (.7) Since this GPE is time-transverse invariant, we can replace ψ ψe ict/ which drops the constant C in the trapping potential and obtain: i ψ(, t) t = ψ + V ( )ψ + β ψ ψ, (.8) ( where V ( ) = + γyy ). In a cigar-shaped condensation where the energies along -ais is much smaller than energies along y- and z-ais, i.e. ω y ω and ω z ω, and there is almost no ecitation along the y- and z-ais as time evolves, we can obtain a one-dimensional GPE. In fact, for any fied β and when γ y and γ z, we set ψ(, t) = ψ (, t)φ 3 (y, z), (.9) φ 3 (y, z) φ ho 3(y, z) = (γ yγ z ) 4 e (γyy +γ zz )/. (.) π

17 .3 Reduction of the GPE to lower dimensions Substituting (.9) into (.8), multiplying both sides by φ ho φ ho 3(y, z) (the conjugate of 3(y, z) ), and integrating both sides in the yz-plane over R, we get: i ψ = ψ t + ( ) ( + C)ψ + β φ 3 (y, z) 4 dydz ψ ψ, (.) R where C = φ 3 (y, z) dydz + R (γyy + γzz ) φ 3 (y, z) dydz. R (.) Ct i Since (.) is time-transverse invariant, we let ψ ψe. This will remove the term containing the constant C and we obtain the GPE in D as: i t ψ(, t) = ψ(, t) + V ()ψ(, t) + β ψ(, t) ψ(, t), (.3) where V () = and β = β R φ 3 (y, z) 4 dydz β R φ ho 3(y, z) 4 dydz = β γ y γ z /π. Thus here we consider the dimensionless GPE with the harmonic potential in d-dimensions (d =,, 3): i ψ(, t) t = ψ + V d ( )ψ + β d ψ ψ, R d, t, (.4) where β d = β γy γ z /π, γz /π, V d ( ) = /, d =, ( + γ yy )/, d =,, ( + γ yy + γ zz )/, d = 3. Similarly, we can obtain the GPE with the bo potential in d-dimensions: i t ψ(, t) = ψ(, t) + V d ( )ψ(, t) + β d ψ(, t) ψ(, t), (.5) where the bo potential, [, ] d, d =,, 3, V d ( ) =, otherwise.

18 .4 Stationary states of GPE Hence, a general d-dimensional (d=,,3) GPE will be as follows: i t ψ(, t) = ψ(, t) + V d ( )ψ(, t) + β d ψ(, t) ψ(, t), Ω, (.6) ψ(, t) =, Ω, where Ω is a bounded domain in R d. Two important invariants of (.6) are the normalization of the wave function N(ψ) = ψ(, t) d Ω Ω ψ(, ) d =, t, (.7) and the energy E β (ψ) = Ω [ ψ(, t) + V d ( ) ψ(, t) + β d ] ψ(, t) 4 d = E β (ψ(, )), t. (.8).4 Stationary states of GPE In order to find stationary state of (.6), we let ψ(, t) = e iµt φ( ), (.9) where φ( ) is a function independent of time t and µ is the chemical potential of the condensate. Substitute (.9) into (.6), we get µφ( ) = φ( ) + V d ( )φ( ) + β d φ( ) φ( ), Ω, (.3) φ( ) =, Ω, (.3) under the normalization condition φ( ) = φ( ) d =. (.3) Ω

19 .4 Stationary states of GPE 3 This is a nonlinear eigenvalue problem with a constraint and the eigenvalue µ can be calculated from the corresponding eignfunction φ( ) by µ = µ β (φ) = = E β (φ) + Ω Ω [ ] φ( ) + V d ( ) φ( ) + β d φ( ) 4 d β d φ( ) 4 d. (.33).4. Ground state The ground state wave function φ g := φ g ( ) of a BEC is found by minimizing the energy functional E β (φ) over the unit sphere S = {φ( ) φ( ) =, E(φ) < }, i.e., find (µ g, φ g S) such that E g := E β (φ g ) = min φ S E β(φ), µ g := µ β (φ g ). (.34) We can easily show that the ground state φ g is an eigenfunction of the nonlinear eigenvalue problem (.3) under the constraint (.3)..4. Ecited states Any eigenfunction φ( ) of (.3) under the constraint (.3) whose energy E β (φ) > E β (φ g ) is usually called as an ecited state in the physics literature. Suppose the eigenfunctions of the eigenvalue problem (.3) under the constraint (.3) are ±φ g ( ), ±φ ( ), ±φ ( ),, (.35) whose energies satisfy E β (φ g ) < E β (φ ) < E β (φ ) <. (.36) Then φ j ( ), j =,, 3,, is called as the j-th ecited state solution.

20 Chapter 3 The singularly perturbed nonlinear eigenvalue problem In this chapter, we derive the singularly perturbed nonlinear eigenvalue problem from the time-independent GPE (.3). When β d, the time-independent GPE, in the bounded domain or whole space, is then rescaled and reduced into semiclassical formulations. We finally obtain the singularly perturbed nonlinear eigenvalue problem under a constraint in a general form. 3. The singularly perturbed nonlinear eigenvalue problem When β d, i.e. the time-independent GPE (.3) is in a strongly repulsive interacting condensation or in the semiclassical regime, we need another scaling for the GPE. 4

21 3. The singularly perturbed nonlinear eigenvalue problem For bounded domain Ω = [, ] d When Ω = [, ] d, the GPE (.3) with bo potential is µφ( ) = φ( ) + β d φ( ) φ( ), Ω = [, ] d, (3.) φ( ) =, Ω. (3.) We let ε = βd and µ represent µ/ε. Divided by β d at both sides, the equation (.3) with the bo potential reduces to the singularly perturbed nonlinear eigenvalue problem and the normalization as µφ( ) = ε φ( ) + φ( ) φ( ), [, ] d, (3.3) φ = φ( ) d =. [,] d (3.4) The chemical potential µ in (3.3) can be computed from its corresponding eigenfunction φ by µ = µ ε (φ) = = E ε (φ) + [,] d [,] d [ ] ε φ( ) + φ( ) 4 d [ ] φ( ) 4 d, and the energy functional reduces to [ ε E ε (φ) = φ( ) + ] φ( ) 4 d. [,] d 3.. For the whole space Ω = R d When Ω = R d is the whole space, the time-independent GPE (.3) with the harmonic potential is as follows, µφ( ) = φ( ) + V d ( )φ( ) + β d φ( ) φ( ), Ω = R d, (3.5) φ( ),. (3.6)

22 3. The singularly perturbed nonlinear eigenvalue problem 6 In order to rescale the GPE, We let = ε /, φ = ε d/4 φ, µ = ε µ, ε = β d/d+ d. (3.7) Substituting the above scaling parameters into (.3), and rearranging the variables, we have the singularly perturbed nonlinear eigenvalue problem µφ( ) = ε φ( ) + V d ( )φ( ) + φ( ) φ( ), (3.8) with the constraint R d φ( ) d =. Again, the chemical potential µ in (3.8) can be computed from its corresponding eigenfunction φ by [ ] ε µ = µ ε (φ) = R φ( ) + V d φ( ) + φ( ) 4 d. d [ ] = E ε (φ) + φ( ) 4 d, and the energy functional becomes [ ε E ε (φ) = φ( ) + V d φ( ) + ] φ( ) 4 d. R d R d 3..3 General formulation In conclusion, we have the following singularly perturbed nonlinear eigenvalue problem whatever the potentials V d ( ) as µφ( ) = ε φ( ) + V d ( )φ( ) + φ( ) φ( ), Ω, (3.9) φ( ) =, Ω, (3.) with the normalization as φ = φ( ) d =. (3.) Ω

