The goal of this work is to improve our understanding of the dynamical aspects of the nonlinear Landau-Ginzburg-Higgs (LGH) equation, which offers a theoretical framework for characterizing several phenomena, including the spontaneous breakdown of symmetries and the emergence of superconducting states. The proposed model integrates the concepts of the Higgs mechanism and the Landau-Ginzburg theory when symmetry breaking appears in phase transitions in particle physics or condensed matter systems. The equation is essential for describing the Higgs field and its constituent particles, such as the Higgs boson. New extended direct algebraic approaches and modified F-expansion techniques are used to address this problem. The obtained solutions, which include kink, anti-kink, bright, dark, and periodic solitons, are essential because they shed light on the stability and nonlinear dynamics of field theories that are pertinent to cosmology and condensed matter physics. To advance the essential propagating features, a few obtained solutions are presented as 3D, contour, and 2D graphics by applying certain values to the parameters under the given constraints. The dynamical insights are examined and significant aspects of the phenomenon under study are discussed through the use of the bifurcation analysis. Additionally, the chaos analysis is carried out to show the quasi-periodic and periodic chaotic patterns. The sensitivity analysis of the studied model is also looked at and presented at different initial conditions. Furthermore, we guarantee that every found solution is precise, effective, and a great addition to the literature of solitary wave theory. To calculate the soliton solutions for nonlinear models in communication engineering and operations research, the methods utilized in this work to derive inclusive and standard solutions are more accessible, effective, and quick.

In this study, we examine several analytical rational solutions using the ansatz transformation. For the (2+1)-dimensional extended KP-Boussinesq (eKP-BO) equation, we acquire several types of lump solutions, such as lump soliton solution (LS), lump one stripe, and lump-periodic solution (LPS). Some of the produced soliton solutions are shown in 3D and 2D, and contour depictions for suitable parameter values are used to demonstrate the dependability, usefulness, and efficacy of the computational technique. Utilizing the Galilean transformation, the dynamical structure of the suggested equation is thus obtained, and the theory of the planar dynamical system is used to carry out its bifurcation. The presence of chaotic behaviors of the eKP-BO equation is examined by taking into account a perturbed term in the derived dynamical system and providing various two- and three-dimensional phase illustrations. Furthermore, the sensitivity analysis of the dynamical system confirms that minor variations to the initial conditions significantly affect the system’s behavior, demonstrating its sensitivity to initial conditions. To the best of our knowledge, this research has not been previously addressed in the literature. The outcomes are expected to be helpful to a wide range of scholars who are interested in fluid mechanics, water wave analysis, and other multidisciplinary fields.

The goal of this research is to utilize some ansatz forms of solutions to obtain novel forms of soliton solutions for the Benney–Luke equation. It is a mathematically valid approximation that describes the propagation of two-way water waves in the presence of surface tension. By using ansatz forms of solutions, with an appropriate set of parameters, the lump soliton, periodic cross-kink waves, multi-waves, breather waves, Ma-breather, Kuznetsov–Ma-breather, periodic waves and rogue waves solutions can be obtained. Breather waves are confined, periodic, nonlinear wave solutions that preserve their amplitude and shape despite alternating between compression and expansion. For some integrable nonlinear partial differential equations, a lump soliton is a confined, stable solitary wave solution. Rogue waves are unusually powerful and sharp ocean surface waves that deviate significantly from the surrounding wave pattern. They pose a threat to maritime safety. They typically show up in solitary, seemingly random circumstances. Periodic cross-kink waves are a particular type of wave pattern that has frequent bends or oscillations that cross at right angles. These waves provide insights into complicated wave dynamics and arise spontaneously in a variety of settings. In order to predict the wave dynamics, certain 2D, 3D and contour profiles are also analyzed. Since these recently discovered solutions contain certain arbitrary constants, they can be used to describe the variation in the qualitative characteristics of wave phenomena.