J. Chen and D.E. Pelinovsky
Traveling periodic waves and breathers in the nonlocal derivative NLS equation
Abstract:
A nonlocal derivative NLS (nonlinear Schrodinger) 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 linear and nonlinear stability of the nonzero constant background
under some restrictions. 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 stable background. A general breather solution
with N solitary waves propagating on the traveling periodic wave background is derived
in a closed determinant form.
Keywords:
nonlocal derivative NLS equation; traveling periodic wave; multi-periodic solutions;
multi-soliton solutions; Lax spectrum; breathers.