23 3. Approimations in D bo potential 7 The chemical potential µ in (3.9) can be computed from its corresponding eigenfunction by [ ] ε µ = µ ε (φ) = Ω φ( ) + V d φ( ) + φ( ) 4 d, = E ε + φ( ) 4 d, (3.) Ω and the energy functional is [ ε E ε (φ) = φ( ) + V d φ( ) + ] φ( ) 4 d, Ω = E kin (φ) + E pot (φ) + E int (φ), (3.3) where E kin, E pot and E int are the kinetic energy, potential energy and interaction energy respectively. They are defined as E kin (φ) = ε φ( ) d, Ω (3.4) E pot (φ) = V d φ( ) d, (3.5) E int (φ) = Ω φ( ) 4 d. (3.6) In addition, the chemical potential µ can also be given by Ω µ ε (φ) = E kin (φ) + E pot (φ) + E int (φ). (3.7) The equation (3.9) with the constraint (3.) is a singularly perturbed nonlinear eigenvalue problem and its solutions are of main interest in this thesis. In the net section, some approimated solutions for the problem in D, which have boundary layer or interior layer for small ε, are summarized. 3. Approimations in D bo potential In this section, we present the matched asymptotic approimations for the ground state and ecited states of BEC confined in a D bo potential, i.e., V () =, for

24 3. Approimations in D bo potential 8 ; V () =, otherwise. We truncate the eigenvalue problem into [, ] with homogeneous Dirichlet boundary condition in this case. 3.. Thomas-Fermi approimation for ground state We first consider (3.9) with bo potential in D. Since < ε, we can drop the first term on the right side and obtain the ground state approimation as: µ TF g φ TF g () = φ TF g φ TF g (), [, ], which implies φ TF g = µ TF g, < <. (3.8) Substituting (3.8) into the normalization condition (3.), we get: φ TF () d = µ TF g d = µ TF g =. (3.9) Hence, the Thomas-Fermi approimation for ground state is given by φ g () φ TF g () =, < <. (3.) However, the approimation for the ground state does not satisfy the zero boundary condition (3.). This suggests the eistence of two boundary layers in the region near = and near = in the ground state of BEC with bo potential when we remove the diffusion term in (3.3). 3.. Matched asymptotic approimations for ground state Since the layers eist at the two boundaries = and = when < ε, we solve (3.8) near = and =, respectively. Let us suppose the boundary layer is of width δ ( < δ < ). We do a rescaling in the region of [, δ] and let = δx, φ() = φ s Φ(X). (3.)

25 3. Approimations in D bo potential 9 We substitute (3.) into (3.8) and obtain µφ(x) = ε δ Φ XX(X) + φ sφ 3 (X), X (, ), (3.) Φ() =, Φ() =. (3.3) In order to solve the above equation, we need to rescale all the terms to O(). We choose δ = ε/ µ and φ s = µ in (3.), the above equation reduces to Φ(X) = Φ XX(X) + Φ 3 (X), X (, ), (3.4) Φ() =, Φ() =. (3.5) All the terms in the equation are now O(). Solving the above equation, we obtain Φ(X) = tanh(x), X (, ). (3.6) Since µ µ TF = for the ground state, we can conclude that the width of boundary layer near = is O(ε). Thus finally we have φ g () ( ) µg µ g tanh ε, (, δ). (3.7) Repeating the similar procedure, we can obtain the approimation near = φ g () ( ) µg µ g tanh ( ), ( δ, ). (3.8) ε Finally, using the matched asymptotic technique, an approimation for the ground state with the bo potential in D can be given by µ MA φ g φ MA g = µ MA g g tanh + tanh ε ε tanh µ MA g ε µ MA g ( ),. (3.9)

26 3. Approimations in D bo potential Using the normalization condition (3.), we find = = µ MA g = µ MA g φ MA g [ tanh () d ) tanh ( µ MA g /ε ( µ MA g /ε ) d + [ ( tanh µ MA g ) tanh ( µ MA g ( )/ε ) /ε ( + tanh µ MA g ( ) ( ) + tanh µ MA g /ε tanh µ MA g ( )/ε d + [ ( ( ) ) ε tanh µ MA g /ε / µ MA g ( ) ( ) 4ε tanh µ MA g /ε ln cosh( µ MA g /ε) / µ MA g ( ( ( )) ) + + εcoth( µ MA g /ε)ln cosh µ MA g /ε / µ MA g µ MA g ε µ MA g εln µ MA g µ MA g = µ MA g d )] ( )/ε d ) tanh ( µ MA g /ε + tanh ( µ MA g ] µ MA g /ε) εln + µ MA g ] d ε µ MA g. (3.3) Solving it, we obtain the chemical potential Moreover, we can obtain µ MA g + ε + ε + ε, < ε. (3.3) Ekin,g MA = E kin (φ MA g ) = ε = ( ) ( µ MA g [sech 3 µma g µ MA [ φ MA () ] d µ MA g g ) ( /ε sech µ MA g ( )/ε)] d g ε. (3.3) Substitute (3.3) into (3.3), we can get the kinetic potential as ( Note that Eint,g MA = µ MA g Ekin,g) MA we can obtain E MA kin,g = 3 ε + ε + ε. (3.33) E MA int,g = + 3 ε + ε, (3.34)

27 3. Approimations in D bo potential and E MA g = E MA kin,g + E MA int,g = ε + ε + ε. (3.35) 3..3 Matched asymptotic approimations for ecited states For the BEC in D bo potential, when < ε, the kth (k N) ecited state not only has boundary layers near = and =, but also has k interior layers at = j, j =,,..., k. (3.36) k + Using the matched asymptotic method described in the previous subsection, we can obtain an approimation for φ MA k, i.e., the kth (k N) ecited states as { [(k+)/] φ k φ MA k = µ MA k + j= tanh [k/] ( µ MA k tanh ( j + ε k + ) j= ( µ MA k ε ) ) ( j k + ) C k tanh ( µ MA k ε ) }, (3.37) where [τ] takes the integer part of the real number τ and the constant C k = when k is odd and C k = when k is even. Plugging equation (3.37) into the normalization condition (3.), we have = Solving it, we obtain φ MA k () d µ MA k [ ] (k + )ε. µ MA k µ k µ MA k = + (k + )ε + (k + ) ε + (k + ) ε, k N, (3.38) where < ε. Similarly, we can obtain E MA kin,k = 3 (k + )ε + (k + ) ε + (k + ) ε, (3.39)

28 3.3 Approimations for D harmonic potential Eint,k MA = ( ) µ MA k Ekin,k MA + 3 (k + )ε + (k + ) ε, (3.4) and E MA k = E MA kin,k + E MA int,k (k + )ε + (k + ) ε + (k + ) ε. (3.4) Based on the above analytical results, we make the following observations for the ground state and ecited states of BEC with bo potential:. Boundary layers are observed at = and = for all ground state and ecited states when < ε. The width of these layers are of O(ε).. For k-th ecited states, interior layers are also observed at = j k+, (j =,..., k) when < ε. The widths of these interior layers are twice the size of widths at the boundary layers. Similarly, we can etend the above asymptotic approimations to ground state and ecited states of BEC with bo potential in higher dimensions. These approimate results will be useful since they tell us the locations and width of the boundary and interior layers of the solutions. These results also help us in choosing the piecewise uniform mesh more effectively, which we will discuss in net chapter. 3.3 Approimations for D harmonic potential In this section, we present some approimations for both ground state and the first ecited states of BEC with D harmonic potential, i.e., V () =.

