Nondegeneracy and Stability of Antiperiodic Bound States for Fractional Nonlinear Schrödinger Equations

Abstract

We consider the existence and stability of real-valued, spatially antiperiodic standing wave solutions to a family of nonlinear Schrödinger equations with fractional dispersion and power-law nonlinearity. As a key technical result, we demonstrate that the associated linearized operator is nondegenerate when restricted to antiperiodic perturbations, i.e. that its kernel is generated by the translational and gauge symmetries of the governing evolution equation. In the process, we provide a characterization of the antiperiodic ground state eigenfunctions for linear fractional Schrödinger operators on $\mathbb{R}$ with real-valued, periodic potentials as well as a Sturm-Liouville type oscillation theory for the higher antiperiodic eigenfunctions.

46 pages, 2 figures

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Cited by in corpus (3)

• New variational characterization of periodic waves in the fractional Korteweg-de Vries equation
• Green's function for the fractional KdV equation on the periodic domain via Mittag-Leffler's function
• Stability of periodic waves for the fractional KdV and NLS equations