Stability of trapped solutions of a nonlinear Schrodinger equation with a nonlocal nonlinear self-interaction potential

This work focuses on the study of the stability of trapped soliton-like solutions of a (1 + 1)-dimensional nonlinear Schrodinger equation (NLSE) in a nonlocal, nonlinear, self-interaction potential of the form [|psi(x,t)vertical bar(2)+vertical bar psi(-x,t)vertical bar(2)](kappa) where kappa is an arbitrary nonlinearity parameter. Although the system with kappa = 1 (i.e. fully integrable case) was first reported by Yang (2018 Phys. Rev. E 98 042202), in the present work, we extend this model to the one in which kappa is arbitrary. This allows us to compare the stability properties of the now trapped solutions to previously found solutions of the more usual NLSE with kappa not equal 1 which are moving soliton solutions. We show that there is a simple, one-component, nonlocal Lagrangian and corresponding action governing the dynamics of the system. Using a collective coordinate method derived from the action as well as assuming the validity of Derrick's theorem, we find that these trapped solutions are stable for 0 < kappa < 2 and unstable when kappa > 2. At the critical value of kappa, i.e. kappa = 2, the solution can either collapse or blowup linearly in time when q(0) = 0, where q(0) is the center of the initial density rho(x, t = 0) = psi(star)psi of the solution. For q(0) not equal 0 the displaced solution collapses. When kappa > 2 initial small displacements from the origin also lead to collapse of the wave function. This phenomenon is not seen in the usual NLSE.

Automated summary

This work focuses on studying the stability of trapped soliton-like solutions in a (1 + 1)-dimensional nonlinear Schrodinger equation with a nonlocal, nonlinear, self-interaction potential. The researchers compare the stability properties of these trapped solutions to previously found solutions with a different parameter and show that the trapped solutions are stable when the parameter is between 0 and 2, and unstable when the parameter exceeds 2.