Asymptotic soliton train solutions of Kaup Boussinesq equations

Transcription

1 Wave Motion 38 (003) Asymptotic soliton train solutions of Kaup Boussinesq equations A.M. Kamchatnov a,b,, R.A. Kraenkel b, B.A. Umarov b,c a Institute of Spectroscopy, Russian Academy of Sciences, Troitsk 490, Moscow Region, Russia b Instituto de Física Teórica, Universidade Estadual Paulista UNESP, Rua Pamplona 45, São Paulo, Brazil c Physical-Technical Institute, Uzbek Academy of Sciences, Tashkent-84, G. Mavlyanov str., -b, Uzbekistan Received 3 February 00; received in revised form 8 April 003; accepted 7 May 003 Abstract Asymptotic soliton trains arising from a large and smooth enough initial pulse are investigated by the use of the quasiclassical quantization method for the case of Kaup Boussinesq shallow water equations. The parameter varying along the soliton train is determined by the Bohr Sommerfeld quantization rule which generalizes the usual rule to the case of two potentials h 0 () and u 0 () representing initial distributions of height and velocity, respectively. The influence of the initial velocity u 0 () on the asymptotic stage of the evolution is determined. Ecellent agreement of numerical solutions of the Kaup Boussinesq equations with predictions of the asymptotic theory is found. 003 Elsevier B.V. All rights reserved.. Introduction In the evolution of an initially large and smooth pulse, it is natural to distinguish two characteristic stages formation of a dissipationless shock wave after the wavebreaking point and asymptotic evolution of soliton trains arising eventually from the initial pulse. Until recently the most attention was paid to the first stage of evolution. First considered in the seminal paper by Gurevich and Pitaevskii [] by means of Whitham s modulation theory [], this problem was treated in many papers [3 9] and may be considered as completely solved for the KdV equation case. It is now being developed for the defocusing NLS equation [0 6], the Toda lattice equation [7,8], and some other equations. Applications of the Whitham modulation method to such modulationally unstable systems as, for eample, the focusing NLS equation is quite complicated and only some particular results have been obtained in this respect until now [9 7]. An introduction to this subject can be found in [8]. The problem of asymptotic distribution functions of the soliton parameters may have many applications in soliton physics, but it has been considered in some detail apparently only for the KdV equation case [,9,30], where it is especially simple due to the fact that solitons velocities and their amplitudes are determined by only one parameter, whose values are equal to the eigenvalues of the Schrödinger spectral problem associated with the KdV equation in the inverse scattering transform method. As we have pointed out recently [3], a similar approach applies effectively to many other equations corresponding to the second order spectral problem with an energy-dependent potential. Corresponding author. Tel./fa: address: kamch@ift.unesp.br (A.M. Kamchatnov) /$ see front matter 003 Elsevier B.V. All rights reserved. doi:0.06/s065-5(03)0006-3

