Sachin Kumar

In this work, we investigate a variety of soliton solutions for the ([Formula: see text])-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani (KdV–SKR) equation, which is a significant class of nonlinear evolution equations that arise in fluid dynamics and plasma physics. This equation is a combination of the SK and KdV equations. Obtaining exact solutions to this equation presents a significant analytical challenge due to its multidimensional nature and high degree of nonlinearity. However, three newly advanced methodologies, viz., the Hirota bilinear (HB) method, the generalized exponential differential rational function (GEDRF) approach, and the improved modified Sardar sub-equation (IMSSE) approach, can be employed to tackle this challenge. We employ a logarithmic transformation to obtain the bilinear equation in terms of an auxiliary function and subsequently formulate it in HB form to derive various soliton solutions. Furthermore, we derive new families of distinct soliton solutions expressed in trigonometric, hyperbolic, rational, and exponential functions using the GEDRF and IMSSE techniques. The obtained solutions provide graphical representations of the wave profiles and their physical relevance, including lump, breather, multi-wave, periodic-lump interactions, solitary-lump interactions, periodic singular, bright, dark, and singular soliton solutions. These solitonic waves show the dynamic properties of the derived solutions displayed through 2D, 3D, density, and contour graphics using Maple software. Solitons and solitonic structures are localized waves that occur across a wide range of disciplines, including plasma physics, oceanography, nonlinear sciences, atmospheric sciences, and earth sciences.

In this original research, we explore new forms of soliton solutions, lumps, and the dynamics of water waves of the considered Generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt (KDKK) and Bogoyavlensky-Konopelchenko (BK) equations in (3+1)-dimensions, utilizing the Hirota bilinear approach. We apply the Painlevé analysis to check the integrability of this model. We formulate a bilinear equation in an auxiliary function using the Cole-Hopf transformation and then develop it into a Hirota bilinear form using a bilinear differential operator. Based on this technique, we derive lump waves, breaking phenomena, solitons, peakons, wave interactions, and wave-to-wave collisions. The solutions are obtained using ansatz functions in quadratic, sine, cosine, and exponential functions. We also illustrate how the lump interacts with solitary, periodic, and breather waves to generate various dynamics of water waves in the obtained solutions, which are visualized graphically via 2D, 3D, and contour plots with the help of symbolic software Maple. We also show the phase plane portraits, bifurcation analysis, and sensitivity analysis based on the bifurcation method with the help of the equilibrium points of the governing model studied.

In this work, we introduce a newly advanced generalized version of the sub-ODE method for constructing soliton solutions and illustrate the dynamic nature of various solitonic structures in a concatenated model, for the first time. We consider the recently proposed concatenated model, which combines the Lakshmanan–Porsezian–Daniel (LPD), Sasa–Satsuma (SS), and nonlinear Schrödinger’s equations. Numerous soliton solutions of the considered model are derived by applying the proposed method together with a traveling-wave reduction. In addition, several versions of the sub-ODE approach are employed, including the new modified sub-ODE method and the newly improved modified generalized sub-ODE method, to construct diverse classes of exact wave solutions. These unified and generalized approaches are utilized to construct novel soliton solutions and wave interactions for the concatenated model, using Computational software-Mathematica. Consequently, we have derived a plethora of exact solutions, namely, bright solitons, dark solitons, singular solitons, periodic wave solutions, Weierstrass elliptic function solutions, as well as Jacobi elliptic function solutions, and additional straddled soliton solutions of the considered model. We also provide the parameter constraints necessary for the existence of these solutions. Furthermore, we illustrate 3D and 2D figures of the acquired solutions for specific selections of arbitrary parameters. We demonstrate the effects of varying the nonlinear parameters of the concatenated model on the evolution of solitons. The solutions obtained in this study are significant for applications in nonlinear dynamics, computational fluid dynamics, plasma physics, and optical physics.

In this paper, we investigated the extraordinarily complex nonlinear evolution equations [ ]- and [ ]-dimensional generalized Camassa–Holm–Kadomtsev–Petviashvili (g-CHKP) equations found in shallow water waves. We use the Jacobi Elliptic Function Expansion (JEFE) method to derive a variety of closed-form analytical soliton solutions, and traveling wave solutions for the nonlinear [ ]- and [ ]-dimensional generalized g-CHKP equations. Several new forms of soliton-like solutions to the governing equations, as well as the dynamics of various types of solutions, are shown graphically for selecting the relevant parameters via numerical simulation. The originality of this work arises from the fact that the results mentioned are entirely new and have not before been studied for the given equation. Furthermore, the applied mathematical approach is more efficient and trustworthy for developing newly constructed soliton solutions in various fields of complex nonlinear phenomena. We anticipate that the findings of this work will have important applications in plasma physics, ocean engineering, optical fibers, quantum mechanics, fluid dynamics, nonlinear dynamics, and soliton theory.

In this work, we investigate the dynamic behavior of the time M-fractional regularized long wave Burger (tM-fRLW-Burger) equation, which describes surface water waves in a channel. The main focus of this research spans several key areas in engineering and mathematical physics. First, we perform a comprehensive bifurcation analysis of the tM-fRLW-Burger equation, methodically identifying equilibrium points and their stability characteristics under diverse parametric settings. We use both linear and nonlinear stability analysis methods to sort these equilibrium sites into groups based on their features, showing the differences between stable nodes, saddle points, and limit cycles. We demonstrate local and global stability through a comprehensive analysis of phase pictures, highlighting the impact of parameter variations on the system’s dynamic behavior. Second, we analyze the chaotic behavior of the system under different initial conditions and perturbation terms using tools such as two- and three-dimensional phase portraits, time series, Poincaré diagrams, and multi-stability. The incorporation of Poincaré maps offers a powerful tool for analyzing the topology of chaotic orbits, enabling a deeper understanding of the system’s underlying dynamical structure. Additionally, we originate the exact traveling wave for the tM-fRLW-Burger model employing the modified simple equation method. For specific values of the free parameters, we conduct a comprehensive analysis of the obtained solutions, accompanied by some unique diagrams. These solutions come in the form of trigonometric, exponential, and hyperbolic functions. For certain parameter values, we identify many wave phenomena, including kink waves, anti-kink waves, interactions between anti-kink and periodic waves, periodic lump and kink interactions, dark bell-shaped waves, and periodic waves, illustrating the model’s intricate solution structure. Finally, we observe the modulation instability of the proposed model.

This paper presents new exact solutions of a nonlinear long–short wave interaction system using two different methods named, the new Jacobi elliptic function expansion (NJEFE) method and the modified Kudryashov method (KM). Different types of solutions are obtained including, hyperbolic and Jacobi elliptic functions. The approaches used produce various dynamical wave structures of soliton solutions in evolutionary dynamical structures of solitary wave solutions. The useful parameter selections are permitted in order to provide a solution. To better comprehend the physical phenomena of these dynamical models that arise in mathematical physics, the physical behavior of these solutions is presented. The methods are efficient and can be used to produce novel solutions for different types of nonlinear partial differential equations.