29 3.3 Approimations for D harmonic potential Thomas-Fermi approimation for ground state We first consider the Thomas-Fermi (TF) approimation for D harmonic oscillator potential. From (3.9), we drop the first term on the right side because < ε and obtain µ TF g φ TF g () = φtf g () + φ TF g () φ TF g (), R, (3.4) which results in the TF approimation for ground state as µ TF φ g () φ TF g g =, µtf g >,, otherwise. (3.43) Plugging (3.43) into (3.), we get = <µ TF g µ ( ) TF φ TF g () g d = µ TF g d. (3.44) µ TF g Solving it, we obtain the TF approimation of the chemical potential in ground state as We also obtain the potential energy Epot,g TF = the interaction energy = µ TF g = < µ TF g () µ TF g () = 4 5 = E TF int,g = µ TF g () ( ) /3 3. (3.45) ( φtf g µ TF g () d ) 4 4 d ( ) µ TF 5/ g ) /3, (3.46) ( 3 < µ TF g φ TF g 4 d

30 3.3 Approimations for D harmonic potential 4 and the ground state energy E TF g = µ TF g () µ TF g () ( ) µ TF g d = 8 ( ) µ TF 5/ g 5 = ( ) /3 3, (3.47) 5 = µ TF g E TF int,g = 3 ( ) /3 3. (3.48) 3.3. Thomas-Fermi approimation for the first ecited state Similarly, using the same approach in the derivation of the TF approimation for the ground state, we can obtain the TF approimation for the st ecited state, µ TF, < < µ TF, φ () φ TF () = µ TF, µ TF < <, (3.49), otherwise. Plugging (3.49) into (3.), we get = <µ TF φ TF () d = µ TF µ TF ( ) µ TF d. (3.5) Solving it, we obtain the approimation of the chemical potential in the first ecited state as µ TF = ( ) /3 3. (3.5) We will then obtain the same potential energy as the ground state Epot, TF = () d = < µ TF µ TF () = 4 5 = µ TF () () φtf ( µ TF ) 4 4 d ( ) µ TF 5/ ) /3, (3.5) ( 3

31 3.3 Approimations for D harmonic potential 5 the interaction energy Eint, TF = φ TF 4 d < µ TF = and the first ecited state energy µ TF () µ TF () ( ) µ TF d = 8 ( ) µ TF 5/ 5 = ( ) /3 3, (3.53) 5 E TF = µ TF E TF int, = 3 ( ) /3 3. (3.54) Matched asymptotic approimations for the first ecited state Since µ TF >, we can deduce that an interior layer eists at =. In order to find the width of this interior layer, we suppose the width of the layer is δ and rescale the equation in the region of ( δ, δ) by setting = δx, φ() = φ s Φ(X). (3.55) Substituting (3.55) into (3.9), we obtain µφ(x) = ε δ Φ XX(X) + δ X Φ() + φ sφ 3 (X), X (, ), (3.56) Since δ is small, we can drop the second term on the right hand side of the equation (3.56) and get µφ(x) = ε δ Φ XX(X) + φ sφ 3 (X), X (, ). (3.57) The equation above is similar to the equation (3.) obtained for the bo potential and the first ecited state is an odd function. In order to solve the above equation

32 3.3 Approimations for D harmonic potential 6 for X <, we need to rescale all the terms to O(). By choosing δ = ε/ µ and φ s = µ, the equation becomes Φ(X) = Φ XX(X) + Φ 3 (X), < X <, (3.58) Φ() =, Φ() =. Solving the above equation, we obtain Φ(X) = tanh(x), X (, ). (3.59) Thus we have φ () = µ tanh( µ ε ), ( δ, δ). (3.6) From (3.6), we can conclude that the width of the interior layer near = is O(ε). In fact the first ecited solution of the equation from = can be approimated by (3.49). Similarly, using the matched asymptotic method, we can get an approimate solution for the first ecited state in BEC with D harmonic potential ( ) µ MA µ tanh MA + ε φ φ MA ( ) = µ MA tanh µ MA ε µ MA µ MA, < < µ MA, µ MA + µ MA, µ MA < <,, otherwise, where µ MA can be determined by the normalization condition (3.). Based on the analytical results obtained, we make the following observations for the ground state and first ecited state of BEC with harmonic potential:. No boundary layer or interior layer is observed for ground state solutions. (3.6). For the st ecited states, an interior layer is observed at =. The width of the interior layer is O(ε).

33 3.3 Approimations for D harmonic potential 7 Similar to the bo potential, we can etend these observations accordingly to higher dimensions. These observations will be useful on how we choose the piecewise uniform mesh when we numerically solve the D eigenvalue problem (3.8) in the subsequent chapters.

34 Chapter 4 Numerical Methods for Singularly Perturbed Eigenvalue Problems In this chapter, we apply the gradient flow with discrete normalization to solve the singularly perturbed nonlinear eigenvalue problem (3.8) under the constraint (3.9). The efficiency and mathematical justification of this numerical method to solve the problem can be found in [4]. The ground state and ecited states of BEC under a bo or harmonic potential are difficult to solve due to the presence of boundary and interior layers. In order to overcome this difficulty, we discretize the gradient flow with a new numerical scheme based on a piecewise uniform mesh also known as Shishkin mesh [49]. 4. Gradient flow with discrete normalization The gradient flow with discrete normalization (GFDN) is one of the most popular techniques for dealing with the normalization constraint (3.9). The key idea of the method is as follows: (i) apply the steepest decent method to an unconstrained minimization problem; (ii) project the solution back to the unit sphere S. For 8

35 4. Discretization with uniform mesh in D 9 simplification of notation, we only consider the following GFDN in D as etension of the method to higher dimension is straightforward: ε φ(, t) = t φ(, t) V ()φ(, t) φ(, t) φ(, t), (4.) Ω = (a, b), t n t t n+, φ(, t n+ ) = φ(, t n+) φ(, t, n, (4.) n+) φ(, ) = φ (), with φ = b a φ ()d =, (4.3) φ(a, t) = φ(b, t) =, (4.4) where V () is the eternal potential given as. Bo potential in D:, < <, V bo () =, otherwise, (4.5). or harmonic oscillator potential in D: V ho () =. (4.6) 4. Discretization with uniform mesh in D In order to discretize the gradient flow equation (4.), we divide the spatial interval Ω = [a, b] into N sub-intervals. Then, the mesh size h, time step k, spatial grid points j and time grid points t n are given by h = = b a, k = t >, (4.7) N j = a + jh, j =,,,..., N, (4.8) t n = nk, n =,,,... (4.9)

