https://doi.org/10.1140/epjp/s13360-025-06693-3
Regular Article
Painlevé analysis, conservation laws and exact solutions of the fourth-order nonlinear Schrödinger equation
1
Department of Applied Mathematics National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), 31 Kashirskoe Shosse, 115409, Moscow, Russian Federation
2
College of Optical and Electrical Engineering Zhejiang Agriculture and Forestly University, 311300, Li’an, China
3
Research Group of Nonlinear Optical Science and Technology, School of Mathematical and Physical Sciences, Wuhan Textile University, 430200, Wuhan, China
Received:
10
December
2024
Accepted:
26
July
2025
Published online:
13
August
2025
The fourth-order nonlinear Schrödinger equation for describing optical solitons is studied. The Painlevé test for nonlinear partial differential equations is used to study integrability properties of equation. It is shown that the equation does not pass the Painlevé test and consequently the Cauchy problem cannot be solved by the inverse scattering transform. A condition is found for the parameter value of the equation at which an analytical solution is possible in traveling wave variables. Using direct algebraic transformations of the system of equations, three independent conservation laws are constructed for the equation under study. The first integrals of the reduction to the nonlinear ordinary differential equation are obtained from the conservation laws. Optical solitons described by the partial differential equation are found under additional conditions on the parameters of the equation. Conserved densities corresponding to the power, linear momentum, and energy of optical solitons are calculated.
Copyright comment Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2025
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
