Nonlinear stability of mKdV breathers

Abstract

Breather solutions of the modified Korteweg-de Vries equation are shown to be globally stable in a natural H^2 topology. Our proof introduces a new Lyapunov functional, at the H^2 level, which allows to describe the dynamics of small perturbations, including oscillations induced by the periodicity of the solution, as well as a direct control of the corresponding instability modes. In particular, degenerate directions are controlled using low-regularity conservation laws.

24 pp., submitted

References in corpus (2)

• The Gardner equation and the L^2-stability of the N-soliton solution of the Korteweg-de Vries equation
• On the variational structure of breather solutions

Cited by in corpus (21)

• Data-driven soliton solutions and model parameters of nonlinear wave models via the conservation-law constrained neural network method
• Nonexistence of small, odd breathers for a class of nonlinear wave equations
• Breathers and the dynamics of solutions to the KdV type equations
• Dynamics of complex-valued modified KdV solitons with applications to the stability of breathers
• Stability of elliptic function solutions for the focusing modified KdV equation
• The multi elliptic-localized solutions and their asymptotic behaviors for the mKdV equation
• Counting unstable eigenvalues in Hamiltonian spectral problems via commuting operators
• Scattering in the energy space for Boussinesq equations
• On the asymptotic stability of the sine-Gordon kink in the energy space
• Variational methods for breather solutions of Nonlinear Wave Equations
• Soliton resolution for the modified KdV equation
• The scattering problem for the $abcd$ Boussinesq system in the energy space
• On the variational structure of breather solutions
• Multisolitons for the cubic NLS in 1-d and their stability
• Stability and instability of breathers in the $U(1)$ Sasa-Satusuma and Nonlinear Schrödinger models
• Global solutions and stability properties of the 5th order Gardner equation
• Nonlinear stability of higher order mKdV breathers
• Dynamics of small solutions in KdV type equations: decay inside the linearly dominated region
• Stability of the multi-solitons of the modified Korteweg-de Vries equation
• Robustness of mKdV Breathers: A Numerical Perspective
• Non dispersive solutions of the generalized KdV equations are typically multi-solitons