36 4. Discretization with uniform mesh in D 3 Let φ n j φ( j, t n ), φ j φ( j, t = t n+) and V j = V ( j ). In order to discretize the time derivative, we use the backward Euler scheme. For the spatial derivative, the second order central finite difference scheme is used. From time t = t n to t = t, the equation (4.) is discretized as φ j φ n j k = ε φ j φ j + φ j+ h V j φ j φ n j φ j, (4.) j =,,..., N, with boundary conditions φ = φ N =. At every time step, normalization step (4.) is discretized as φ n+ j = φ j, j =,,..., N, Φ N Φ = h (φ j ), j= with the initial condition (4.3) discretized as φ j = φ ( j ), j =,,,..., N. The above is known as the backward Euler finite difference scheme (BEFD) and it preserves the energy diminishing property of the normalized gradient flow [4]. The method is implicit and the solution can be obtained by solving the following linear system using Thomas algorithm at every time step, AΦ = Φ n, (4.)

37 4. Discretization with uniform mesh in D 3 where Φ n R N, Φ R N and A is a (N ) (N ) symmetric tridiagonal matri, i.e. Φ = φ φ. φ N, Φ n = φ n φ n. φ n N, (4.) A = d φ N with the diagonal entries of A as φ n N ε k h ε k h d ε k h ε k h d d N 3 ε k h ε k h d N ε k h ε k h d N, ( ) ε d j = + k h + V j + φ n j, j =,,, N. (4.3) After solving the linear system for Φ followed by normalization to obtain Φ n+ for time t n+, we can repeat the same procedure to calculate the solution for the net time step. The energy functional is discretized as E ε (φ) = = b a N j= h ( ε φ () + V () φ() + ) φ() 4 d j+ ε N j φ j+ () d + [ N j= ε ( φj+ φ j h j= ) + N j= j [V () φ() + φ() 4 ] d ( V j φ j + φ j 4 ) ]. (4.4)

38 4.3 Discretization with piecewise uniform mesh in D 3 Here, we used the composite midpoint rule for the first term and the composite trapezoidal quadrature rule for the second term. Both quadrature rules are second order accuracy. Similarly to energy, the chemical potential is discretized as µ ε (φ) = = b a N j= h ( ) ε φ () + V () φ() + φ() 4 d j+ ε N j φ j+ () d + [ N j= ε ( φj+ φ j h j= ) + N j= j [ V () φ() + φ() 4] d ( Vj φ j + φ j 4)]. (4.5) 4.3 Discretization with piecewise uniform mesh in D In this section, we discretize the gradient flow equation (4.) by adapting the piecewise uniform mesh proposed by Shishkin [4] for the singular perturbed two-point boundary value problem. We divide the spatial interval Ω = [a, b] into N subintervals, i.e., a = < <... < N = b is a partition of the interval [a, b]. Let j = j+ j for j =,,..., N. The piecewise uniform mesh { j} N j= for the interval [a,b] will be adapted to the features of the solution, i.e., there will be more mesh points in regions where there are boundary or interior layers The full discretization with piecewise uniform mesh Based on the new mesh, assuming that φ j φ( j, t = t n+) and φ n j φ( j, t n ), the numerical scheme to discretize the gradient flow equation (4.) at time t = t n+ is given by φ j φ n j k = ε ( φ j j ( j + j ) φ j j j + φ ) j+ j ( j + j ) V j φ j φ n j φ j, j =,,..., N, (4.6)

39 4.3 Discretization with piecewise uniform mesh in D 33 with boundary conditions φ = φ N =. (4.7) V j and j are defined as V j = V ( j), j = j+ j, j =,,,..., N. (4.8) At every time step, normalization of the solution is done by letting φ n+ j = Φ = = φ j, j =,,,..., N. (4.9) Φ b a N j= (φ ()) d = [ j + j N j= j+ j (φ ()) d N j= [ (φ j ) + (φ j+) ] j (φ j) ]. (4.) Similarly, here we can solve the linear system by using the Thomas algorithm, AΦ = Φ n, (4.) where Φ n R N and Φ R N and A is a (N ) (N ) symmetric tridiagonal matri Φ = φ φ. φ N, Φ n = φ n φ n. φ n N, φ N φ n N

40 4.3 Discretization with piecewise uniform mesh in D 34 A = d e c d e c 3 d d N 3 e N 3 c N d N e N c N d N, with the matri entries of A as ε k c j =, j =, 3,, N, j ( j + j ) ( ) ε d j = + k + V j + φ n j, j =,,, N, j j e j = ε k, j =,,, N. j ( j + j ) Similarly, we can calculate the discretized energy as follows E ε (φ) = b ε φ () d + a ( ε j φj+ φ j j a N j= b [V () φ() + φ() 4 ] d ) + N j= j + j ( V j φ j + φ j 4 ). (4.) Again, here we used the composite midpoint rule for the first term and the composite trapezoidal quadrature rule for the second term. Both quadrature rules are second order accuracy. Similarly, the chemical potential is discretized as µ ε (φ) = b ε j= b a φ d + V φ + φ 4 d a N ( ) ε N j φj+ φ j + j j= j + j ( V j φ j + φ j 4). (4.3)

41 4.3 Discretization with piecewise uniform mesh in D Piecewise uniform mesh for ground state with bo potential Recall from the subsection 3.., we know that for BEC in ground state of bo potential, there are boundary layers in the region near = and = and that the width of these boundary layers are of O(ε). Taking [a, b] = [, ] for computation, we choose the mesh as j = j + h, j = j + h, < j N/4, N/4 < j 3N/4, j = j + h, 3N/4 < j N, (4.4) where { } N/4 = min 4, εlnn, 3N/4 = N/4, (4.5) h = 4 N/4 N, h = ( 3N/4 N/4 ). N In fact, we have used N 4 points for the boundary layer near =, N 4 points for the boundary layer near =, and N points for the remaining middle portion of the interval [,] Piecewise uniform mesh for first ecited state with bo potential Recall from the subsection 3..3, we know that for BEC in the first ecited state of bo potential, there are boundary layers in the region near = and = and one interior layer in the region at =.5. By taking [a, b] = [, ] for computation,

42 4.3 Discretization with piecewise uniform mesh in D 36 we choose the mesh as j = j + h, < j N/8. j = j + h, N/8 < j 3N/8. j = j + h, 3N/8 < j 5/8. j = j + h, 5N/8 < j 7N/8. j = j + h, 7N/8 < j N, (4.6) where { } N/8 = min 8, εlnn, 3N/8 = N/8, 5N/8 = + N/8, (4.7) 7N/8 = N/8. h = 8 N/8 N, h = 4( 3N/8 N/8 ). N In fact, we have used N 8 points for the boundary layer near =, N 8 points for the boundary layer near =, N 4 points for the interior layer near =.5 and the remaining N points for the remaining portion of the interval [,] which do not contain any boundary or interior layers. By etending the above idea, we can obtain the piecewise uniform mesh for the other ecited states in bo potential Piecewise uniform mesh for first ecited state with harmonic potential Recall from the subsection 3.3.3, we know that for the first ecited state of BEC with harmonic potential, there is an interior layer in the region of = when < ε.

