Influence of dissipation on solitary wave solution to generalized Boussinesq equation

doi:10.1007/s10483-023-2954-8

摘要/Abstract

摘要： This paper uses the theory of planar dynamic systems and the knowledge of reaction-diffusion equations, and then studies the bounded traveling wave solution of the generalized Boussinesq equation affected by dissipation and the influence of dissipation on solitary waves. The dynamic system corresponding to the traveling wave solution of the equation is qualitatively analyzed in detail. The influence of the dissipation coefficient on the solution behavior of the bounded traveling wave is studied, and the critical values that can describe the magnitude of the dissipation effect are, respectively, found for the two cases of $b_{3}<0$ and $b_{3}>0$ in the equation. The results show that, when the dissipation effect is significant (i.e., $r$ is greater than the critical value in a certain situation), the traveling wave solution to the generalized Boussinesq equation appears as a kink-shaped solitary wave solution; when the dissipation effect is small (i.e., $r$ is smaller than the critical value in a certain situation), the traveling wave solution to the equation appears as the oscillation attenuation solution. By using the hypothesis undetermined method, all possible solitary wave solutions to the equation when there is no dissipation effect (i.e., $r=0$) and the partial kink-shaped solitary wave solution when the dissipation effect is significant are obtained; in particular, when the dissipation effect is small, an approximate solution of the oscillation attenuation solution can be achieved. This paper is further based on the idea of the homogenization principles. By establishing an integral equation reflecting the relationship between the approximate solution of the oscillation attenuation solution and the exact solution obtained in the paper, and by investigating the asymptotic behavior of the solution at infinity, the error estimate between the approximate solution of the oscillation attenuation solution and the exact solution is obtained, which is an infinitesimal amount that decays exponentially. The influence of the dissipation coefficient on the amplitude, frequency, period, and energy of the bounded traveling wave solution of the equation is also discussed.

关键词: generalized Boussinesq equation, influence of dissipation, qualitative analysis, solitary wave solution, oscillation attenuation solution, error estimation

Abstract: This paper uses the theory of planar dynamic systems and the knowledge of reaction-diffusion equations, and then studies the bounded traveling wave solution of the generalized Boussinesq equation affected by dissipation and the influence of dissipation on solitary waves. The dynamic system corresponding to the traveling wave solution of the equation is qualitatively analyzed in detail. The influence of the dissipation coefficient on the solution behavior of the bounded traveling wave is studied, and the critical values that can describe the magnitude of the dissipation effect are, respectively, found for the two cases of $b_{3}<0$ and $b_{3}>0$ in the equation. The results show that, when the dissipation effect is significant (i.e., $r$ is greater than the critical value in a certain situation), the traveling wave solution to the generalized Boussinesq equation appears as a kink-shaped solitary wave solution; when the dissipation effect is small (i.e., $r$ is smaller than the critical value in a certain situation), the traveling wave solution to the equation appears as the oscillation attenuation solution. By using the hypothesis undetermined method, all possible solitary wave solutions to the equation when there is no dissipation effect (i.e., $r=0$) and the partial kink-shaped solitary wave solution when the dissipation effect is significant are obtained; in particular, when the dissipation effect is small, an approximate solution of the oscillation attenuation solution can be achieved. This paper is further based on the idea of the homogenization principles. By establishing an integral equation reflecting the relationship between the approximate solution of the oscillation attenuation solution and the exact solution obtained in the paper, and by investigating the asymptotic behavior of the solution at infinity, the error estimate between the approximate solution of the oscillation attenuation solution and the exact solution is obtained, which is an infinitesimal amount that decays exponentially. The influence of the dissipation coefficient on the amplitude, frequency, period, and energy of the bounded traveling wave solution of the equation is also discussed.

Key words: generalized Boussinesq equation, influence of dissipation, qualitative analysis, solitary wave solution, oscillation attenuation solution, error estimation

中图分类号:

Weiguo ZHANG, Siyu HONG, Xingqian LING, Wenxia LI. Influence of dissipation on solitary wave solution to generalized Boussinesq equation[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(3): 477-498.

