Breathers and the Dynamics of Solutions in KdV Type Equations

Abstract

In this paper our first aim is to identify a large class of non-linear functions f (·) 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 ∩ 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 ≪ 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.

Asymptotic Behavior of Solutions of the Dispersion Generalized Benjamin–Ono Equation

“…Remark 1.3. We recall that in [21] for the case of the KdV a similar result was established in a space region with lesser growth but with the whole limit as t ↑ ∞ instead of the limit infimum.…”

“…Remark 1.1. As in [21] our approach was inspired by the works of Kowalczyk, Martel and Muñoz [14]- [15] concerning the decay of solutions in 1 + 1 dimensional scalar field models.…”

On the Dynamics of Zero-Speed Solutions for Camassa–Holm-Type Equations

“…A similar proof works for the DP case. See [74] for similar proofs in the gKdV case. However, Theorem 1.2 not only proves nonexistence of zero speed solutions, but also proves local in space decay to zero of such entities.…”

“…The proof of Theorem 1.2 is based in elementary techniques employed in [74] (see also [75]) for the gKdV equation ∂ t u + ∂ x (∂ 2…”

“…As a consequence, we also show scattering and decay in CH type equations with long range nonlinearities. Our proof relies in the introduction of suitable Virial functionalsà la Martel-Merle in the spirit of the works [74,75] and [50] adapted to CH, DP and BBM type dynamics, one of them placed in L 1x , and a second one in the energy space H 1 x . Both functionals combined lead to local in space decay to zero in |x| t 1/2− as t → +∞.…”