43 4.4 Choice of initial data 37 Taking [a, b] = [ c, c] for computation, we choose the mesh as j = j + h, < j N/4, j = j + h, N/4 < j 3N/4, j = j + h, 3N/4 < j N, (4.8) where = c <, N = c >, { c } N/4 = min, εlnn, (4.9) 3N/4 = N/4, h = N/4 N, h = (c + N/4 ). N In fact, we have used N points for the interior layer near = and the remaining N points for the remaining portion of the interval [-c,c] which do not contain any boundary or interior layers. 4.4 Choice of initial data In this section, we apply the proposed adaptive numerical scheme to solve the singularly perturbed nonlinear eigenvalue problems (3.8) under the constraint (3.9) in D. We consider the following cases:. Ground state of BEC in D bo potential;. First, third and ninth ecited states of BEC in D bo potential; 3. First ecited state of BEC in D harmonic potential. The initial data are carefully chosen for different potentials. For the bo potential, the problem is solved on Ω = [, ]. The initial condition in (4.3) for finding

44 4.4 Choice of initial data 38 the ground state is taken as φ () = sin(π), [, ], (4.3) and the boundary conditions are φ(, t) = φ(, t) =. (4.3) as In order to obtain the kth ecited state, we choose the initial conditions in (4.3) φ () = sin((k + )π), [, ], (4.3) and the boundary conditions is the same as (4.3). For the harmonic potential, the problem is solved on Ω = [ c, c] (where c is some large enough positive constant). The initial condition for finding the first ecited state in (4.3) is taken as and the boundary conditions are φ( c, t) = φ(c, t) =. / φ () = e, (4.33) π/4 In our numerical calculations presented in the net three sections, an eact solution φ() is defined as a solution generated using our adaptive schemes with 4 + mesh points. This solution φ() is used as the basis to validate the numerical accuracy of the solutions obtained by using the piecewise uniform mesh or uniform mesh methods. Let φ ε,n () be the numerical solution with parameters ε and N + mesh points. In the net three sections, error plots refer to the plot of φ() φ ε,n () for different ε and N values using the two different meshes presented in this chapter.

45 4.5 analysis of uniform mesh analysis of uniform mesh In this section, by means of the scheme based on uniform mesh, we calculate and compare the ground state of BEC with bo potential, first ecited state of BEC with bo potential and first ecited state of BEC with harmonic potential. For the results related to ground state of BEC with bo potential based on the uniform mesh scheme, we refer to Figure 4. and Figure 4.. We observe that most of the errors are concentrated near the boundary i.e., near = and =. This is epected as there are boundary layers at = and =. When ε becomes smaller, the pointwise errors increase. When more mesh points are used, the pointwise errors in general are reduced. However, this reduction is very insignificant for smaller values of ε. For the results related to first ecited state of BEC with bo potential based on the uniform mesh scheme, we refer to Figure 4.3 and Figure 4.4. The results are similar to results of the ground state of BEC with bo potential. We observe that most of the errors are concentrated near the boundary, i.e., near = and = and near =.5. This is epected as there are boundary layers at = and = and an interior layer eist at =.5. When ε becomes smaller, the pointwise errors increase. When more mesh points are used, the pointwise errors in general are reduced. However, this reduction is very insignificant for smaller values of ε. For the results related to first ecited state of BEC with harmonic potential based on the uniform mesh scheme, we refer to Figure 4.5 and Figure 4.6. We observe that the largest errors are concentrated near = and the rest of the errors are concentrated at =. and =.. This is epected as an interior layer eists at =. The second largest errors are mainly at =. and =. because the gradient of φ() at those points are steep. When ε becomes smaller, the change in errors is not noticeable. When more mesh points are used, the pointwise errors in

46 4.5 analysis of uniform mesh 4 general are reduced. However, this reduction is very insignificant for smaller values of ε, especially near = where there is an interior layer.

47 4.5 analysis of uniform mesh ε=..6 ε= ε= ε= Figure 4.: plot for ground state of BEC in bo potential with fied mesh points using uniform mesh scheme. N, the total number of mesh points used is N = 4 + ( full line) and N = 6 + (dotted line). A comparison is made for increasing values of ε from ε =. 8 to ε =..

48 4.5 analysis of uniform mesh N= 4 + N= N= N= Figure 4.: plot for ground state of BEC in bo potential with fied ε values using uniform mesh scheme. The ε values used are ε =. ( full line) and ε =. 6 (dotted line). A comparison is made for increasing values of N, the total number of mesh points used from N = 4 + to N = +.

49 4.5 analysis of uniform mesh ε=..6 ε= ε= ε= Figure 4.3: plot for first ecited state of BEC in bo potential with fied mesh points using uniform mesh scheme. N, the total number of mesh points used is N = 4 + (full line) and N = 6 + (dotted line). A comparison is made for increasing values of ε from ε =. 8 to ε =..

50 4.5 analysis of uniform mesh N= N= N= 8 +. N= Figure 4.4: plot for first ecited state of BEC in bo potential with fied ε values using uniform mesh scheme. The ε values used are ε =. ( full line) and ε =. 6 (dotted line). A comparison is made for increasing values of N, the total number of mesh points used from N = 4 + to N = +.

51 4.5 analysis of uniform mesh ε= ε= ε= ε=. 8.. Figure 4.5: plot for first ecited state of BEC in harmonic potential with fied mesh points using uniform mesh scheme. N, the total number of mesh points used is N = 4 + (full line) and N = 6 + (dotted line). A comparison is made for increasing values of ε from ε =. 8 to ε =..

52 4.5 analysis of uniform mesh N= 4 + N= N= 8 + N= Figure 4.6: plot for first ecited state of BEC in harmonic potential with fied ε using uniform mesh scheme. The ε values used are ε =. ( full line) and ε =. 6 (dotted line). A comparison is made for increasing values of N, the total number of mesh points used from N = 4 + to N = +.

53 4.6 analysis of piecewise uniform mesh analysis of piecewise uniform mesh In this section, by means of the scheme based on piecewise uniform mesh, we calculate and compute the ground state of BEC with bo potential, first ecited state of BEC with bo potential and first ecited state of BEC with harmonic potential. For the results related to ground state of BEC with bo potential based on the piecewise uniform mesh scheme, we refer to Figure 4.7 and Figure 4.8. We observe that most of the errors are concentrated near the boundary i.e., near = and =. This is epected as there are boundary layers at = and =. When ε becomes smaller, the maimum error remained unchanged near.8. When more mesh points are used, the maimum errors at the boundary layers are significantly reduced. For the results related to first ecited state of BEC with bo potential based on the piecewise uniform mesh scheme, we refer to Figure 4.9 and Figure 4.. We observe that most of the errors are concentrated near =, =.5 and =. This is epected as there are boundary layers at = and = and an interior layer at =.5. When ε becomes smaller, the maimum error remained unchanged near.7. When more mesh points are used, the maimum errors at the boundary and interior layers are significantly reduced. For the results related to first ecited state of BEC with harmonic potential based on the piecewise uniform mesh scheme, we refer to Figure 4. and Figure 4.. We observe that the largest errors are concentrated near =. and =.. Using the piecewise uniform has significantly reduced the errors at the interior layer. When ε becomes smaller, the maimum error remained unchanged near.7. When more mesh points are used, the maimum errors at the boundary and interior layers are significantly reduced.

