F. Natali, U. Le and D.E. Pelinovsky
Periodic waves in the fractional modified Korteweg-de Vries equation
Journal of Dynamics and Differential Equations 34 (2022) 1601-1640
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 L4 norm: the sign-definite
solutions arise in the subspace of even functions, whereas the sign-indefinite solutions are
obtained in the subspace of odd functions. Morse index is computed for both solutions
and the spectral stability criterion in the evolution of the mKdV equation is obtained. 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.
Keywords:
Fractional modified Korteweg-de Vries equation, traveling periodic waves, varatiational characterization,
stability of periodic waves.