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

• Published in: Mathematics (Basel) Vol. 12; no. 4; p. 580
• Main Authors: Kulagin, Anton E., Shapovalov, Alexander V.
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01-02-2024
• Subjects: Analysis; Approximation; Asymptotic methods; asymptotic solution; atom laser; Atom lasers; Atoms; Cauchy problems; dissipation; Maslov’s complex germ method; Methods; Numerical analysis; open quantum systems; Optics; Partial differential equations; Phase transitions; Schrodinger equation; semiclassically concentrated solutions; Russia
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math12040580

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.