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We consider the $L^{2}$ -critical focusing non-linear Schrödinger equation in $1 + 1$ -d. We demonstrate the existence of a large set of initial data close to the ground state soliton resulting in the pseudo-conformal type blow-up behavior. More specifically, we prove a version of a conjecture of Perelman, establishing the existence of a codimension one stable blow-up manifold in the measurable category.

Nonlinear stability and convergence of finite-differnce methods for the "good" Boussinesq equation.

J.M. Sanz-Serna, T. Ortega (1990/1991)

Numerische Mathematik

Nontrivial solitary waves of GKP equation in multi-dimensional spaces.

Xuan, Benjin (2003)

Revista Colombiana de Matemáticas

Note on coupled linear systems related to two soliton collision for the quartic gKdV equation.

Yvan Martel, Frank Merle (2008)

Revista Matemática Complutense

Numerical comparisons of two long-wave limit models

Stéphane Labbé, Lionel Paumond (2004)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The Benney-Luke equation (BL) is a model for the evolution of three-dimensional weakly nonlinear, long water waves of small amplitude. In this paper we propose a nearly conservative scheme for the numerical resolution of (BL). Moreover, it is known (Paumond, Differential Integral Equations 16 (2003) 1039–1064; Pego and Quintero, Physica D 132 (1999) 476–496) that (BL) is linked to the Kadomtsev-Petviashvili equation for almost one-dimensional waves propagating in one direction. We study here numerically...

Numerical comparisons of two long-wave limit models

Stéphane Labbé, Lionel Paumond (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The Benney-Luke equation (BL) is a model for the evolution of three-dimensional weakly nonlinear, long water waves of small amplitude. In this paper we propose a nearly conservative scheme for the numerical resolution of (BL). Moreover, it is known (Paumond, Differential Integral Equations16 (2003) 1039–1064; Pego and Quintero, Physica D132 (1999) 476–496) that (BL) is linked to the Kadomtsev-Petviashvili equation for almost one-dimensional waves propagating in one direction. We study here numerically...

Numerical computation of solitons for optical systems

Laurent Di Menza (2009)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper, we present numerical methods for the determination of solitons, that consist in spatially localized stationary states of nonlinear scalar equations or coupled systems arising in nonlinear optics. We first use the well-known shooting method in order to find excited states (characterized by the number $k$ of nodes) for the classical nonlinear Schrödinger equation. Asymptotics can then be derived in the limits of either large $k$ are large nonlinear exponents $σ$ . In a second part, we compute...

Numerical computation of solitons for optical systems

Laurent Di Menza (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we present numerical methods for the determination of solitons, that consist in spatially localized stationary states of nonlinear scalar equations or coupled systems arising in nonlinear optics. We first use the well-known shooting method in order to find excited states (characterized by the number k of nodes) for the classical nonlinear Schrödinger equation. Asymptotics can then be derived in the limits of either large k are large nonlinear exponents σ. In a second part, we compute...

Numerical investigation for solitary solutions of the Boussinesq equation.

Al-Khaled, Kamel, Nusier, Ameina S. (2008)

Applied Mathematics E-Notes [electronic only]

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