Breathers and the dynamics of solutions to the KdV type equations

Abstract

In this paper our first aim is to identify a large class of non-linear functions $f(\cdot )$ for which the IVP for the generalized Korteweg-de Vries equation does not have breathers or "small" breathers solutions. Also we prove that all small, uniformly in time $L^1\cap H^1$ bounded solutions to KdV and related perturbations must converge to zero, as time goes to infinity, locally in an increasing-in-time region of space of order $t^{1/2}$ around any compact set in space. This set is included in the linearly dominated dispersive region $x\ll t$. Moreover, we prove this result independently of the well-known supercritical character of KdV scattering. In particular, no standing breather-like nor solitary wave structures exists in this particular regime.

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Cited by in corpus (5)

• Exponentially fitted methods that preserve conservation laws
• Variational methods for breather solutions of Nonlinear Wave Equations
• On small breathers of nonlinear Klein-Gordon equations via exponentially small homoclinic splitting
• Full family of flattening solitary waves for the mass critical generalized KdV equation
• Robustness of mKdV Breathers: A Numerical Perspective