A Semiclassical Approach to the Nonlocal Nonlinear Schrödinger Equation with a Non-Hermitian Term

Abstract

The nonlinear Schrödinger equation (NLSE) with a non-Hermitian term is the model for various phenomena in nonlinear open quantum systems. We deal with the Cauchy problem for the nonlocal generalization of multidimensional NLSE with a non-Hermitian term. Using the ideas of the Maslov method, we propose the method of constructing asymptotic solutions to this equation within the framework of semiclassically concentrated states. The semiclassical nonlinear evolution operator and symmetry operators for the leading term of asymptotics are derived. Our approach is based on the solutions of the auxiliary dynamical system that effectively linearizes the problem under certain algebraic conditions. The formalism proposed is illustrated with the specific example of the NLSE with a non-Hermitian term that is the model of an atom laser. The analytical asymptotic solution to the Cauchy problem is obtained explicitly for this example.

Quasiparticles for the one-dimensional nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov equation

“…The case of a single quasiparticle in the FKPP equation can also be treated as a particular case of work [45] in a manner. We are encouraged to generalized the method [45] for few quasiparticles using ideas of this work.…”

Numerical Simulation of Soliton Propagation Behavior for the Fractional-in-Space NLSE with Variable Coefficients on Unbounded Domain

“…Moritz Braun [17] numerically solved the one-dimensional fractional-order NLSE. A semiclassical method is used by Kulagin and Shapovalov [18] to solve the NLSE with a non-Hermitian term. Aldhafeeri and Nuwairan [19] gave some solutions for the fractional-in-time-modified NLSE.…”