Stability of periodic waves for the fractional KdV and NLS equations

Abstract

We consider the focusing fractional periodic Korteweg-deVries (fKdV) and fractional periodic nonlinear Schrödinger 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 traveling 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 traveling wave solution to fKdV and fNLS $\phi :\Vert \phi {\Vert }_{L^2[-T,T]}^2=\lambda$, which is non-degenerate and spectrally stable, as well as orbitally stable. This is done completely rigorously, without any {\it a priori} assumptions on the smoothness of the waves or the Lagrange multipliers.

References in corpus (4)

• Uniqueness and Nondegeneracy of Ground States for $(-\Delta {)}^{s}Q+Q-Q^{\alpha +1}=0$ in $\mathbb{R}$
• Local well-posedness and blow up in the energy space for a class of L2 critical dispersion generalized Benjamin-Ono equations
• A para-differential renormalization technique for nonlinear dispersive equations
• Nondegeneracy and Stability of Antiperiodic Bound States for Fractional Nonlinear Schrödinger Equations

Cited by in corpus (1)

• Green's function for the fractional KdV equation on the periodic domain via Mittag-Leffler's function