2 356 A.M. Kamchatnov et al. / Wave Motion 38 (003) Here we shall study in some detail by this method the Kaup Boussinesq system describing the shallow water motion and show that the quasiclassical approach gives a very simple and effective solution of the problem.. Soliton solutions of Kaup Boussinesq system The Kaup Boussinesq system [3]: h t + (hu) ± 4 ε u = 0, u t + uu + h = 0, () where h(, t) denotes the height of the water surface above a horizontal bottom and u(, t) denotes its velocity averaged over depth, describes motion of shallow water in the same approimation as the well-known Boussinesq equation in the lowest order in small parameters controlling weak dispersion and nonlinearity effects (see, e.g. [,33]). The upper sign here corresponds to the case when the gravity force dominates over the capillary one, and the lower sign corresponds to the opposite case when the capillarity plays the main role. The analytic theory developed below applies to both cases. In the dispersionless limit ε 0 the system () transforms into the system of so-called shallow water equations [,8]. We have written the system () in standard dimensionless units. In a linear approimation the system () leads to the following dispersion relation: ω = k ( 4 ε k ), () k and ω being wavenumber and frequency of the linear wave, respectively. The parameter ε controls the dispersion effects and since for an initially large and smooth pulse these effects are small, we consider the so-called weak dispersion limit ε. The Kaup Boussinesq system () is completely integrable and can be presented as a compatibility condition of two linear equations [3]: ε ψ =±[(λ u) h]ψ, (3) ψ t = 4 u ψ (λ + u)ψ. This permits one to find its soliton and periodic solutions [3 34], and Whitham s modulation equations with application to the theory of dissipationless shock waves, due to decay of an initial step-like discontinuity [33]. Here we shall consider an asymptotic stage of evolution of an initially localized pulse with large and smooth enough distributions of h 0 () and u 0 () at t = 0 in the above mentioned weak dispersion limit ε. Now, on the contrary to the KdV equation case, the soliton solution is not parameterized by a single eigenvalue of the linear problem (3), and hence we have to develop in some detail the theory of soliton solutions of the Kaup Boussinesq system (). We suppose that the asymptotic stage can be described as soliton trains, that is, modulated periodic solutions of the system (). The strictly periodic solution can be obtained by the well-known finite-gap integration method (see, e.g. [8,33]) in the following way. Let ψ + and ψ be two basis solutions of the second order linear differential equation (3). Then their product g = ψ + ψ (5) satisfies the third order equation: ε g + ( uu h λu )g + 4[(λ u) h]g = 0. (6) Upon multiplication by g, this equation can be integrated once to give ε ( gg 4 g ) + [(λ u) h]g = P(λ), (7) where the integration constant P(λ) can only depend on λ. The time dependence of g(, t) is determined by the equation: g t = ug (λ + u)g. (8) (4)

3 This equation readily gives the relation: ( ) ( = λ + u/ ) g g t, A.M. Kamchatnov et al. / Wave Motion 38 (003) which can be considered as a generating function of an infinite sequence of conservation laws. The periodic solutions of the system () are distinguished by the condition that P(λ) in (7) be a polynomial in λ. The one-phase periodic solution, which we are interested in, corresponds to the fourth degree polynomial: P(λ) = 4 (λ λ i ) = λ 4 s λ 3 + s λ s 3 λ + s 4. (0) i= Then we find from Eq. (7) that g(, t) is the first-degree polynomial: g(, t) = λ µ(, t) and µ(, t) is connected with u(, t) and h(, t) by the relations: u(, t) = s µ(, t), h(, t) = 4 s s µ + s µ, () following from comparison of coefficients of λ i on both sides of Eq. (7). The spectral parameter λ is arbitrary and on substitution of λ = µ into Eq. (7) we obtain an equation for µ: εµ = ±P(µ), and a similar substitution into Eq. (8) gives µ t = (µ + u)µ = s µ. (9) () Hence, µ(, t) as well as u(, t) and h(, t) depend only on the phase: θ = s t (3) and the dependence µ(θ) is determined by the equation: εµ θ = ±P(µ). (4) For the fourth degree polynomial (0) the solution of this equation is readily epressed in terms of elliptic functions. Let the zeros λ i, i =,, 3, 4 of the polynomial P(λ) be real and ordered according to the rule: λ 4 <λ 3 <λ <λ. (5) Then the real variable µ oscillates in the interval where the epression under the square root in (4) is positive. Obviously, this depends on the choice of the sign in the system (). We shall consider the two possible cases separately... Bright soliton solution This case corresponds to the upper sign in the system () and hence in Eq. (4). Then µ oscillates in the interval: λ 3 µ λ, (6) where P(λ) > 0 and soliton solutions correspond to either λ 4 = λ 3,orλ = λ. In the case λ 4 = λ 3 easy integration of (4) with initial condition µ(0) = λ yields µ + s (, t) = λ (λ λ )(λ λ 3 ) λ λ + (λ λ 3 )/ cosh ( ), (7) (λ λ 3 )(λ λ 3 )θ/ε

