Soliton interactions for the extended Korteweg-de Vries (KdV) equation are examined. It is shown that the extended KdV equation can be transformed (to its order of approximation) to a higher-order member of the KdV hierarchy of integrable equations. This transformation is used to derive the higher-order, two-soliton solution for the extended KdV equation. Hence it follows that the higher-order ...

We use a spectral method to solve numerically two nonlocal, nonlinear, dispersive, integrable wave equations, the Benjamin-Ono and the Intermediate Long Wave equations. The proposed numerical method is able to capture well the dynamics of the solutions; we use it to investigate the behaviour of solitary wave solutions of the equations with special attention to those, among the properties usuall...

We investigate envelope solitary waves on square with two degrees of freedom and nonlinear nearest and next-nearest neighbor interactions. We consider solitary waves which are localized in the direction of their motion and periodically modulated along the perpendicular direction. In the quasi– monochromatic approximation and low–amplitude limit a system of two coupled nonlinear Schrödinger equa...

The stability of the bright solitary wave solution to the perturbed cubicquintic Schrödinger equation is considered. It is shown that in a certain region of parameter space these solutions are unstable, with the instability being manifested as a small positive eigenvalue. Furthermore, it is shown that in the complimentary region of parameter space there are no small unstable eigenvalues. The pr...

We consider systems of equations which arise in modelling strong interactions of weakly nonlinear long waves in dispersive media. For a certain class of such systems, we prove the existence and stability of localized solutions representing coupled solitary waves travelling at a common speed. Our results apply in particular to the systems derived by Gear and Grimshaw and by Liu, Kubota, and Ko a...

The complete analytical solution of the governing nonlinear vector equations is found which describes periodic and solitary propagating-rotating waves in an initially helical fiber. A detailed description of various types of these waves is given. The solitary wave velocity is found to be proportional to the square root of the amplitude of the internal force, and the effective wave length is sho...

We establish soliton-like asymptotics for finite energy solutions to the Schrödinger equation coupled to a nonrelativistic classical particle. Any solution with initial state close to the solitary manifold, converges to a sum of traveling wave and outgoing free wave. The convergence holds in global energy norm. The proof uses spectral theory and the symplectic projection onto solitary manifold ...

We consider some Boussinesq systems of water wave theory, which are of coupled KdV type. After a brief review of the theory of existence-uniqueness of solutions of the associated initial-value problems, we turn to the numerical solution of their initialand periodic boundary-value problems by unconditionally stable, highly accurate methods that use Galerkin/finite element type schemes with perio...

[1] A sequence of three internal solitary waves of elevation were observed propagating shoreward along a near-bottom density interface over Oregon’s continental shelf. These waves are highly turbulent and coincide with enhanced optical backscatter, consistent with increased suspended sediments in the bottom boundary layer. Nonlinear solitary wave solutions are employed to estimate wave speeds a...

We establish a long time soliton asymptotics for a nonlinear system of wave equation coupled to a charged particle. The coupled system has a six-dimensional manifold of soliton solutions. We show that in the large time approximation, any solution, with an initial state close to the solitary manifold, is a sum of a soliton and a dispersive wave which is a solution to the free wave equation. It i...