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Blow up and near soliton dynamics for the L 2 critical gKdV equation

Yvan Martel, Frank Merle, Pierre Raphaël (2011/2012)

Séminaire Laurent Schwartz — EDP et applications

These notes present the main results of [22, 23, 24] concerning the mass critical (gKdV) equation u t + ( u x x + u 5 ) x = 0 for initial data in H 1 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 H 1 , construction of various exotic blow up rates in H 1 , including grow up in infinite time.

Blow up for the critical gKdV equation. II: Minimal mass dynamics

Yvan Martel, Frank Merle, Pierre Raphaël (2015)

Journal of the European Mathematical Society

We consider the mass critical (gKdV) equation u t + ( u x x + u 5 ) x = 0 for initial data in H 1 . 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].

L 2 -stability of multi-solitons

Claudio Muñoz (2011/2012)

Séminaire Laurent Schwartz — EDP et applications

The aim of this note is to give a short review of our recent work (see [5]) with Miguel A. Alejo and Luis Vega, concerning the L 2 -stability, and asymptotic stability, of the N -soliton of the Korteweg-de Vries (KdV) equation.

On the numerical solutions of some fractional ordinary differential equations by fractional Adams-Bashforth-Moulton method

Haci Mehmet Baskonus, Hasan Bulut (2015)

Open Mathematics

In this paper, we apply the Fractional Adams-Bashforth-Moulton Method for obtaining the numerical solutions of some linear and nonlinear fractional ordinary differential equations. Then, we construct a table including numerical results for both fractional differential equations. Then, we draw two dimensional surfaces of numerical solutions and analytical solutions by considering the suitable values of parameters. Finally, we use the L2 nodal norm and L∞ maximum nodal norm to evaluate the accuracy...

Solitons and Gibbs Measures for Nonlinear Schrödinger Equations

K. Kirkpatrick (2012)

Mathematical Modelling of Natural Phenomena

We review some recent results concerning Gibbs measures for nonlinear Schrödinger equations (NLS), with implications for the theory of the NLS, including stability and typicality of solitary wave structures. In particular, we discuss the Gibbs measures of the discrete NLS in three dimensions, where there is a striking phase transition to soliton-like behavior.

Stabilité des solitons de l’équation de Landau-Lifshitz à anisotropie planaire

André de Laire, Philippe Gravejat (2014/2015)

Séminaire Laurent Schwartz — EDP et applications

Cet exposé présente plusieurs résultats récents quant à la stabilité des solitons sombres de l’équation de Landau-Lifshitz à anisotropie planaire, en particulier, quant à la stabilité orbitale des trains (bien préparés) de solitons gris [16] et à la stabilité asymptotique de ces mêmes solitons [2].

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