Semiclassical approach to the nonlocal nonlinear Schrödinger equation with a non-Hermitian term

Abstract

The nonlinear Schödinger 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 linearize 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.

32 pages, 2 figure

References in corpus (6)

• Observation of vortices and vortex stripes in a dipolar Bose-Einstein condensate
• (Semi)classical limit of the Hartree equation with harmonic potential
• Variational approach to multidimensional solitons in highly nonlocal nonlinear media
• On Nonlocal Gross-Pitaevskii Equations with Periodic Potentials
• Family of asymptotic solutions to the two-dimensional kinetic equation with a nonlocal cubic nonlinearity
• On Meijer's $G$ function $G_{p,p}^{m,n}$ for $m+n=p$

Cited by in corpus (2)

• Quasiparticles for the one-dimensional nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov equation
• Quasiparticle solutions for the nonlocal NLSE with an anti-Hermitian term in semiclassical approximation