Breather lattice and its stabilization for the modified Korteweg–de Vries equation

Abstract

We obtain an exact solution for the breather lattice solution of the modified Korteweg-de Vries (MKdV) equation. Numerical simulation of the breather lattice demonstrates its instability due to the breather-breather interaction. However, such multi-breather structures can be stabilized through the concurrent application of ac driving and viscous damping terms.

On the variational structure of breather solutions I: Sine-Gordon equation

“…We rigorously show that these breathers satisfy a suitable elliptic equation, and we also show numerical spectral stability. However, we also identify the source of nonlinear instability in the case described in [41], and we conjecture that, even if spectral stability is satisfied, nonlinear stability/instability depends only on the sign of a suitable discriminant function, a condition that is trivially satisfied in the case of non-periodic breathers. Finally, we present a new class of breather solution for mKdV, believed to exist from geometric considerations, and which is periodic in time and space, but has nonzero mean, unlike standard breathers.…”

“…Using numerical methods, we confirm that all spectral assumptions leading to the H 2 × H 1 stability of SG breathers are numerically satisfied, even in the ultra-relativistic, singular regime. In a second part, we study the periodic mKdV case, where a periodic breather is known from the work of Kevrekidis et al [41]. We rigorously show that these breathers satisfy a suitable elliptic equation, and we also show numerical spectral stability.…”

Nonlinear dispersion relations

“…and is a model that appears in the context of ion acoustic solitons, van Alfvén waves in collisionless plasma, Schottky barrier transmission lines, models of traffic congestion as well as phonons in anharmonic lattices among others (see, e.g., [5] and …”

Focusing mKdV Breather Solutions with Nonvanishing Boundary Condition by the Inverse Scattering Method

“…First results on the existence of periodic breathers were obtained by Kevrekidis, Khare and Saxena in [13,14]. In these works, they showed regular and singular periodic breathers for the focusing and defocusing mKdV by using suitable ansätze with free parameters to be adjusted.…”