4 358 A.M. Kamchatnov et al. / Wave Motion 38 (003) where θ = (/)(λ + λ + λ 3 ) and corresponding distributions of u(, t) and h(, t) can be found by substitution of (7) into Eq. (). In particular, at ± we get u = λ + λ, h = 4 (λ λ ). (8) In a similar way, for the case λ = λ we obtain µ + s (, t) = λ (λ λ 4 )(λ 3 λ 4 ) 4 + λ 3 λ 4 (λ λ 3 )/ cosh ( ), (9) (λ λ 3 )(λ λ 4 )θ/ε where θ = (/)(λ + λ 3 + λ 4 ) and u = λ 3 + λ 4, h = 4 (λ 3 λ 4 ). (0) It is easy to find that these formulas describe bright solitons of h(, t) over a constant background h... Dark soliton solution This case corresponds to the lower sign in () and (4). Then µ oscillates in the intervals: λ 4 λ λ 3 or λ λ λ. () In the soliton limit λ = λ 3, we obtain µ s (, t) = λ (λ λ )(λ λ 4 ) λ λ 4 + (λ λ 4 )/ cosh ( (λ λ )(λ λ 4 )θ/ε ) () and µ s (, t) = λ (λ λ 4 )(λ λ 4 ) 4 + λ λ 4 (λ λ )/ cosh ( ), (3) (λ λ )(λ λ 4 )θ/ε where θ = (/)(λ + λ + λ 4 )t. The corresponding distributions of velocity u(, t) and height h(, t) can be found again from Eq. (). In particular, we find u = λ + λ 4, h = 4 (λ λ 4 ). (4) As we see, in each case two of the three parameters are determined by the asymptotic values of u and h.we suppose that they are known from the initial conditions u 0 () and h 0 () as u 0 ( ) and h 0 ( ). Our aim now is to find values of the third parameter for solitons arising eventually from the large pulse with given initial conditions u 0 () and h 0 (). It is worthwhile to note that though soliton solutions of the Kaup Boussinesq system () can be obtained by elementary methods, only their parametrization in terms of λ i permits one to find these parameters as eigenvalues of the linear equation (3). 3. Asymptotic stage of evolution Karpman s theory [9,30] of asymptotic solution of the KdV equation u t + 6uu + ε u = 0 is based on the fact that the parameters which characterize the solitons in the train coincide with the eigenvalues of the Schrödinger spectral problem associated with the KdV equation in the inverse scattering transform method and this spectrum does not change during the KdV evolution. Hence, it can be calculated from the initial distribution u 0 () = u(, 0). In the limit of weak dispersion ε, the eigenvalues can be calculated with the use of the Bohr Sommerfeld quantization rule. Here we shall apply the same method to the Kaup Boussinesq system connected with the second order spectral problem with energy-dependent potential.

5 A.M. Kamchatnov et al. / Wave Motion 38 (003) Eigenvalues λ of Eq. (3) are determined in this approimation by the well-known Bohr Sommerfeld quantization rule [35]: [ ( λ ) ( ε u 0() h 0 ()] d = π n + ), n = 0,,,...,N, (5) where integration is taken over the cycle around two turning point where the integrand function vanishes. Eigenvalues found in this way are equal to values of the third parameter in soliton solutions, when these solutions are well separated from each other in the asymptotic soliton train. The regions of possible values of λ are determined by the condition that the epression under the square root in Eq. (5) is positive and has two real turning points. Thus the plots of the potentials : r ± = u 0() ± h 0 () (6) permit one to find approimately (with accepted here quasiclassical accuracy), the maimal and minimal values of λ. Note that r ± coincide with the Riemann invariants for shallow water equations. It is instructive to note that the rule (5) corresponds to the quasiclassical quantization of a mechanical system with the Hamiltonian: ( H(p, ) = p + h 0 () + 0()) u, (7) where is a coordinate and p a momentum. At the turning points (6) the momentum p vanishes. Eq. (5) states that the area inside the contour H(p, ) = const. in the phase plane (, p) can take only half-integer values in units πε. Differentiation of Eq. (5) with respect to λ yields the number of eigenvalues in the interval (λ, λ + dλ): ( dn = f(λ) dλ = πε ) (λ u 0 ()/) [(λ (/)u0 ()) h 0 ()] d dλ. (8) The analytic theory can be illustrated by comparison with results of numerical solution of the system (). To be definite, we choose in all formulae the lower sign, because in this case the dispersive relation () is stable which simplifies numerical work. As is obvious from above, the analytic theory describes both cases. Now we specify the initial conditions so that u 0 () 0as and h 0 ( ) h. Then from Eq. (4) we get λ = λ 4 = h and hence the soliton solution takes the form: u s (θ) = for the case (), and (λ λ ) λ cosh ( λ λ θ/ε ) (λ + λ ) (λ u s (θ) = λ ( ) ) λ cosh λ λ θ/ε (λ λ ) for the case (3), where in both cases: θ = λ t (9) (30) (3) (3)

