Linear stability of traveling periodic waves in the nonlocal derivative NLS equations

Establish the linear stability, with respect to small perturbations, of the traveling periodic wave solutions of the nonlocal derivative nonlinear Schrödinger equations i u_t = u_{xx} + σ u (i + H) (|u|^2)_x for σ ∈ {+1, −1}, where H denotes the Hilbert transform. Specifically, for all parameter regimes in which these traveling periodic waves exist (including the nonzero-background waves in both the defocusing and focusing cases and the zero-background waves in the focusing case), show that small perturbations of the traveling periodic wave remain bounded uniformly in time in appropriate function spaces.

Background

The paper constructs and analyzes traveling periodic waves for both the defocusing (σ = +1) and focusing (σ = −1) versions of the nonlocal derivative NLS (NDNLS) equations, using Hirota’s bilinear method. It further derives the Lax spectrum and exact breather solutions propagating on these periodic backgrounds and observes no rogue-wave phenomena on the nonzero constant background.

While linear and nonlinear stability of the nonzero constant background is established under various conditions, the authors do not prove the stability of the traveling periodic wave backgrounds themselves. Motivated by the observed steady propagation of solitary waves on these backgrounds and the absence of modulational instability in their explicit constructions, the authors conjecture that the traveling periodic wave is linearly stable with respect to small perturbations and leave this question for future paper.