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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MDPI AG
2024-02-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/12/4/580 |
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| author | Anton E. Kulagin Alexander V. Shapovalov |
| author_facet | Anton E. Kulagin Alexander V. Shapovalov |
| author_sort | Anton E. Kulagin |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-07T22:23:17Z |
| format | Article |
| id | doaj.art-ec8c93208ef4496d855737cbfcbf0f0a |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-07T22:23:17Z |
| publishDate | 2024-02-01 |
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| series | Mathematics |
| spelling | doaj.art-ec8c93208ef4496d855737cbfcbf0f0a2024-02-23T15:26:12ZengMDPI AGMathematics2227-73902024-02-0112458010.3390/math12040580A Semiclassical Approach to the Nonlocal Nonlinear Schrödinger Equation with a Non-Hermitian TermAnton E. Kulagin0Alexander V. Shapovalov1Division for Electronic Engineering, Tomsk Polytechnic University, 30 Lenina av., 634050 Tomsk, RussiaDepartment of Theoretical Physics, Tomsk State University, 1 Novosobornaya Sq., 634050 Tomsk, RussiaThe 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.https://www.mdpi.com/2227-7390/12/4/580semiclassically concentrated solutionsMaslov’s complex germ methodopen quantum systemsasymptotic solutiondissipationatom laser |
| spellingShingle | Anton E. Kulagin Alexander V. Shapovalov A Semiclassical Approach to the Nonlocal Nonlinear Schrödinger Equation with a Non-Hermitian Term Mathematics semiclassically concentrated solutions Maslov’s complex germ method open quantum systems asymptotic solution dissipation atom laser |
| title | A Semiclassical Approach to the Nonlocal Nonlinear Schrödinger Equation with a Non-Hermitian Term |
| title_full | A Semiclassical Approach to the Nonlocal Nonlinear Schrödinger Equation with a Non-Hermitian Term |
| title_fullStr | A Semiclassical Approach to the Nonlocal Nonlinear Schrödinger Equation with a Non-Hermitian Term |
| title_full_unstemmed | A Semiclassical Approach to the Nonlocal Nonlinear Schrödinger Equation with a Non-Hermitian Term |
| title_short | A Semiclassical Approach to the Nonlocal Nonlinear Schrödinger Equation with a Non-Hermitian Term |
| title_sort | semiclassical approach to the nonlocal nonlinear schrodinger equation with a non hermitian term |
| topic | semiclassically concentrated solutions Maslov’s complex germ method open quantum systems asymptotic solution dissipation atom laser |
| url | https://www.mdpi.com/2227-7390/12/4/580 |
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