Kink Solitary Wave Research Articles

Novel complexiton, rational wave, multi-lumps and the kink solitary wave solutions to the new (3+1)-dimensional integrable fourth-order equation for shallow water waves

In this exploration, we aim to seek a number of new exact solutions to the new (3+1)-dimensional integrable fourth-order nonlinear equation, which is widely used to describe the shallow water waves. Employing the Cole-Hopf transformation, we develop its bilinear form. Then, taking advantage of the ansatz function method, a new functional form is utilized to probe the singular complexiton solutions. Based on which, the non-singular complexiton solutions are derived by imposing the constraint conditions. In addition, we find the rational wave solutions and multi-lumps solutions wielding the rational function method and new homoclinic method respectively. At the end, we investigate the kink solitary wave solutions using the variational approach that is based on the variational principle and Ritz method. Meanwhile, the Hamiltonian of the system is also elaborated. Correspondingly, the graphic descriptions of the extracted results are presented to unfold their dynamic behaviors through Maple. As we all know, the findings of this paper are firstly reported and can enlarge the exact solutions of the considered PDE.

Classical and nonclassical Lie symmetries, bifurcation analysis, and Jacobi elliptic function solutions to a 3D-modified nonlinear wave equation in liquid involving gas bubbles

The current paper undertakes an in-depth exploration of the dynamics of nonlinear waves governed by a 3D-modified nonlinear wave equation, a significant model in the study of complex wave phenomena. To this end, the study employs both classical and nonclassical Lie symmetries for rigorously deriving invariant solutions of the governing equation. These symmetries enable the formal construction of exact solutions, which are crucial for understanding the complex behavior of the model. Furthermore, the research extends into the realm of bifurcation analysis through the application of planar dynamical system theory. Such an analysis reveals the conditions under which the 3D-modified nonlinear wave equation admits Jacobi elliptic function solutions. The study also delves into the impact of the nonlinear parameter on the physical characteristics of bright and kink solitary waves as well as continuous periodic waves using Maple. Overall, the comprehensive analysis presented not only enhances the understanding of complex nonlinear wave dynamics but also sets the stage for future advancements in vast areas of fluid dynamics and plasma physics.

Multi-lump, resonant Y-shape soliton, complex multi kink solitons and the solitary wave solutions to the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation for incompressible fluid

The major contribution in this paper is to inquire into some new exact solutions to the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation (BLMPE) which plays a major role in area of the incompressible fluid. Taking advantage of the Cole-Hopf transform, we extract its bilinear form. Then two different kinds of the multi-lump solutions are probed by applying the new homoclinic approach. Secondly, the Y-shape soliton solutions are explored via assigning the resonance conditions to the N-soliton solutions. Additionally, the complex multi kink soliton solutions (CMKSSs) are investigated through the Hirota bilinear method. Lastly, some other wave solutions including the kink and anti-kink solitary wave solutions are developed with the aid of two efficacious approaches, namely the variational method and Kudryashov method. In the meantime, the profiles of the accomplished solutions are displayed graphically via Maple.

Modulation instability and extraction of fractional optical solitons in the presence of generalized Kudryashov’s law and dual form of non-local nonlinearity

This research delves into the study of optical solitons governed by Kudryashov’s law of nonlinear refractive index, incorporating a conformable fractional derivative, perturbed terms, and a quadrupled-power law in optical fibers. Employing a novel approach known as the generalized exp(− S(ζ) ) expansion method, we identify novel optical soliton solutions characterized by exponential, rational, and hyperbolic functions featuring a conformable fractional derivative. These solutions manifest as unified bright-dark, singular, singular periodic, kink, anti-kink, and novel solitary waves. We employ 3D, contour, and 2D plots to analyze the behavior of these solutions across different temporal and fractional-order derivative parameters, offering an understanding of their physical interpretations. Furthermore, we perform modulation instability (MI) analysis that depends on standard linear stability analysis to derive the MI gain spectrum, demonstrating how well our methodology works for nonlinear problems in the natural sciences and engineering domains. This approach provides an effective way to investigate optical solutions in various integer and fractional Schrödinger equations.

