Long-time dynamics of variable coefficient modified Korteweg-de Vries solitary waves

2005/03/08 by S. I. Dejak, B. L. G. Jonsson · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #math-ph #math.MP #msc:35K40 #msc:35Q53

paper · pdf · doi:10.1063/1.2217809

published as J. Math. Phys. 47, 072703 (2006) · 19 pages

arxiv created 2005/03/08 · openalex publication_date 2006/07/01 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We study the long-time behavior of solutions to the Korteweg-de Vries-type equation ∂tu=−∂x(∂x2u+f(u)−b(t,x)u), with initial conditions close to a stable, b=0 solitary wave. The coefficient b is a bounded and slowly varying function, and f is a nonlinearity. For a restricted class of nonlinearities, we prove that for long time intervals, such solutions have the form of the solitary wave, whose center and scale evolve according to a certain dynamical law involving the function b(t,x), plus an H1(R)-small fluctuation. The result is stronger than those previously obtained for general nonlinearities f.

Citations

• Solitary Wave Dynamics in an External Potential
• Existence of solitary waves in higher dimensions
• Asymptotic dynamics of nonlinear Schrödinger equations: Resonance‐dominated and dispersion‐dominated solutions
• Nonlinear scalar field equations, I existence of a ground state
• The stability of solitary waves
• Long-Time Dynamics of KdV Solitary Waves over a Variable Bottom

Cited by

• Effective dynamics for boson stars