Periodic waves of the modified KdV equation as minimizers of a new variational problem
Exactly Solvable and Integrable Systems
2021-12-13 v1 Mathematical Physics
Analysis of PDEs
Dynamical Systems
math.MP
Pattern Formation and Solitons
Abstract
Periodic waves of the modified Korteweg-de Vries (mKdV) equation are identified in the context of a new variational problem with two constraints. The advantage of this variational problem is that its non-degenerate local minimizers are stable in the time evolution of the mKdV equation, whereas the saddle points are unstable. We explore the analytical representation of periodic waves given by Jacobi elliptic functions and compute numerically critical points of the constrained variational problem. A broken pitchfork bifurcation of three smooth solution families is found. Two families represent (stable) minimizers of the constrained variational problem and one family represents (unstable) saddle points.
Keywords
Cite
@article{arxiv.2112.05231,
title = {Periodic waves of the modified KdV equation as minimizers of a new variational problem},
author = {Uyen Le and Dmitry E. Pelinovsky},
journal= {arXiv preprint arXiv:2112.05231},
year = {2021}
}
Comments
17 pages; 9 figures;