54 4.6 analysis of piecewise uniform mesh ε=..5. ε= ε= ε= Figure 4.7: plot for first ecited state of BEC in bo potential with fied mesh points using piecewise uniform mesh scheme. N, the total number of mesh points used is N = 4 + (full line) and N = 6 + (dotted line). A comparison is made for increasing values of ε from ε =. 8 to ε =..

55 4.6 analysis of piecewise uniform mesh N= 4 + N= N= N= Figure 4.8: plot for ground state of BEC in bo potential with fied ε values using piecewise uniform mesh scheme. The ε values used are ε =. ( full line) and ε =. 6 (dotted line). A comparison is made for increasing values of N, the total number of mesh points used from N = 4 + to N = +.

56 4.6 analysis of piecewise uniform mesh ε=..5 ε= ε=. 6.5 ε= Figure 4.9: plot for first ecited state of BEC in bo potential with fied mesh points using piecewise uniform mesh scheme. N, the total number of mesh points used is N = 4 + (full line) and N = 6 + (dotted line). A comparison is made for increasing values of ε from ε =. 8 to ε =..

57 4.6 analysis of piecewise uniform mesh N= 4 + N= N= 8 + N= Figure 4.: plot for first ecited state of BEC in bo potential with fied ε values using piecewise uniform mesh scheme. The ε values used are ε =. ( full line) and ε =. 6 (dotted line). A comparison is made for increasing values of N, the total number of mesh points used from N = 4 + to N = +.

58 4.6 analysis of piecewise uniform mesh ε=.. ε= ε=. 6. ε= Figure 4.: plot for first ecited state of BEC in harmonic potential with fied mesh points using piecewise uniform mesh scheme. N, the total number of mesh points used is N = 4 + (full line) and N = 6 + (dotted line). A comparison is made for increasing values of ε from ε =. 8 to ε =..

59 4.6 analysis of piecewise uniform mesh N= 4 + N= N= 8 + N= Figure 4.: plot for first ecited state of BEC in harmonic potential with fied ε using piecewise uniform mesh scheme. The ε values used are ε =. ( full line) and ε =. 6 (dotted line). A comparison is made for increasing values of N, the total number of mesh points used from N = 4 + to N = +.

60 4.7 Numerical comparisons Numerical comparisons In order to compare the new numerical scheme based on piecewise uniform mesh against the classical uniform mesh scheme, we used a fied number of mesh points (i.e. 4 + mesh points) with decreasing ε values. (i.e. ε =.,. 4,. 6 or. 8.) From Figure 4.3, Figure 4.4 and Figure 4.5, we observe that the largest errors mainly occurs at the boundary layers or the interior layers when using the uniform mesh method. These errors found at the boundary or interior layers are significantly reduced when the piecewise uniform mesh method is being applied. Furthermore, all errors at other regions using piecewise uniform mesh scheme is also smaller than the those using uniform mesh scheme. Based on the numerical results for the 3 types of potentials, we noted the maimum errors, i.e. ma a b φ() φ ε,n (), for different values of ε and the different numbers of mesh points used for both piecewise uniform mesh and uniform mesh methods. A summary of these results are given in Tables 4. to 4.6. From these Tables, we observe that as ε decreases, the maimum error increases. This is epected because the gradient change in the boundary layer or interior layer is larger for smaller values of ε. Comparing the results from these Tables, we also observe that the advantage of using piecewise uniform mesh scheme over uniform mesh scheme is more significant when comparing solutions with smaller values of ε. When ε is very small, adding more mesh points with the uniform mesh method does not reduce the maimum error significantly. However, when we use more mesh points with the piecewise uniform mesh scheme, the reduction in maimum errors is much more significant. From Tables 4.7 to 4.9, we observe that the numerical scheme based on piecewise uniform mesh is uniformly convergent. From Tables 4. to 4., we observe that

61 4.7 Numerical comparisons 55 the numerical scheme based on uniform mesh is not convergent. In the net chapter, we will apply the piecewise uniform mesh scheme to find the ground state and ecited states of BEC with bo potential and first ecited state of BEC with harmonic potential in D. We will also etend the method to solve the singularly perturbed problems (3.8) under the constraint (3.9) in D, in order to find the ground state or ecited states of BEC with bo potential or harmonic potential or harmonic plus optical potential in D.

62 4.7 Numerical comparisons ε=..6 ε= ε= ε= Figure 4.3: comparison between piecewise uniform mesh and uniform mesh obtained from ground state of BEC with bo potential. Piecewise uniform mesh (full line), uniform mesh (dotted line).

63 4.7 Numerical comparisons ε= ε= ε= ε= Figure 4.4: comparison between piecewise uniform mesh and uniform mesh obtained from first ecited state of BEC with bo potential. Piecewise uniform mesh (full line), uniform mesh (dotted line).

64 4.7 Numerical comparisons ε= ε= ε= ε=. 8.. Figure 4.5: comparison between piecewise uniform mesh and uniform mesh obtained from first ecited state of BEC with harmonic potential.piecewise uniform mesh (full line), uniform mesh (dotted line).

65 4.7 Numerical comparisons 59 Number of mesh points used ε e e-6.6e-4.7e-3.64e-. 3.6e-6 4.3e-5 4.4e-4 4.e-3.7e e-6 4.6e-5 4.7e e-3.7e e-6 4.e-5 4.3e-4 3.8e-3.7e-. 8.6e-5 4.7e-5 4.e e-3.69e-. 7.7e-5 9.6e-5 4.e e-3.69e e-4 3.7e e e-3.7e e-3.39e-3.43e e-3.7e- Table 4.: Maimum errors for ground state of BEC with bo potential using piecewise uniform mesh. Number of mesh points used ε e e-6.6e-4.7e-3.64e-. 5.8e e-5.4e-3.43e- 3.9e e-5.36e-3.37e- 3.8e- 7.68e e-3.35e- 3.79e- 7.68e- 9.4e e- 3.79e- 7.67e- 9.4e- 9.76e e- 7.67e- 9.4e- 9.77e- 9.93e e- 9.4e- 9.77e- 9.93e- 9.98e e- 9.77e- 9.93e- 9.98e- 9.99e- Table 4.: Maimum errors for ground state of BEC with bo potential using uniform mesh.

