https://doi.org/10.1140/epjp/s13360-025-07256-2
Regular Article
Phase plane dynamics and exact soliton solutions of a modified Ginzburg–Landau equation with multiple nonlinear laws
Department of Mathematics, College of Science, University of Zakho, Zakho, Iraq
a
This email address is being protected from spambots. You need JavaScript enabled to view it.
Received:
6
November
2025
Accepted:
22
December
2025
Published online:
19
January
2026
Abstract
In this study, a generalized form of the complex Ginzburg–Landau equation incorporating various nonlinear response laws namely, the Kerr law, power law, parabolic, and dual-power nonlinearities is examined. By applying a suitable traveling wave transformation, the governing partial differential equation is reduced to a nonlinear ordinary differential equation. After additional variable shifts, the system is expressed in Hamiltonian form, from which the corresponding first integral is derived to analyze the equilibrium structure. The qualitative behavior of the system is investigated by constructing phase portraits under different parameter regimes, revealing the existence and stability characteristics of both singular and symmetric equilibria. Subsequently, the modified Kudryashov method is employed to obtain several new classes of exact solutions for the considered nonlinear models. These solutions are presented in hyperbolic, exponential, and rational function forms, corresponding to bright, dark, singular, kink, and anti-kink wave profiles. To illustrate the physical features and propagation dynamics of the obtained solutions, two-dimensional and three-dimensional graphical representations are plotted for appropriate parameter choices. The analytical and graphical results demonstrate the effectiveness of the proposed approach in describing nonlinear wave evolution across diverse nonlinear optical media.
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 2026
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.
