U. Le and D.E. Pelinovsky
Periodic waves of the modified KdV equation
as minimizers of a new variational problem
SIAM J. Applied Dynamical Systems 21 (2022) 2518-2534
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:
Modified Korteweg-de Vries equation, traveling periodic waves, varatiational characterization,
stability of periodic waves.