Strong Interactions between Solitary Waves Belonging to Different Wave Modes

Abstract

Strong interactions between weakly nonlinear long waves are studied. Strong interactions occur when the linear long wave phase speeds are nearly equal although the waves belong to different modes. Specifically we study this situation in the context of internal wave modes propagating in a density stratified fluid. The interaction is described by two coupled Korteweg-deVries equations, which possess both dispersive and nonlinear coupling terms. It is shown that the coupled equations possess an exact analytical solution involving the characteristic" sech 2 "profile of the Korteweg-deVries equation. It is also shown that when the coefficients satisfy some special conditions, the coupled equations possess an n-soliton solution analogous to the Korteweg-deVries n-soliton solution. In general though the coupled equations are found not to be amenable to solution by the inverse scattering transform technique, and thus a numerical method has been employed in order to find solutions. This method is described in detail in Appendix A. Several numerical solutions of the coupled equations are presented.

Coupled system of Korteweg–de Vries equations type in domains with moving boundaries

“…Now we present some numerical results for the solution of the coupled system of equations described in [16,15] which is the same system (2.1)-(2.4) with the extra term rv…”

“…For this, we consider a semi-implicit finite difference scheme based on unconditionally stable schemes similar to the one described in [10,30] for the KdV and the KdV-Kawahara equations. Our method is different from the one presented in [16,15] where the authors used Fourier transform for the x-variable. The advantage of our method is that it is better adapted to our nonperiodic boundary conditions, and to the treatment of the system with variable coefficients in the cylindrical domain given by the problem with moving boundaries.…”

Complete integrability and the Miura transformation of a coupled KdV equation

“…The coupled KdV equations have many applications in several physical fields, such as those of shallow stratified liquids [1,2], atmospheric dynamical systems [3], Bose-Einstein condensates [4], and so on. In [5], Lou et al proved that the coupled KdV equation…”

Duality relation among the Hamiltonian structures of a parametric coupled Korteweg-de Vries system

“…Here and in the sequel u and v belong to the real Schwartz space de ned by The system (1),(2) for λ = − describes a two-layer liquid model studied in references [2][3][4]22]. It is a very interesting evolution system.…”

“…Hirota and Satsuma Alvaro Restuccia: Departamento de Física, Universidad de Antofagasta, E-mail: arestu@usb.ve Alvaro Restuccia: Departamento de Física, Universidad Simón Bolívar, E-mail: arestu@usb.ve *Corresponding Author: Adrián Sotomayor: Departamento de Matemáticas, Universidad de Antofagasta, E-mail: adrian.sotomayor@uantof.cl [1] proposed a model that describes interactions of two long waves with di erent dispersion relations. Gear and Grimshaw [2,3] considered a coupled KdV system to describe linearly-stable internal waves in a strati ed uid.…”