Dynamics of transformed nonlinear waves for the (2+1)-dimensional pKP-BKP equation: interactions and molecular waves

In this paper, the dynamical behaviors of transformed nonlinear waves for the (2+1)-dimensional combined potential Kadomtsev-Petviashvili and B-type Kadomtsev-Petviashvili (pKP-BKP) equation are investigated, which can be used to reveal the nonlinear wave phenomena in nonlinear optics, plasma physics and hydrodynamics. The breath-wave and the lump solutions are constructed by means of the soliton solutions. The conversion mechanism for the breath-wave is systematically analyzed, which leads to several new kink-shaped nonlinear waves. The gradient relationships of these transformed waves are revealed by a Riemannian circle. Through the analysis of the nonlinear superposition between the periodic wave component and the kink solitary wave component, the dynamical characteristics including the formation mechanism, oscillation and locality for the nonlinear waves are investigated. The time-varying properties of transformed waves are shown by the study of time variables. By virtue of the two breath-wave solutions, several interactions including elastic and inelastic collisions between two nonlinear waves are studied. In particular, some transformed molecular waves encompassing the non-, semi- and full-transition modes are presented with the aid of velocity resonance. The results can help us further understand the complex nonlinear waves existing in the integrable systems.

Construction of new wave structures and stability analysis for the nonlinear Klein-Gordon equation

In this study, the nonlinear Klein–Gordon equation the relativistic equivalents of the nonlinear Schrödinger equations is presented which characterizes the connection between relativistic energy-momentum in a quantized manner. A variety of exact solutions are developed by employing the extended Fan sub-equation approach and the Sardar sub-equation method. The study includes three dimensional surface plots of some derived solutions like dark and bright solitary waves, kink solitary waves, anti-kink solitary waves, periodic solitary waves and hyperbolic functions, with several solutions being novel. Furthermore, the stability analysis of the observed solutions is also established to validate the scientific computations. The 3D, 2D and contour visualizations of the wave dynamics are demonstrated using Mathematica for a suitable choice of parameters. The applied strategies are recognized as a significant mathematical tool for obtaining analytical solutions to partial differential equations in mathematical physics.

The kink solitary wave and numerical solutions for conformable non-linear space–time fractional differential equations

This paper is about the conformable non-linear space–time fractional Telegraph and the non-linear space–time fractional Kolmogorov–Petrovskii–Piskunov (KPP) equations. To construct some real and analytical wave solutions of both equations the simplest equation method is applied. As a consequence, a series of analytical wave solutions are obtained as well as verify these solutions by using Mathematica software. These solutions denote that this method is useful and straightforward. We show some graphs in 2-D and 3-D of our solution to illustrate the behaviour of waves. Finally, we introduce the numerical solutions for the conformable non-linear space–time fractional Telegraph and the non-linear space–time fractional KPP equations using the finite difference method and make comparisons with analytical solutions.

Computation of soliton structure and analysis of chaotic behaviour in quantum deformed Sinh-Gordon model.

Soliton dynamics and nonlinear phenomena in quantum deformation has been investigated through conformal time differential generalized form of q deformed Sinh-Gordon equation. The underlying equation has recently undergone substantial amount of research. In Phase 1, we employed modified auxiliary and new direct extended algebraic methods. Trigonometric, hyperbolic, exponential and rational solutions are successfully extracted using these techniques, coupled with the best possible constraint requirements implemented on parameters to ensure the existence of solutions. The findings, then, are represented by 2D, 3D and contour plots to highlight the various solitons' propagation patterns such as kink-bright, bright, dark, bright-dark, kink, and kink-peakon solitons and solitary wave solutions. It is worth emphasizing that kink dark, dark peakon, dark and dark bright solitons have not been found earlier in literature. In phase 2, the underlying model is examined under various chaos detecting tools for example lyapunov exponents, multistability and time series analysis and bifurcation diagram. Chaotic behavior is investigated using various initial condition and novel results are obtained.