66 4.7 Numerical comparisons 6 Number of mesh points used ε e e-6.57e-4.5e e-. 5.6e-6 9.5e-5.5e-3.86e-.84e-. 4.4e-5.5e-4.66e-3.69e-.66e e-5.5e-4.6e-3.66e-.6e-. 8.3e-5.5e-4.6e-3.65e-.6e-. 8.6e-5.79e-4.6e-3.65e-.6e-..86e e-4.63e-3.64e-.6e e-3.e-3.84e-3.66e-.6e- Table 4.3: Maimum errors for st ecited state of BEC with bo potential using piecewise uniform mesh. Number of mesh points used ε e e-6.57e-4.5e e-. 5.6e-6 9.5e-5.5e-3.5e- 4.5e e-5.38e-3.39e- 3.85e- 7.69e e-3.36e- 3.8e- 7.68e- 9.3e e- 3.79e- 7.67e- 9.4e- 9.76e e- 7.67e- 9.4e- 9.77e- 9.93e e- 9.4e- 9.77e- 9.93e- 9.98e e- 9.77e- 9.93e- 9.98e- 9.99e- Table 4.4: Maimum errors for st ecited state of BEC with bo potential using uniform mesh.

67 4.7 Numerical comparisons 6 Number of mesh points used ε e-5.67e e e-.6e e-5.e-3.e- 9.3e-.5e e-4.49e-3.84e-.8e-.6e e e-3.7e-.e-.63e e e-3.9e-.e-.63e e e-3.8e-.e-.63e-. 3.4e e-3.8e-.e-.63e e e-3.8e-.e-.63e- Table 4.5: Maimum errors for first ecited state of of BEC with harmonic potential using piecewise uniform mesh. Number of mesh points used ε e-.4e-.44e- 5.89e- 7.37e-..57e-.44e- 5.88e- 7.36e- 7.87e e- 5.89e- 7.36e- 7.87e- 8.3e e- 7.36e- 7.87e- 8.3e- 8.7e e- 7.87e- 8.3e- 8.7e- 8.9e e- 8.3e- 8.7e- 8.9e- 8.9e-. 8.3e- 8.7e- 8.9e- 8.9e- 8.9e e- 8.9e- 8.9e- 8.9e- 8.9e- Table 4.6: Maimum errors for st ecited state of BEC with harmonic potential using uniform mesh.

68 4.7 Numerical comparisons 6 N ε e- 4.e-3 4.7e-4 4.e-5.6e-5 Table 4.7: Maimum errors for ground state of BEC with bo potential using piecewise uniform mesh. N ε e-.86e-.66e-3.5e-4.3e-4 Table 4.8: Maimum errors for first ecited state of BEC with bo potential using piecewise uniform mesh. N ε e- 9.3e-.84e- 3.49e e-4 Table 4.9: Maimum errors for first ecited state of BEC with harmonic potential using piecewise uniform mesh.

69 4.7 Numerical comparisons 63 N ε e-.43e-.37e-.35e-.35e- Table 4.: Maimum errors for ground state of BEC with bo potential using uniform mesh. N ε e-.5e-.39e-.36e-.35e- Table 4.: Maimum errors for first ecited state of BEC with bo potential using uniform mesh. N ε e- 7.36e- 7.36e- 7.36e- 7.37e- Table 4.: Maimum errors for first ecited state of BEC with harmonic potential using uniform mesh.

70 Chapter 5 Numerical Applications In this chapter, we first apply the newly proposed numerical scheme based on piecewise uniform mesh to find the ground state and ecited states of BEC with bo potential in D or with harmonic potential in D. We net etend the gradient flow with discrete normalization based on adaptive mesh method shown in Chapter 4 to solve the singularly perturbed nonlinear eigenvalue problems (3.8) under the constraint (3.9) in two dimensions (D). We are particularly interested in the ground state and various ecited states for BEC with bo potential in D, or harmonic potential in D, or harmonic plus optical lattice potential in D. These stationary states are particularly interesting because the particle number of the BEC at equilibrium is usually very large or the BEC is in a semiclassical regime (this corresponds to that ε goes to zero). This is also to illustrate the capability of the piecewise uniform mesh method in solving singularly perturbed problems and find boundary layers or interior layers in higher dimensions. The problem now is solved in two dimensions and there are different ecited states in both the - and y-direction. For given positive integers j and k, a (j,k)-th ecited state is where the BEC is in the j-th ecited state in the -direction and k-th ecited state in the y-direction. The (,)-th state is the ground state. 64

71 5. Numerical results in D Numerical results in D 5.. Ground state and ecited states with bo potential Figure 5. shows the numerical result for the ground state using our adaptive mesh numerical scheme based on piecewise uniform mesh with 4 + mesh points. There are boundary layers near = and = respectively when ε goes near. These agree well with the asymptotic approimation presented in the subsection 3... Figure 5. shows the numerical result for the first ecited state using our adaptive mesh numerical scheme based on piecewise uniform mesh with 4 + mesh points. There are boundary layers near = and = respectively when ε goes near. Moreover, there is an interior layer near =. Figure 5.3 shows the numerical result for the third ecited state using our adaptive mesh numerical scheme based on piecewise uniform mesh with 6 + mesh points. There are boundary layers near = and = respectively when ε goes near. There are interior layers near = 4,, 3 4 respectively. Figure 5.4 shows the numerical result for the ninth ecited state using our adaptive mesh numerical scheme based on piecewise uniform mesh with 8 mesh points. There are boundary layers near = and = respectively when ε goes near. There are interior layers near =, 5,, 9 respectively. Table 5. shows the energy and chemical potential values for different values of ε in the different states of the BEC in bo potentials. We observe that for larger values of ε, the corresponding energy and chemical potentials are also higher. Also, bo potentials in higher ecited states have higher energy and chemical potentials levels compared to bo potentials in lower ecited states. All these numerical results agree well with the asymptotic approimation presented in the subsection 3..3.

72 5. Numerical results in D 66 φ().8.6 ε =. X ε =. X ε =. X 4 ε =. X Figure 5.: Solution for ground state with bo potential in D, 4 + mesh points..5 ε =. X ε =. X ε =. X 4 ε =. X 8 φ() Figure 5.: Solution for first ecited state of BEC with bo potential in D, 4 + mesh points.

73 5. Numerical results in D 67.5 ε =. X ε =. X ε =. X 4 ε =. X 8.5 φ() Figure 5.3: Solution for third ecited state of BEC with bo potential in D, 6 + mesh points..5.5 ε =. X ε =. X ε =.. X 4 ε =. X 8 φ() Figure 5.4: Solution for ninth ecited state of BEC with bo potential in D, 8 mesh points.

74 5. Numerical results in D 68 ε E g µ g E µ E µ E µ Table 5.: Energy and chemical potential of different states of BEC with bo potential in D for different ε.

75 5. Numerical results in D 69 ε E g µ g E µ Table 5.: Energy and chemical potential of different states of BEC with harmonic potential in D for different ε. 5.. Ground state and ecited states with harmonic potential in D In this subsection, we calculate the ground state and first ecited state with harmonic potential in D. Figure 5.5 shows the numerical result for ground state by using our uniform mesh numerical scheme with 5 + mesh points. The uniform mesh is used because there are neither boundary layers nor interior layers inside the computed domain [, ]. Figure 5.6 shows the numerical result for the first ecited state by using our adaptive numerical scheme based on piecewise uniform mesh with 5 + mesh points. There is an interior layer near =. Table 5. shows the energy and chemical potential values for different values of ε in the different states of the BEC in harmonic potentials. We observe that for larger values of ε, the corresponding energy and chemical potentials are also higher. Also, the energy and chemical potential levels in the first ecited ecited states are higher than the energy and chemical potential levels in the ground state. All these numerical results agree well with the asymptotic approimation presented in the subsection

76 5. Numerical results in D ε = ε = ε = ε = 3.5 φ() Figure 5.5: Solution for ground state of BEC with harmonic potential in D, 5 + mesh points φ()..4 ε =. X ε =. X ε =. X Figure 5.6: Solution for first ecited state of BEC with harmonic potential in D, 5 + mesh points.

