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The damped Korteweg-de Vries (D-KdV) equation is a significant extension of the Korteweg–de Vries equation, which plays an important role in understanding the complex dissipative wave system. This study explores the historical evolution and mathematical properties of the damped KdV equation. The auxiliary equation approach is utilized on nonlinear damped KdV equation to extract the newly exact solitary wave results. New solutions are extracted in bright solitons, kink wave solitons, mixed dark-bright solitons, anti-kink wave solitons, periodic solitons, dark solitons, and solitary wave structures. The physical behavior of secured solutions are visualizing in contour, 3-D, and 2-D plots based on numerical simulation with the computational software Mathematica. The proposed approach is employed, offering a powerful mathematical tool for analyzing the effects of damping on wave dynamics. Through this investigation, the study sheds light on the captivating interplay between physics and mathematics in the realm of wave phenomena. These newly constructed solutions are important in nonlinear acoustics, optical fiber, nonlinear optics, soliton wave theory, plasma physics, and ion acoustic wave phenomena. As a result, our proposed method is directed, concise, and practical for various kinds of nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

This work aims to enhance our comprehension of the dynamical features of the nonlinear Landau-Ginzburg-Higgs evolution equation, which provides a theoretical framework for identifying various phenomena, such as the formation of superconducting states and the spontaneous breakdown of symmetries. When symmetry breaking is involved in phase transitions in particle physics or condensed matter systems, the Landau-Ginzburg-Higgs model combines the ideas of the Landau-Ginzburg theory and the Higgs mechanism. The equation plays a crucial role in characterizing the Higgs field and its related particles, including Higgs boson. In a standard model of the particle physics, Higgs mechanism explains precisely how mass is acquired. The Lie invariance requirements are taken into account by the symmetry generators. The method produces a 3 - dimensional Lie algebra of the Landau-Ginzburg-Higgs model with translational symmetry (dilation or scaling) and translations in the space and the time associated with the mass and energy conservation. It is shown to be the optimal sub-algebraic system after similarity reductions are also performed. The next wave transformation method reduces the governing system to ordinary differential equations and yields a large number of exact travelling wave solutions. The Khater approach is used to solve an ordinary differential equation and investigate the closed-form analytical travelling wave solutions for the considered diffusive system. The obtained results include a singular, mixed singular, periodic, mixed trigonometric, complex combo, trigonometric, mixed hyperbolic, plane, and combined bright-dark soliton solution. The results of the sensitivity analysis demonstrate how vulnerable the suggested equation is to various initial conditions. The findings are visually displayed in contour, three-dimensional, and two-dimensional forms to emphasize the features of pulse propagation. [ABSTRACT FROM AUTHOR]

The study aims to explore the nonlinear Landau-Ginzburg-Higgs equation, which describes nonlinear waves with long-range and weak scattering interactions between tropical tropospheres and mid-latitude, as well as the exchange of mid-latitude Rossby and equatorial waves. We use the recently enhanced rising procedure to extract the important, applicable and further general solitary wave solutions to the formerly stated nonlinear wave model via the complex travelling wave transformation. Exact travelling wave solutions obtained include a singular wave, a periodic wave, bright, dark and kink-type wave peakon solutions using the generalized projective Riccati equation. The obtained findings are represented as trigonometric and hyperbolic functions. Graphical comparisons are provided for Landau-Ginzburg-Higgs equation model solutions, which are presented diagrammatically by adjusting the values of the embedded parameters in the Wolfram Mathematica program. The propagating behaviours of the obtained results display in 3-D, 2-D and contour visualization to investigate the impact of different involved parameters. The velocity of soliton has a stimulating effect on getting the desired aspects according to requirement. The sensitivity analysis is demonstrated for the designed dynamical structural system's wave profiles, where the soliton wave velocity and wave number parameters regulate the water wave singularity. This study shows that the method utilized is effective and may be used to find appropriate closed-form solitary solitons to a variety of nonlinear evolution equations (NLEEs). [ABSTRACT FROM AUTHOR]

The generalized Calogero–Bogoyavlenskii–Schiff equation (GCBSE) is examined and analyzed in this paper. It has several applications in plasma physics and soliton theory, where it forecasts the soliton wave propagation profiles. In order to obtain the analytically exact solitons, the model under consideration is a nonlinear partial differential equation that is turned into an ordinary differential equation by using the next traveling wave transformation. The new extended direct algebraic technique and the modified auxiliary equation method are applied to the generalized Calogero–Bogoyavlenskii–Schiff equation to get new solitary wave profiles. As a result, novel and generalized analytical wave solutions are acquired in which singular solutions, mixed singular solutions, mixed complex solitary shock solutions, mixed shock singular solutions, mixed periodic solutions, mixed trigonometric solutions, mixed hyperbolic solutions, and periodic solutions are included with numerous soliton families. The propagation of the acquired soliton solution is graphically presented in contour, two- and three-dimensional visualization by selecting appropriate parametric values. It is graphically demonstrated how wave number impacts the obtained traveling wave structures. [ABSTRACT FROM AUTHOR]

A generalized (3 + 1)-dimensional nonlinear wave is investigated, which defines many nonlinear phenomena in liquid containing gas bubbles. Basic theories of the natural phenomenons are usually described by nonlinear evolution equations, for example, nonlinear sciences, marine engineering, fluid dynamics, scientific applications, and ocean plasma physics. The new extended algebraic method is applied to solve the model under consideration. Furthermore, the nonlinear model is converted into an ordinary differential equation through the next wave transformation. A well-known analytical approach is used to obtain more general solutions of different types with the help of Mathematica. Shock, singular, mixed-complex solitary-shock, mixed-singular, mixed-shock singular, mixed trigonometric, periodic, mixed-periodic, mixed-hyperbolic solutions are obtained. As a result, it is found that the energy-carrying capacity of liquid with gas bubbles and its propagation can be increased. The stability of the considered model is ensured by the modulation instability gain spectrum generated and proposed with acceptable constant values. Two-dimensional, three-dimensional, and contour surfaces are plotted to see the physical properties of the obtained solutions. [ABSTRACT FROM AUTHOR]

The (2 + 1)-dimensional Nizhnik–Novikov–Veselov equation is a significant model that describes the behaviour of conserved scalar nucleons and their connection to neutral scalar masons. This system exhibits self-formation and depends on various free parameters. It is commonly used to study the dynamics of scalar nucleons and neutral scalar masons. This research article used an extended direct algebraic technique to obtain exact travelling wave solutions for the Nizhnik–Novikov–Veselov system. The obtained soliton solutions include various types, such as multi-periodic, periodic, plane, singular, bright, dark, and flat kink-type wave solitons. We present these soliton solutions graphically by varying the involved parameters using the advanced software program Wolfram Mathematica. The graphical representations in 3D, contour, and 2D surfaces allow us to visualise the behaviour of the solutions as the parameters change and the physical interpretation for these solutions is obtained. Additionally, we conduct a sensitivity analysis to examine the wave profiles of the newly designed dynamical framework. The results of this analysis demonstrate the reliability and efficiency of the proposed method, which can be applied to find closed-form travelling wave solitary solutions for a wide range of nonlinear evolution equations. [ABSTRACT FROM AUTHOR]