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.
Analytical and numerical methods for the CMKdV-II equation.
Darboux transformation for classical acoustic spectral problem.
Global attraction to solitary waves for Klein-Gordon equation with mean field interaction
Global attraction to solitary waves in models based on the Klein-Gordon equation.
Integrability and diffeomorphisms on target space.
Invariant manifolds for one-dimensional parabolic partial differential equations of second order
Mechanical analogy for the wave-particle: Helix on a vortex filament.
Multi solitary waves for nonlinear Schrödinger equations
-wave equations with orthogonal algebras: and reductions and soliton solutions.
New exact traveling wave solutions for compound KdV-Burgers equations in mathematical physics.
New explicit and exact solutions for the Nizhnik-Novikov-Vesselov equation.
New solitons and periodic solutions for Thekadomtsev-Petviashvili equation.
On asymptotic stability of solitary waves for nonlinear Schrödinger equations
On Numerical Solution of the Gardner–Ostrovsky Equation
A simple explicit numerical scheme is proposed for the solution of the Gardner–Ostrovsky equation (ut + cux + α uux + α1u2ux + βuxxx)x = γu which is also known as the extended rotation-modified Korteweg–de Vries (KdV) equation. This equation is used for the description of internal oceanic waves affected by Earth’ rotation. Particular versions of this equation with zero some of coefficients, α, α1, β, or γ are also known in numerous applications....
On planar selfdual electroweak vortices
On Spectral Stability of Solitary Waves of Nonlinear Dirac Equation in 1D⋆⋆
We study the spectral stability of solitary wave solutions to the nonlinear Dirac equation in one dimension. We focus on the Dirac equation with cubic nonlinearity, known as the Soler model in (1+1) dimensions and also as the massive Gross-Neveu model. Presented numerical computations of the spectrum of linearization at a solitary wave show that the solitary waves are spectrally stable. We corroborate our results by finding explicit expressions for...
On the local spectral radius in partially ordered Banach spaces
On the long time behavior of KdV type equations
In a series of recent papers, Martel and Merle solved the long-standing open problem on the existence of blow up solutions in the energy space for the critical generalized Korteweg- de Vries equation. Martel and Merle introduced new tools to study the nonlinear dynamics close to a solitary wave solution. The aim of the talk is to discuss the main ideas developed by Martel-Merle, together with a presentation of previously known closely related results.