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.
Bilinear estimates related to the KP equations
We survey some recent results for the KP-II equation. We also give an idea for treating the “bad frequency interactions” of the bilinear estimates in the Fourier transform restriction spaces related to the KP-I equation.
Blow up and near soliton dynamics for the critical gKdV equation
These notes present the main results of [22, 23, 24] concerning the mass critical (gKdV) equation for initial data in close to the soliton. These works revisit the blow up phenomenon close to the family of solitons in several directions: definition of the stable blow up and classification of all possible behaviors in a suitable functional setting, description of the minimal mass blow up in , construction of various exotic blow up rates in , including grow up in infinite time.
Blow up for the critical gKdV equation. II: Minimal mass dynamics
We consider the mass critical (gKdV) equation for initial data in . We first prove the existence and uniqueness in the energy space of a minimal mass blow up solution and give a sharp description of the corresponding blow up soliton-like bubble. We then show that this solution is the universal attractor of all solutions near the ground state which have a defocusing behavior. This allows us to sharpen the description of near soliton dynamics obtained in [29].
Blow-up solutions and strong instability of standing waves for the generalized Davey-Stewartson system in
Constructing soliton and kink solutions of PDE models in transport and biology.
Construction de solutions pour les équations de Korteweg-de Vries généralisées
Darboux transforms of Dupin surfaces
We present a Möbius invariant construction of the Darboux transformation for isothermic surfaces by the method of moving frames and use it to give a complete classification of the Darboux transforms of Dupin surfaces.