D.E. Pelinovsky and M. Plum
Dynamics of black solitons in a regularized nonlinear Schrodinger equation
Proceedings of AMS 152 (2024) 1217-1231
Abstract:
We consider a family of regularized defocusing nonlinear Schrodinger (NLS) equations
proposed in the context of the cubic NLS equation with a bounded dispersion relation.
The time evolution is well-posed if the black soliton is perturbed by a small perturbation in
the Sobolev space Hs(R) with s > 1/2. We prove that the black soliton is spectrally stable
(unstable) if the regularization parameter is below (above) some explicitly specified threshold.
We illustrate the stable and unstable dynamics of the perturbed black solitons by using the
numerical finite-di erence method. The question of orbital stability of the black soliton is left
open due to the mismatch of the function spaces for the energy and momentum conservation.
Keywords:
nonlinear Schrodinger equation; dark solitons; local well-posedness; energetic stability; spectral stability; conserved quanitites;