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.

In this paper, a generalized nonlinear Schrödinger-type equation with higher-order nonlocal nonlinearities and Kudryashov’s arbitrary refractive index is studied. A wide range of physical phenomena, such as nonlinear optics, plasma dynamics, and wave propagation in dispersive media, are modeled by this equation. By applying a traveling wave transformation, the system is reduced to a singular planar dynamical system, which is subsequently regularized to facilitate a comprehensive bifurcation analysis. The equilibrium points are classified based on system parameters, and corresponding phase portraits are constructed to illustrate the qualitative dynamics across various bifurcation scenarios. The unified Riccati equation expansion method and the sine-Gordon expansion method are used to obtain explicit analytical soliton solutions, which are expressed in trigonometric and hyperbolic forms and capture a range of wave structures with different physical properties. Graphic representations in 2D and 3D are presented to illustrate the propagation dynamics. The results contribute to the current solution landscape of nonlocal nonlinear systems and provide new information on the Hamiltonian structure and bifurcation behavior of such singular wave models. To the best of our knowledge, the dynamical classification and exact solutions presented here are new and have not been published before.

In this study, we explore the bifurcation and optical soliton solutions in weakly nonlocal media with cubic-quintic nonlinearity, which are significant for understanding optical soliton propagation in nonlocal nonlinear systems. The cubic-quintic nonlinear Schrödinger equation, including weak nonlocality, is introduced to model the evolution of soliton trains in optical fibers under the influence of a nonlocal medium. Using a traveling wave transformation, the equation is reduced to a singular dynamical system and further transformed into a regular dynamical system through changing variables. The study confirms the equivalence of the first integrals for both systems and provides a detailed analysis of phase portraits, emphasizing their geometric and topological features. Additionally, the unified Riccati equation expansion method is applied to derive exact solutions, including periodic, dark, and singular soliton solutions. 2D and 3D graphical representations of the solutions are presented to illustrate their physical properties, with parameters chosen to highlight the effects of cubic-quintic nonlinearity and weak nonlocality. These findings offer insights into the dynamics of nonlinear wave propagation in optical systems and contribute to advancements in nonlocal nonlinear optics and soliton theory.

In this work, we investigate the periodic solutions for non-linear system of differential equations by using the method of periodic solutions of ordinary differential equations which are given by A.M.Samoilenko. Additionally, the existence and uniqueness theorem have been proved for second differential equations system by using Banach fixed point theorem.

In this paper, we give a reconstruction formula for the potential q for a second order differential equation with boundary condition which contains spectral parameter. For this as methodology, we use Prufer substitution ¨ that has an advantage different from other methods. Because in this method, we do not need any information of eigenfunctions.

The shallow water equations have a wide range of applications in the ocean, atmospheric modeling, and pneumatic computing, which can also be used to modeling flows in rivers and coastal areas. In this study, we build the analytic traveling wave solution of the (1+1) dimensional coupled Whitham-Broer-Kaup (WBK) equations, by using the Bernoulli sub-equation function method. The system of (1+1)-dimensional (CWBK) partial differential equation is converted to the ordinary differential equation for obtaining new exponential prototype structures. We obtained new results using this technique. We plotted two and three-dimensional surfaces of the results using Wolfram Mathematica software. At the end of this study, we submitted a conclusion in a comprehensive manner.