NONLINEAR SCHRÖDINGER EQUATION OF GENERAL FORM: MULTIFUNCTIONAL MODEL, REDUCTIONS AND EXACT SOLUTIONS
https://doi.org/10.26583/vestnik.2025.1.3
EDN: DXKPEC
Abstract
A new mathematical model based on the nonlinear Schrödinger equation with six arbitrary functions and allowing for various factors is presented. This multifunctional model is a broad generalization of numerous simpler related nonlinear models that are commonly encountered in various areas of theoretical physics, including nonlinear optics, superconductivity, and plasma physics. To analyze the nonlinear equation under consideration, a combination of the method of functional constraints and methods of generalized separation of variables is used. One-dimensional non-symmetry reductions are described, which lead the studied complex partial differential equation to simpler ordinary differential equations or systems of such equations. A number of exact solutions of the nonlinear Schrödinger equation of general form have been found, which are expressed in quadratures or elementary functions. Both periodic solutions in time and in spatial variable are obtained. Special attention is paid to some narrower classes of nonlinear PDEs with a smaller number of arbitrary functions. The described general multifunctional model allows one to effectively analyze numerous simpler models by specifying a specific particular forms of arbitrary functions. The exact solutions obtained in this work can be used as test problems intended to check the adequacy and assess the accuracy of numerical and approximate analytical methods for integrating nonlinear equations of mathematical physics.
Keywords
nonlinear Schrödinger equation,
general nonlinear PDEs,
nonlinear optics,
exact solutions,
solutions in quadratures,
generalized separable solutions,
non-symmetry reductions
Supporting agencies: The work was carried out on the topics of the state assignment (state registration no. 124012500440-9 and FSWU-2023-0031).
About the Authors
A. D. Polyanin
Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences
Russian Federation
Russian Federation
N. A. Kudryashov
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)
Russian Federation
Russian Federation
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Review
For citations:
Polyanin A.D., Kudryashov N.A. NONLINEAR SCHRÖDINGER EQUATION OF GENERAL FORM: MULTIFUNCTIONAL MODEL, REDUCTIONS AND EXACT SOLUTIONS. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2025;14(1):24-36. (In Russ.) https://doi.org/10.26583/vestnik.2025.1.3. EDN: DXKPEC
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