Shape analysis of bounded traveling wave solutions and solution to the generalized Whitham-Broer-Kaup equation with dissipation terms

Abstract

This paper deals with the problem of the bounded traveling wave solutions’ shape and the solution to the generalized Whitham-Broer-Kaup equation with the dissipation terms which can be called WBK equation for short. The authors employ the theory and method of planar dynamical systems to make comprehensive qualitative analyses to the above equation satisfied by the horizontal velocity component u(ξ) in the traveling wave solution (u(ξ), H(ξ)), and then give its global phase portraits. The authors obtain the existent conditions and the number of the solutions by using the relations between the components u(ξ) and H(ξ) in the solutions. The authors study the dissipation effect on the solutions, find out a critical value r*, and prove that the traveling wave solution (u(ξ),H(ξ)) appears as a kink profile solitary wave if the dissipation effect is greater, i.e., |r| ≥ r*, while it appears as a damped oscillatory wave if the dissipation effect is smaller, i.e., |r| < r*. Two solitary wave solutions to the WBK equation without dissipation effect is also obtained. Based on the above discussion and according to the evolution relations of orbits corresponding to the component u(ξ) in the global phase portraits, the authors obtain all approximate damped oscillatory solutions (ũ(ξ), \tilde H(ξ)) under various conditions by using the undetermined coefficients method. Finally, the error between the approximate damped oscillatory solution and the exact solution is an infinitesimal decreasing exponentially.

Keywords

Generalized Whitham-Broer-Kaup equation / Shape analysis / Solitary wave solution / Damped oscillatory solution / Error estimate

References

[1] Whitham G. B.. Variational methods and applications to water waves. Proc. R. Soc. A, 1967, 299: 6-25

[2] Broer L. J.. Approximate equations for long water waves. Appl. Sci. Res., 1975, 31: 377-395

[3] Sachs R. L.. On the integrable variant of the Boussinesq system: Painlevé property, rational solutions, a related many-body system, and equivalence with the AKNS hierarchy. Phys. D, 1988, 30: 1-27

[4] Kaup D. J.. A higher-order water-wave equation and the method for solving it. Progr. Theoret. Phys., 1975, 54: 396-408

[5] Ablowitz M. J., Clarkson P. A.. Solitons, Nonlinear Evolution Equations and Inverse Scattering, 1991, Cambridge: Cambridge University Press

[6] Kupershmidt B. A.. Mathematics of dispersive water waves. Comm. Math. Phys., 1985, 99: 51-73

[7] Wang M. L.. Solitary wave solitions for variant Boussinesq equations. Phys. Lett. A, 1995, 199: 169-172

[8] Wang M. L., Zhou Y. B., Li Z. B.. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Phys. Lett. A, 1996, 216: 67-75

[9] Fan E. G., Zhang H. Q.. Backlund transformation and exact solution for Whitham-Broer-Kaup equations in shallow water. Appl. Math. Mech., 1998, 19: 667-670

[10] Yan Z. Y., Zhang H. Q., Fan E. G.. New explicit and travelling wave solutions for a class of nonlinear evolution equations. Acta. Phys. Sin., 1999, 48: 1962-1967

[11] Salah M. E., Dogan K.. Exact and numerical traveling wave solutions of Whitham-Broer-Kaup equations. Appl. Math. Comput., 2005, 167: 1339-1349

[12] Rafei M., Daniali H.. Application of the variational iteration method to the Whitham-Broer-Kaup equations. Comp. Math. Appl., 2007, 54: 1079-1085

[13] Whitham G. B.. Linear and Nonlinear Wave, 1974, New York: Wiley

[14] Nemytskii V., Stepanov V.. Qualitative Theory of Differential Equations, 1989, New York: Dover

[15] Aronson D. G., Weiberger H. F.. Multidimentional nonlinear diffusion arising in population genetics. Adv. Math., 1978, 30: 33-76

[16] Fife P. C.. Mathematical Aspects of Reacting and Diffusing Systems, 1979, New York: Springer-Verlag

[17] Ye Q. X., Li Z. Y.. Introduction of Reaction Diffusion Equations (in Chinese), 1990, Beijing: Science Press

