Traveling periodic waves and breathers in the nonlocal derivative NLS equation

A nonlocal derivative nonlinear Schrödinger equation describes modulations of waves in a stratified fluid and a continuous limit of the Calogero–Moser–Sutherland system of particles. For the defocusing version of this equation, we prove the linear stability of the nonzero constant background for decaying and periodic perturbations and the nonlinear stability for periodic perturbations. For the focusing version of this equation, we prove the linear stability of the nonzero constant background under a non-resonance condition on the initial data and the nonlinear stability for sufficiently small periods. For both versions, we characterize the traveling periodic wave solutions by using Hirota’s bilinear method, both on the nonzero and zero backgrounds. For each family of traveling periodic waves, we construct families of breathers which describe solitary waves moving across the periodic background. A general breather solution with N solitary waves propagating on the periodic background is derived in a closed determinant form.