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 Schr¨odinger equation (NLSE) in a nonlocal, nonlinear, self-interaction potential of the form [ |ψ(x, t)|2 +|ψ(−x, t)|2] κ where κ is an arbitrary nonlinearity parameter. Although the system with κ = 1 (i.e., fully integrable case) was first reported by Yang (Phys. Rev. E 98 (2018) 042202), in the present work, we extend this model to the one in which κ 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 κ 6= 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 < κ < 2 and unstable when κ > 2. At the critical value of κ, i.e. κ = 2, the solution can either collapse or blowup linearly in time when q0 = 0, where q0 is the center of the initial density ρ(x, t = 0) = ψ*ψ of the solution. For q0 6= 0 the displaced solution collapses. When κ > 2 initial small displacements from the origin also lead to collapse of the wave function. This phenomenon is not seen in the usual NLSE.