Nonlinear stability of periodic waves in the Korteweg-de Vries equation under localized perturbations

Abstract

We investigate the stability and asymptotic behavior of spatially periodic cnoidal waves in the Korteweg-de Vries equation subject to localized perturbations. Standard stability arguments in Hamiltonian systems break down in this setting, since localized perturbations preclude a characterization of stable periodic waves as strict minimizers of a suitable energy functional subject to finitely many constraints. As a result, the nonlinear stability of periodic waves under localized perturbations has remained a long-standing open problem in Hamiltonian systems, with previous results only addressing plane waves that can be reduced to constant states by passing to polar coordinates. In this paper, we develop a novel method that resolves this obstruction by combining variational arguments, Floquet-Bloch theory, and Duhamel-based estimates with spatiotemporal modulation. Our framework applies to general periodic waves in Hamiltonian systems with symmetry and reduces the nonlinear stability problem to verifying diffusive spectral stability conditions for the second variation of a suitable conserved energy. Applying our approach to cnoidal waves in the Korteweg-de Vries equation, we obtain the first nonlinear stability result for periodic waves in Hamiltonian systems under localized perturbations that cannot be reduced to constant states.