77 5. Numerical results in D for bo potential 7 5. Numerical results in D for bo potential In this section, we calculate the ground state and ecited states solutions for the BEC confined in bo potential. The bo potential in D is given as:, <, y <, V (, y) =, otherwise. (5.) The problem is solved on the domain Ω = [, ] [, ] and the mesh size is Choice of mesh The mesh used is based on the idea mentioned previously in section 4.3. Using the same idea, we etend the mesh in both and y directions. Hence, for a D BEC under bo potential in the (,)th ecited state, we will choose the mesh in the -direction as j = j + h, < j 64, j = j + h, 64 < j 9, (5.) j = j + h, 9 < j 56, where { } 64 = min 4, εln(56), 9 = 64, (5.3) h = 64 64, h = Similarly, we choose the mesh in the y-direction as yk = y k + h y, < k 3, yk = y k + h y, 3 < k 96, yk = y k + h y, 96 < k 6, yk = y k + h y, 6 < k 4, yk = y k + h y, 4 < k 56, (5.4)

78 5. Numerical results in D for bo potential 7 where { } y3 = min 8, εln(56), y96 = y 3, y6 = + y 3, (5.5) y4 = y3, h y = y N/8 3, h y = y 96 y3. 64 By etending the idea accordingly, we can obtain the piecewise uniform mesh for the other states in D bo potential. 5.. Choice of initial data The initial data used is based on the idea mentioned previously in section 4.4. By etension of the same idea, the initial data used for finding the (j, k)th ecited state of a D BEC under bo potential is given as φ (, y) = sin((j + )π) sin((k + )πy),, y [, ], (5.6) 5..3 Results For the ground state in the D bo potential, Figures 5.7 and 5.8 show the surface plot and image plot of (,)th state with bo potential in D with ε = 3, respectively. It is clearly seen that ground state with bo potential in D has boundary layers near the boundary of the domain Ω. For the various ecited states in the D bo potential, Figures 5.9 and 5. show the surface plot and image plot of (,)-th state with bo potential in D with ε = 3, respectively. Figures 5. and 5. show the surface plot and image plot

79 5. Numerical results in D for bo potential 73 Figure 5.7: Surface plot of ground state with bo potential in D, ε = 3. of (,3)-th state with bo potential in D with ε = 3, respectively. Figures 5.3 and 5.4 show the surface plot and image plot of (9,9)-th state with bo potential in D with ε = 3, respectively. Figures 5.5 and 5.6 show the surface plot and image plot of (9,9)-th state with bo potential in D with ε = 3, respectively. It is clearly seen that ecited states in the D bo potential not only have boundary layers near the boundary of the domain Ω but also have interior layer inside the domain Ω. Table 5.3 shows the energy and chemical potential values for different values of ε in the different states of the BEC in D bo potentials. The trends are similar to those observed in Table 5.. We observe that for larger values of ε, the corresponding energy and chemical potentials are also higher. Also, bo potentials in higher ecited states have higher energy and chemical potential levels compared to bo potentials in lower ecited states.

80 5. Numerical results in D for bo potential Figure 5.8: Image plot of ground state with bo potential in D, ε = 3. Figure 5.9: Surface plot of (,)-th ecited state with bo potential in D, ε = 3.

81 5. Numerical results in D for bo potential Figure 5.: Image plot of (,)-th ecited state with bo potential in D, ε = 3. Figure 5.: Surface plot of (,3)-th ecited state with bo potential in D, ε = 3.

82 5. Numerical results in D for bo potential Figure 5.: Image plot of (,3)-th state with bo potential in D, ε = 3. Figure 5.3: Surface plot of (9,9)-th state with bo potential in D, ε = 3.

83 5. Numerical results in D for bo potential Figure 5.4: Image plot of (9,9)-th state with bo potential in D, ε = 3. Figure 5.5: Surface plot of (9,9)-th ecited state with bo potential in D, ε = 3.

84 5. Numerical results in D for bo potential Figure 5.6: Image plot of (9,9)-th ecited state with bo potential in D, ε = 3. ε... E g µ g E, µ, E, µ, Table 5.3: Energy and chemical potential of different states of BEC with bo potential in D for different ε

85 5.3 Numerical results in D for harmonic potential Numerical results in D for harmonic potential In this section, we show the ground state and ecited states solutions for the BEC in D harmonic oscillator potential. The potential for the D harmonic oscillator potential is: V () = + y, (, y) R. The problem is solved on the domain Ω = [, ] [, ] Choice of mesh Similar to the previous section, the mesh used is based on the idea mentioned previously in section 4.3. Using the same idea, we etend the mesh in both and y directions. Hence, for a D BEC under harmonic potential in the (,)th ecited state, we will use a uniform mesh in the -direction as there are no interior or boundary layers. For the y-direction, we choose the mesh as yk = y k + h y, < k 64, yk = y k + h y, 64 < k 9, yk = y k + h y, 9 < k 56, (5.7) where y =, y 56 =, y 64 = min {, εln(56)}, (5.8) y 9 = y 96, h y = y 96 56, h y = + y 96 8.

86 5.3 Numerical results in D for harmonic potential Choice of initial data The choice of initial data used is based on the idea mentioned previously in section 4.4. The initial data used for finding the (j, k)th (j, k can take values or ) ecited state of a D BEC under harmonic potential is given as φ (, y) = j y k e ( +y )/ π. (5.9) Results For the ground state with harmonic potential, Figures 5.7 and 5.8 show the surface plot and image plot of ground state with harmonic potential with ε =.56 3, respectively. It is clearly seen that ground state in D harmonic potential has no boundary layers in the whole domain Ω = [, ] [, ], even if ε is very small. For various ecited states in the D harmonic potential, Figures 5.9 and 5. show the surface plot and image plot of (,) ecited state with harmonic potential with ε =.56 3, respectively. Figures 5. and 5. show the surface plot and image plot of -th ecited state with harmonic potential with ε =.56 3, respectively. It is clearly seen that ecited states with harmonic potential in D do not have boundary layers near the boundary of the domain Ω but have interior layers inside the computed domain Ω. Table 5.4 shows the energy and chemical potential values for different values of ε in the different states of the BEC in harmonic potentials. The trends are similar to those observed in Table 5.. We observe that for larger values of ε, the corresponding energy and chemical potentials are also higher. Also, the energy and chemical potential levels in the first ecited ecited states are higher than the energy and chemical potential levels in the ground state.

87 5.3 Numerical results in D for harmonic potential 8 Figure 5.7: Surface plot of,-th state with harmonic potential in D, ε = Figure 5.8: Image plot of ground state with harmonic potential in D, ε =.56 3.