6 360 A.M. Kamchatnov et al. / Wave Motion 38 (003) and h(, t) = u(, t)(λ (u(, t))) + λ. (33) The values of the velocity u s (0, 0) and height h s (0, 0) at the center of soliton are equal to u s (0, 0) = (λ + λ ), h s (0, 0) = λ (λ + λ ), λ λ (λ ) (34) for the solution (), and u s (0, 0) = (λ λ ), h s (0, 0) = λ (λ λ ), (λ ) λ λ (35) for the solution (3). The inequalities indicated in these formulas follow from Eqs. (5) and (9) and the condition that h s (0, 0) >0. Hence, the solution () with negative values of λ describes solitons moving in the negative direction of the ais, and the solution (3) with positive values of λ describes solitons moving in the positive direction. Since in this case λ is equal by Eq. (3) to the velocity of solitons, formula (8) gives actually the distribution function of velocities of solitons well separated from each other at the asymptotic stage of evolution. At this stage, we can neglect the initial positions of solitons and take the coordinate of he nth soliton at the moment t equal to = λ (n) t, where λ (n) is the nth eigenvalue determined by Eq. (5). Then the number of solitons in the interval (, + d) is given by ( ) ( ) dn = f d, (37) t t (36) where f(λ) is defined by Eq. (8). For numerical simulation, we choose the initial distribution of the height h 0 () in the form: h 0 () = cosh (0.) and the initial distribution of u 0 () as either u 0 () = 0 (38) (39a) or 0.3 u 0 () = cosh (0.). (39b) The parameter ε controlling the dispersion effects is chosen equal as ε = 0.3. In Fig., the plots of Riemann invariants (6) are shown for (39b) nonzero initial velocities u 0 (). The possible values of λ are located inside the intervals: r min <λ<r ma, r+ min <λ<r+ ma. (40) They must satisfy the quantization rule (5) which selects contours H(p, ) = const. = λ in the phase plane (, p). These contours corresponding to even values of n are depicted in Fig. for (39b) with nonzero initial velocity and λ>0. The dependence of n on λ is shown in Fig. 3. In(39a) with u 0 () = 0, the function n(λ) is even and the branches with λ>0 and λ<0 coincide with each other. In (39b) with nonzero u 0 (), the number of eigenvalues with positive λ is greater than that with negative λ. This means that the number of solitons moving in the positive

7 A.M. Kamchatnov et al. / Wave Motion 38 (003) r + _ 0.5 _ Fig.. Plots of initial Riemann invariants r ± (see Eq. (6)) as functions of for h 0 () and u 0 () given by Eqs. (38) and (39b). The turning points ± corresponding to the eigenvalue λ =. are shown. The possible values of λ are given by Eq. (40). p Fig.. Contours H(p, ) = const. = λ in the phase plane (, p) of the mechanical system described by Hamiltonian (7) with h 0 () and u 0 () given by Eqs. (38) and (39b) and for values of λ determined by the quantization rule (5) with even n. 6 8 n 4 (a) l n 6 4 (b) 0 l > l < l Fig. 3. The dependence of n on λ defined by (5) for h 0 () and u 0 () given by Eqs. (38) (39b), respectively. Upper curve in (b) corresponds to λ>0, and lower curve to λ<0.

