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

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 propo...

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Bibliographic Details
Published in:Mathematics (Basel) 2024-02, Vol.12 (4), p.580
Main Authors: Kulagin, Anton E., Shapovalov, Alexander V.
Format: Article
Language:English
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
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Summary: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.
ISSN:2227-7390
2227-7390
DOI:10.3390/math12040580