Stability of exact solutions of a nonlocal and nonlinear Schrödinger equation with arbitrary nonlinearity
This work focuses on the study of solitary wave solutions to a nonlocal, nonlinear Schrödinger system in $1$+$1$ dimensions with arbitrary nonlinearity parameter $κ$. Although the system we study here was first reported by Yang (Phys. Rev. E, 98 (2018), 042202) for the fully integrable case $κ=1$, we extend its considerations and offer criteria for soliton stability and instability as a function of $κ$. In particular, we show that for $κ<2$ the solutions are stable whereas for $κ>2$ they are subject to collapse or blowup. At the critical point of $κ=2$, there is a critical mass necessary for blowup or collapse. Furthermore, we show there is a simple one-component nonlocal Lagrangian governing the dynamics of the system which is amenable to a collective coordinate approximation. To that end, we introduce a trial wave function with two collective coordinates to study the small oscillations around the exact solution. We obtain analytical expressions for the small oscillation frequency for the width parameter in the collective coordinate approximation. We also discuss a four collective coordinate approximation which in turn breaks the symmetry of the exact solution by allowing for translational motion. The ensuing oscillations found in the latter case capture the response of the soliton to a small translation. Finally, our results are compared with numerical simulations of the system.