Periodic waves in the fractional modified Korteweg--de Vries equation

TL;DR

This paper investigates periodic wave solutions of the fractional modified Korteweg-de Vries equation, characterizing their stability, bifurcations, and energy minimization properties in both local and fractional cases.

Contribution

It extends the analysis of periodic solutions to the fractional mKdV equation, including stability, bifurcation, and energy minimization in the fractional setting.

Findings

Sign-definite solutions exhibit a fold bifurcation at lower regularity.

Sign-indefinite solutions undergo symmetry-breaking bifurcation.

Spectral stability criterion is derived for both solution types.

Abstract

Periodic waves in the modified Korteweg-de Vries (mKdV) equation are revisited in the setting of the fractional Laplacian. Two families of solutions in the local case are given by the sign-definite dnoidal and sign-indefinite cnoidal solutions. Both solutions can be characterized in the general fractional case as global minimizers of the quadratic part of the energy functional subject to the fixed $L^4$ norm: the sign-definite (sign-indefinite) solutions are obtained in the subspace of even (odd) functions. Morse index is computed for both solutions and the spectral stability criterion is derived. We show numerically that the family of sign-definite solutions has a generic fold bifurcation for the fractional Laplacian of lower regularity and the family of sign-indefinite solutions has a generic symmetry-breaking bifurcation both in the fractional and local cases.