8 36 A.M. Kamchatnov et al. / Wave Motion 38 (003) u (a) h (b) Fig. 4. Soliton trains obtained by numerical solution of the Kaup Boussinesq system () with ε = 0.3 and initial conditions (38) and (39a) with zero initial velocity. Asymptotic distributions of velocity (a) and height (b) are shown as functions of space coordinate at the moment t = 00. direction is greater than the number of solitons moving in the negative direction for our choice of u 0 (). The soliton trains arising from the initial pulse are located at the moment t in the intervals: r min t<<r ma t, r+ min t<<r+ mat. (4) We have solved the Kaup Boussinesq system () with the initial conditions (38) (39b) numerically and the results are shown in Fig. 4 for (39a) with zero u 0 () and in Fig. 5 for nonzero u 0 (). As one can see, the results are in ecellent agreement with the predicted intervals. The amplitudes of the deepest solitons in the trains are also close to the theoretical values calculated according to Eqs. (34) and (35). To make a comparison of the theory with numerical results clearer, we have calculated velocities of those solitons in the trains, which are well separated from each other, and according to Eq. (3) have taken these velocities as values of λ calculated numerically. In Tables and, the values are given of λ λ BS calculated from the Bohr Sommerfeld quantization rule (5) and of λ λ simulation calculated from velocities of the soliton trains. Table corresponds to zero initial velocity (only u (a) h (b) Fig. 5. Soliton trains obtained by numerical solution of the Kaup Boussinesq system () with ε = 0.3 and initial conditions (38) and (39b) with nonzero initial velocity. Asymptotic distributions of velocity (a) and height (b) are shown as functions of space coordinate at the moment t = 75.

9 A.M. Kamchatnov et al. / Wave Motion 38 (003) Table Eigenvalues for u 0 () = 0 n λ BS λ simulation positive values of λ are presented) and Table to nonzero initial velocity. The agreement between the two methods of calculation is quite good. The difference does not eceed 3% and is maimal at very small n, where the accuracy of the quasiclassical calculation of the eigenvalues cannot generally speaking be etremely high, and at large n, where a numerical estimate has poor accuracy because solitons here are not separated well enough from each other and their motion does not obey the formula (3) yet. Table Eigenvalues for u 0 () = 0.3/ cosh (0.) n Positive eigenvalues Negative eigenvalues λ BS λ simulation λ BS λ simulation

10 364 A.M. Kamchatnov et al. / Wave Motion 38 (003) Conclusions The quasiclassical approach to calculation of the eigenvalues of the problem (3) with an energy-dependent potential yields a simple and effective description of the asymptotic stage of evolution of an initially large and smooth pulse in the weak dispersion limit. One may hope that this approach can be generalized to other classes of evolutional equations. Acknowledgements AMK and BAU are grateful to the staff of Instituto de Física Teórica, UNESP, where this work was done, for kind hospitality. The authors were partially supported by FAPESP (Brazil). AMK thanks also RFBR (grant ), RAK thanks CNPq (Brazil) and BAU thanks CRDF (grant ZM-095) for partial support. References [] A.V. Gurevich, L.P. Pitaevskii, Nonstationary structure of a collisionless shock wave, Zh. Eksp. Teor. Fiz. 65 (973) [Sov. Phys. JETP 38 (973) 9]. [] G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 974. [3] P.D. La, C.D. Levermore, The small dispersion limit of the Korteweg de Vries equation, Commun. Pure Appl. Math. 36 (983) 53 90, and [4] S. Venakides, The zero dispersion limit of the Korteweg de Vries equation for initial potentials with nontrivial reflection coefficient, Commun. Pure Appl. Math. 38 (985) [5] G.V. Potemin, Algebro-geometric construction of self-similar solutions of the Whitham equations, Usp. Math. Nauk 43 (988) [Russ. Math. Surv. 43 (988) 5 53]. [6] A.V. Gurevich, A.L. Krylov, G.A. El, Evolution of a Riemann wave in the dispersive hydrodynamics, Zh. Eksp. Teor. Fiz. 0 (99) [Sov. Phys. JETP 74 (99) 957]. [7] O.C. Wright, Korteweg de Vries zero dispersion limit through first breaking for cubic-like analytic initial data, Commun. Pure Appl. Math. 46 (993) [8] F.-R. Tian, Oscillations of the zero dispersion limit of the Korteweg de Vries equation, Commun. Pure Appl. Math. 46 (993) [9] F.-R. Tian, The initial-value problem for the Whotham averaged system, Commun. Math. Phys. 66 (994) [0] M.G. Forest, J.E. Lee, Geometry and modulation theory for periodic nonlinear Schrödinger equation, in: C. Dadermos, et al. (Eds.), Oscillation Theory, Computation, and Methods of Compensated Compactness, IMA Volumes on Mathematics and its Applications, Springer, New York, 986. [] M.V. Pavlov, The nonlinear Schrödinger equation and the Bogoliubov Whitham method of averaging, Teor. Mat. Fiz. 7 (987) [Theor. Math. Phys. 9 (987) 684]. [] A.V. Gurevich, A.L. Krylov, Nondissipative shock waves in media with positive dispersion, Zh. Eksp. Teor. Fiz. 9 (987) [3] G.A. El, V.V. Geogjaev, A.V. Gurevich, A.L. Krylov, Decay of an initial discontinuity in the defocusing NLS hydrodynamics, Physica D 87 (995) [4] Y. Kodama, The Whitham equations for optical communications: mathematical theory of NRZ, SIAM J. Appl. Math. 59 (999) [5] S. Jin, C.D. Levermore, D.W. McLaughlin, The semiclassical limit of the defocusing NLS hierarchy, Commun. Pure Appl. Math. 5 (999) [6] F.-R. Tian, J. Ye, On the Whitham equation for the semiclassical limit of the defocusing nonlinear Schrödinger equation, Commun. Pure Appl. Math. 5 (999) [7] S. Venakides, P. Deift, R. Oba, The Toda shock problem, Commun. Pure Appl. Math. 43 (99) 7 4. [8] A.M. Bloch, Y. Kodama, Dispersive renormalization of the Whitham equations for the Toda lattice, SIAM J. Appl. Math. 5 (99) [9] A.M. Kamchatnov, Periodic solutions and Whitham equations for the Heisenberg continuous classical spin model, Phys. Lett. A 6 (99) [0] G.A. El, A.V. Gurevich, V.V. Khodorovsky, A.L. Krylov, Modulational instability and formation of a nonlinear oscillatory structure in a focusing medium, Phys. Lett. A 77 (993) [] R.F. Bikbaev, V.R. Kudashev, Eample of shock waves in unstable media: the focusing nonlinear Schrödinger equation, Phys. Lett. A 90 (994) 55 60; R.F. Bikbaev, V.R. Kudashev, Eample of shock waves in unstable media: the focusing nonlinear Schrödinger equation, Erratum A 96 (994) 454.

