Breathing patterns in nonlinear relaxation

Abstract

In numerical experiments involving nonlinear solitary waves propagating through nonhomogeneous media one observes "breathing" in the sense of the amplitude of the wave going up and down on a much faster scale than the motion of the wave. In this paper we investigate this phenomenon in the simplest case of stationary waves in which the evolution corresponds to relaxation to a nonlinear ground state. The particular model is the popular ${\delta }_0$ impurity in the cubic nonlinear Schroedinger equation on the line. We give asymptotics of the amplitude on a finite but relevant time interval and show their remarkable agreement with numerical experiments. We stress the nonlinear origin of the "breathing patterns" caused by the selection of the ground state depending on the initial data, and by the non-normality of the linearized operator.

References in corpus (6)

• Resonances, Radiation Damping and Instability in Hamiltonian Nonlinear Wave Equations
• Solitary Wave Dynamics in an External Potential
• Fast soliton scattering by delta impurities
• Enhanced Quantum Reflection of Matter-Wave Solitons
• Soliton splitting by external delta potentials
• Dynamics of Nonlinear Schrodinger / Gross-Pitaevskii Equations; Mass Transfer in Systems with Solitons and Degenerate Neutral Modes

Cited by in corpus (8)

• Stability of small solitary waves for the 1$d$ NLS with an attractive delta potential
• Initial boundary value problem for the focusing NLS equation with Robin boundary condition: half-line approach
• Long-Time Asymptotics for Solutions of the NLS Equation with a Delta Potential and Even Initial Data
• Fifty Years of KdV: An Integrable System
• Modified scattering for the $1d$ cubic NLS with a repulsive delta potential
• Mathematical Study of Scattering Resonances
• On stability of small solitons of the 1--D NLS with a trapping delta potential
• Differential Equations of Quantum Mechanics