In this paper, we investigate the generalized coupled Zakharov system (GCZS), a fundamental model in plasma physics that describes the nonlinear interaction between high-frequency Langmuir waves and low-frequency ion-acoustic waves, including the influence of magnetic fields on weak ion-acoustic wave propagation. This research aims to achieve three main objectives. First, we uncover soliton solutions of the coupled system in hyperbolic, trigonometric, and rational forms, both in single and combined expressions. These results are obtained using the extended rational sinh-Gordon expansion method and the $\left({\displaystyle \frac{G^{\prime }}{G}},{\displaystyle \frac{1}{G}}\right)$-expansion method. Second, we analyze the dynamic characteristics of the model by performing bifurcation and sensitivity analyses and identifying the corresponding Hamiltonian function. To understand the mechanisms of intricate physical phenomena and dynamical processes, we plot 2D, 3D, and contour diagrams for appropriate parameter values. We also analyze the bifurcation of phase portraits of the ordinary differential equations corresponding to the investigated partial differential equation. The novelty of this study lies in the fact that the proposed model has not been previously explored using these advanced methods and comprehensive dynamical analyses.

This study investigates soliton solutions and dynamic wave structures in the complex Ginzburg-Landau (CGL) equation, which is crucial for understanding wave propagation in various physical systems. We employ three analytical methods: the Kumar-Malik method, the generalized Arnous method, and the energy balance method to derive novel closed-form solutions. These solutions exhibit diverse solitonic phenomena, including multi-wave solitons, complex solitons, singular solitons, periodic oscillating waves, dark-wave, and bright-wave profiles. Our results reveal new families of exact solitary waves via the generalized Arnous method and diverse soliton solutions through the Kumar-Malik method, including hyperbolic, trigonometric, and Jacobi elliptic functions. Verification is ensured through back-substitution to the considered model using Mathematica software. Additionally, we plot the various graphs with the appropriate parametric values under the influence of the M-truncated fractional derivative to visualize the solution behaviors with varying parameter values. This research contributes significantly to understanding wave dynamics in physical oceanography, and the unique outcomes explored in this research will play a vital role for the forthcoming investigation of nonlinear equations.

The purpose of this study is to construct diverse forms of exact soliton solutions and conduct a comprehensive qualitative analysis. For this aim, we use the Gross–Pitaevskii system, which belongs to the family of nonlinear Schrödinger equations. This model is considered to be iconic and significant because it has potential applications in applied sciences, such as in physics, where it is used to exemplify quantum systems like Bose–Einstein condensates and illustrate the propagation of waves in optical fibers. Employing analytical techniques, the modified sine–cosine/sinh–cosh and extended rational sinh–Gordon expansion methods, we extract several waves from solutions in the shape of trigonometric, hyperbolic, and rational forms. To further deepen our insights related to the system’s behavior, we execute a detailed dynamical analysis, including sensitivity, bifurcation, and chaos, using the corresponding Hamiltonian structure. We also derive the instability modulation using linear stability theory. Using Mathematica, we systematically simulate and verify all constructed results and present some solutions for appropriate parameter values using 2D, 3D, and contour plots. The outcomes provide fruitful insights relevant to multiple scientific domains, including optical fiber technology, plasma, and condensed matter physics. This work contributes to the ongoing study of nonlinear models by applying novel solution techniques and offering a broader perspective on the complex behavior of such systems. The novelty of this study lies in the fact that the proposed model has not been previously explored using the aforementioned advanced methods and comprehensive dynamical analyses.

This work investigates the coupled Ivancevic option-pricing model, a nonlinear wave alternative to the Black–Scholes model. By utilizing the recently developed Kumar-Malik method, modified Sardar sub-equation method and the generalized Arnous method, the substantial results of this research are the successful derivation of novel exact soliton solutions, including bright, singular, dark, combined dark–bright, singular-periodic, complex solitons, exponential and Jacobi elliptic functions. A detailed analysis of option price wave functions and modulation instability analysis is conducted, with the conditions for valid solutions outlined. Additionally, a mathematical framework is established to capture market price fluctuations. Numerical simulations, illustrated through 2D, 3D and contour graphs, highlight the effects of parameter variations. Our findings demonstrate the effectiveness of the coupled Ivancevic model as a fractional nonlinear wave system, providing valuable insights into stock volatility and returns. This study contributes to creating new option-pricing models, which affect financial market analysis and risk management.