11 A.M. Kamchatnov et al. / Wave Motion 38 (003) [] J.C. Bronski, D.W. McLaughlin, Semiclassical behavior in the NLS equation: optical shock focusing instabilities, in: N.M. Ercolani, et al. (Eds.), Singular Limit of Dispersion Waves, Plenum Press, New York, 994. [3] J.C. Bronski, Semiclassical eigenvalue distribution of the Zakharov Shabat eigenvalue problem, Physica D 97 (996) [4] P.D. Miller, S. Kamvissis, On the semiclassical limit of the focusing nonlinear Schrödinger equation, Phys. Lett. A 47 (998) [5] J.C. Bronski, J.N. Kutz, Numerical simulation of the semiclassical limit of the focusing nonlinear Schrödinger equation, Phys. Lett. A 54 (999) [6] P.D. Miller, Some remarks on a WKB method for the nonselfadjoint Zakharov Shabat eigenvalue problem, Physica D 5 (00) [7] J.C. Bronski, Spectral instability of the semiclassical Zakharov shabat eigenvalue problem, Physica D 5 (00) [8] A.M. Kamchatnov, Nonlinear Periodic Waves and Their Modulations An Introductory Course, World Scientific, Singapore, 000. [9] V.I. Karpman, An asymptotic solution of the Korteweg de Vries equation, Phys. Lett. A 5 (967) [30] V.I. Karpman, Nonlinear Waves in Dispersive Media, Pergamon Press, Oford, 975. [3] A.M. Kamchatnov, R.A. Kraenkel, B.A. Umarov, On asymptotic solutions of integrable wave equations, Phys. Lett. A 87 (00) 3 3. [3] D.J. Kaup, Higher-order water-wave equation and method for solving it, Progr. Theor. Phys. 64 (976) [33] G.A. El, R.H.J. Grimshaw, M.V. Pavlov, Integrable shallow-water equations and undular bores, Stud. Appl. Math. 06 (00) [34] V.B. Matveev, M.I. Yavor, Almost periodical solutions of nonlinear hydrodynamic equation of Kaup, Ann. Inst. H. Poincaré, Sect. A 3 (979) 5 4. [35] L.D. Landau, E.M. Lifshitz, Quantum Mechanics, Pergamon Press, London, 959.