A further improved tanh function method exactly solving the -dimensional dispersive long wave equations.
A generalization for Peakon's solitary wave and Camassa-Holm equation.
A look on some results about Camassa–Holm type equations
We present an overview of some contributions of the author regarding Camassa--Holm type equations. We show that an equation unifying both Camassa--Holm and Novikov equations can be derived using the invariance under certain suitable scaling, conservation of the Sobolev norm and existence of peakon solutions. Qualitative analysis of the two-peakon dynamics is given.
A new conservative finite difference scheme for Boussinesq paradigm equation
A family of nonlinear conservative finite difference schemes for the multidimensional Boussinesq Paradigm Equation is considered. A second order of convergence and a preservation of the discrete energy for this approach are proved. Existence and boundedness of the discrete solution on an appropriate time interval are established. The schemes have been numerically tested on the models of the propagation of a soliton and the interaction of two solitons. The numerical experiments demonstrate that the...
A nonlinear Schrödinger equation with magnetic field effect: solitary waves with internal rotation
A -scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shadow water system
The goal of this paper is to construct a first-order upwind scheme for solving the system of partial differential equations governing the one-dimensional flow of two superposed immiscible layers of shallow water fluids. This is done by generalizing a numerical scheme presented by Bermúdez and Vázquez-Cendón [3, 26, 27] for solving one-layer shallow water equations, consisting in a -scheme with a suitable treatment of the source terms. The difficulty in the two layer system comes from the coupling...
A Q-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system
The goal of this paper is to construct a first-order upwind scheme for solving the system of partial differential equations governing the one-dimensional flow of two superposed immiscible layers of shallow water fluids. This is done by generalizing a numerical scheme presented by Bermúdez and Vázquez-Cendón [3, 6, 27] for solving one-layer shallow water equations, consisting in a Q-scheme with a suitable treatment of the source terms. The difficulty in the two layer system comes from the coupling...
Adiabatic Evolution of Coupled Waves for a Schrödinger-Korteweg-de Vries System
The effective dynamics of interacting waves for coupled Schrödinger-Korteweg-de Vries equations over a slowly varying random bottom is rigorously studied. One motivation for studying such a system is better understanding the unidirectional motion of interacting surface and internal waves for a fluid system that is formed of two immiscible layers. It was shown recently by Craig-Guyenne-Sulem [1] that in the regime where the internal wave has a large...
An interpretation of the Vakulenko-Kapitanskiĭ estimate.
Andrew Lenard: a mystery unraveled.
Applications of an extended -expansion method to find exact solutions of nonlinear PDEs in mathematical physics.
Asymptotic stability of solitary waves for nonlinear Schrödinger equations
Auto-Bäcklund transformation and exact analytical solutions for the Kupershmidt equation.