Stability of periodic waves for the fractional KdV and NLS equations

We consider the focussing fractional periodic Korteweg-deVries (fKdV) and fractional periodic non-linear Schrodinger equations (fNLS) equations, with L-2 sub-critical dispersion. In particular, this covers the case of the periodic KdV and Benjamin-Ono models. We construct two parameter family of bell-shaped travelling waves for KdV (standing waves for NLS), which are constrained minimizers of the Hamiltonian. We show in particular that for each lambda > 0, there is a travelling wave solution to fKdV and fNLS phi :parallel to phi parallel to(2)(L2[-T, T]) = lambda, which is non-degenerate. We also show that the waves are spectrally stable and orbitally stable, provided the Cauchy problem is locally well-posed in H-alpha/2[-T, T] and a natural technical condition. This is done rigorously, without any a priori assumptions on the smoothness of the waves or the Lagrange multipliers.

Automated summary

This study focuses on the fractional periodic KdV and fractional periodic non-linear Schrodinger equations, constructing a family of bell-shaped travelling wave solutions that are non-degenerate, spectrally stable, and orbitally stable. The results are rigorously established without any a priori assumptions on the smoothness of the waves or the Lagrange multipliers.