参考文献

[1] BOUSSINESQ, J. Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquid contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. Journal de Mathématiques Pures et Appliquées, 17, 55-108(1872)
[2] ZHAKAROV, V. E. On stochastization of one-dimensional chains of nonlinear oscillators. Soviet Physics--JETP, 38(1), 108-110(1974)
[3] MCKEAN, H. P. Boussinesq's equation on the circle. Communications on Pure and Applied Mathematics, 34(5), 599-691(1981)
[4] WEISS, J., TABOR, M., and CARNEVALE, G. The Painlevé property for partial differential equations. Journal of Mathematical Physics, 24(3), 522-526(1983)
[5] WEISS, J. The Painlevé property for the partial differential equations, II: Bäcklund transformation, Lax pairs, and the Schwarzian derivative. Journal of Mathematical Physics, 24(6), 1405-1413(1983)
[6] WEISS, J. The Painlevé property and Bäcklund transformations for the sequence of Boussinesq equations. Journal of Mathematical Physics, 26(2), 258-269(1985)
[7] MANORANJAN, V. S., ORTEGA, T., and SANZ-SERNA, J. M. Soliton and antisoliton interaction in the "good" Boussinesq equation. Journal of Mathematical Physics, 29(9), 1964-1968(1988)
[8] ZAKHAROV, V. E. The Inverse Scattering Method, Springer, Berlin/Heidelberg, 243-285(1980)
[9] CLARKSON, P. New exact solutions of the Boussinesq equation. European Journal of Applied Mathematics, 1(3), 279-300(1990)
[10] BONA, J. L. and SACHS, R. L. Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation. Communications in Mathematical Physics, 118(1), 15-29(1988)
[11] LIU, Y. C. and XU, R. Z. Global existence and blow up of solutions for Cauchy problem of generalized Boussinesq equation. Physica D: Nonlinear Phenomena, 237(6), 721-731(2008)
[12] LIN, Q., WU, Y., and LOXTON, R. On the Cauchy problem for a generalized Boussinesq equation. Journal of Mathematical Analysis and Applications, 353(1), 186-195(2009) R. Hirota, Exact N-soliton solutions of the wave equation of long waves in shallow-water and in nonlinear lattices, Journal of Mathematical Physics. 14(7), (1973)
[13] WHITHAM, G. B. Linear and nonlinear waves. Pure and Applied Mathematics, Wiley, U.S.A. (1999)
[14] VARLAMOV, V. V. On the Cauchy problem for the damped Boussinesq equation. Differential Integral Equations, 9(3), 619-634(1996)
[15] POLAT, N., KAYA, D., and TUTALAR, H. I. Blow-up of solutions for the damped Boussinesq equation. Zeitschrift fur Naturforschung A, 60(7), 473-476(2005)
[16] CLARKA, H. R., COUSINB, A. T., FROTAB, C. L., and LIMACO, J. On the dissipative Boussinesq equation in a non-cylindrical domain. Nonlinear Analysis}: Theory, Methods and Applications, 67(8), 2321-2334(2007)
[17] YANG, H. W., YIN, B. S., and SHI, Y. L. Forced dissipative Boussinesq equation for solitary waves excited by unstable topography. Nonlinear Dynamics, 70(2), 1389-1396(2012)
[18] ZHANG, Z. F., DING, T. R., HUANG, W. Z., and DONG, Z. X. it Qualitative Theory of Differential Equations, American Mathematical Society, U.S.A. (1992)
[19] NEMYTSKII, V. and STEPANOV, V. Qualitative theory of differential equations. The Mathematical Gazette, 46(356), 159(1962)
[20] ARONSON, D. G. and WEIBERGER, H. F. Multidimensional nonlinear diffusion arising in population genetics. Advances in Mathematics, 30(1), 33-76(1978)
[21] FIFE, P. C. Mathematical aspects of reacting and diffusing systems. Lecture Notes in Biomathematics, Springer-Verlag, New York (1979)
[22] YE, Q. X., LI, Z. Y., WANG, M. X., and WU, Y. P. Introduction to Reaction Diffusion Equations, 2nd ed., Science Press, Beijing (1990)
[23] ZHANG, W., CHANG, Q., and JIANG, B. Explicit exact solitary-wave solutions for compound KdV-type and compound KdV-Burgers-type equations with nonlinear terms of any order. Chaos, Solitons and Fractals, 13(2), 311-319(2002)
[24] MEI, C. C. The Applied Dynamics of Ocean Surface Waves, 2nd ed., World Scientific Publishing Company, Singapore (1989)
[25] DEAN, R. G. and DALRYMPLE, R. A. Water wave mechanics for engineers and scientists. Advanced Series on Ocean Engineering, World Scientific Publishing Company, Singapore (1991)