[18] Zhang W. G., Chang Q. S., Jiang B. G.. Explicit exact solitary-wave solitons for compound KdV-type and compound KdV-burgers-type equations with nonlinear terms of any order. Chaos Solitons Fractals, 2002, 13: 311-319

[19] Benney D. J.. Long waves on liquid films. J. Math. Phys., 1996, 45: 150-155

[20] Grad H., Hu P. N.. Unified shock profile in a plasma. Phys. Fluids, 1967, 10: 2596-2602

[21] Bona J. L., Schonbek M. E.. Travelling wave solutions to the Korteweg-de Vries Burgers equation. Proc. Roy. Soc. Edinburgh Sect. A, 1985, 101: 207-226

[22] Johnson R. S.. A nonlinear incorporating damping and dispersion. J. Fluid Mech., 1970, 42: 49-60

[23] Johnson R. S.. Shallow water waves on a viscous fluid—the undular bore. Phys. Fluids, 1972, 15: 1693-1699

[24] Wijngaarden L. V.. On the motion of gas bubbles in a perfect fluid. Annu. Rev. Fluid Mech., 1972, 4: 369-373

[25] Hu P. N.. Collisional theory of shock and nonlinear waves in a plasma. Phys. Fluids, 1972, 15: 854-864

[26] Canosa J., Gazdag J.. The Korteweg-de Vries Burgers equation. J. Comput. Phy., 1977, 23: 393-403

[27] Guan K. Y., Gao G.. Qualitative analysis of traveling wave solutions of mixed Burgers-KdV equation. Sci. China Ser. A, 1987, 30: 64-73

[28] Xiong S. L.. A class of analytical solutions of Burgers-KdV equation. Chinese Sci. Bull., 1989, 1: 26-29

[29] Liu S. D., Liu S. K.. KdV-Burgers equation modelling of turbulence. Sci. China Ser. A, 1992, 35: 576-586

[30] Benjamin T. B., Bona J. L., Mahony J. J.. Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lon. Ser. A, 1972, 272: 47-78

[31] Bona J. L., Dougalia V. A.. An initial- and boundary-value problem for a model equation for propagation of long waves. J. Math. Anal. Appl., 1980, 75: 513-522

[32] Dood R. K.. Solitons and Nonlinear Wave Equations, 1982, London: Academic Press Inc. Ltd.

[33] Ablowitz M. J., Segur H.. Solitons and the Inverse Scattering Transform, 1981, Philiadelphia: SIAM

[34] Zhou Y. B., Wang M. L., Wang Y. M.. Periodic wave solutions to a coupled KdV equations with variable coefficients. Phys. Lett. A, 2003, 308: 31-36

[35] Chen Y. X., Lu X. H.. Spatiotemporal similaritons in (3+1)-dimensional inhomogeneous nonlinear medium with cubic-quintic nonlinearity. Commun. Theor. Phys., 2011, 55: 871-877

[36] Lai X. J.. Chirped waves for a generalized (2+1)-dimensional nonlinear Schrödinger equation. Commun. Theor. Phys., 2011, 55: 555-559

[37] Dai C. Q., Wang X. G., Zhang J. F.. Nonautonomous spatiotemporal localized structures in the inhomogeneous optical fibers: interaction and control. Ann. Phys., 2011, 326: 645-656

[38] Dai C. Q., Wang Y. Y., Lang C. J.. Analytic investigation on the similariton transmission control in the dispersion decreasing fiber. Optics Commun., 2011, 284(13): 3440-3444

[39] Wu J. P.. A generalized hirota ansatz to obtain soliton-like solutions for a (3+1)-dimensional nonlinear evolution equation. Commun. Theor. Phys., 2011, 56: 297-300

[40] Liu W. J., Tian B., Xu T. Bright and dark solitons in the normal dispersion regime of inhomogeneous optical fibers: soliton interaction and soliton control. Ann. Phys., 2011, 325(8): 1633-1643

[41] Gambo B., Kuetche K. V., Bouetou B. T. Travelling wave solutions to stretched beams equation: phase portraits survey. Commun. Theor. Phys., 2011, 55: 605-608

[42] Maccari A.. Chaotic and fractal patterns for interacting nonlinear waves. Chaos Solitons Fractals, 2010, 43: 86-95