Soliton molecules, interaction and other wave solutions of the new (3+1)-dimensional integrable fourth-order equation for shallow water waves

In the present work, we aim to explore the new (3+1)-dimensional integrable fourth-order nonlinear equation(IFNE) for describing the shallow water waves. First, we study its N-soliton solutions via the bilinear form which is constructed by applying the Cole-Hopf transform. The resonance conditions of the soliton molecular are extracted and the soliton molecules are obtained. Second, the ansatz function method together with the symbolic computation, is implemented to develop the interaction wave solutions(IWSs). Finally, we take advantage of the Bernoulli sub-equation function method(BSFM) to look into the travelling wave solutions(TWSs). Different kinds of the TWSs like the singular-kink and kink solitary wave solutions are found. Correspondingly, the dynamic performances of the solutions are depicted graphically to present the physical interpretations. And for all we know, the solutions got in this work are all new and can be regarded as an extension of the solutions for the new (3+1) dimensional IFNE, which are expected to have practical significance for the application of these equations in physics.

Dynamic nature of analytical soliton solutions of the (1+1)-dimensional Mikhailov-Novikov-Wang equation using the unified approach

Abstract In this work, we investigate the dynamical study of the (1+1)-dimensional Mikhailov-Novikov-Wang (MNW) equation via the unified method is investigated. This technique is used to obtain the soliton solutions, including the trigonometric function solution, the periodic function solution, the exponential function solution, the elliptic function solution, and other soliton-form solutions. All the obtained results in this work utilizing an effective unified method help gain a better understanding of the physical meaning and behavior of the equation, thus sheding light on the significance of investigating diverse nonlinear wave phenomena in physics and ocean engineering. These derived results are entirely new and never repeated in the previous works done by the other authors. For the interest of visual presentation and physical illustrations, we plot the graphical demonstrations of some of the specified solutions in 3-dimensional, contour, and 2-dimensional plots by using Mathematica software. Consequently, we observe that the acquired solutions of the MNW equations are anti-bell-shape, kink wave solution, solitary wave, periodic solution, multisoliton, and different types of soliton solutions.

Study on chirped dark, bright solitons conversion and the effect of intermodal dispersion, self-frequency shift, and self-steepening effect on the chirping of bright, dark, and kink solitary waves

This study reveals the dynamics of short optical pulses in nonlinear media. Then, for the survival of different types of solitary waves in the nonlinear medium, the constraint relations to the physical parameters and chirps associated with each of the solitary waves are explicitly shown. We authenticate numerically all those obtained from the analytical method and show the chirping effect is saturative in nature and can be managed by changing coefficients of inter-modal dispersion, stimulated Raman scattering (SRS), and self-steepening effect. We also verify the more operative condition of intermodal dispersion.

Applications of the generalized nonlinear evolution equation with symbolic computation approach

In this work, we will try to find lump solutions, interaction between lump wave and solitary wave solutions, kink-solitary wave solutions and shock wave-type solutions to [Formula: see text]-dimensional generalized nonlinear evolution equation arising in the shallow water waves. The lump solutions, the interaction between lump wave and solitary wave solutions and kink-solitary wave solutions are derived with symbolic computation based on a logarithmic derivative transform which is derived by the help of Hirota’s simple method. The shallow water waves in this equation are associated with some natural problems such as tides, storms, atmospheric currents and tsunamis. For the physical presentation of the solutions, we draw 3D and counter graphics by giving the suitable values to include the free parameters. We believe that disciplines such as mathematical physics, nonlinear dynamics, fluid mechanics and engineering sciences can benefit from this study.

Numerical Investigation of Time-Fractional Phi-Four Equation via Novel Transform

This paper examines two methods for solving the nonlinear fractional Phi-four problem with variable coefficients. One of the distinct states of the Klein–Gordon model yields the Phi-four equation. It is also used to simulate the kink and anti-kink solitary wave connections that have recently emerged in biological systems and nuclear particle physics. The approaches that are being suggested consist of the Yang transform, the homotopy perturbation approach, the decomposition approach, and the fractional derivative as stated by Caputo. The advantages of the proposed techniques are their capability of combining two dominant approaches for attaining precise and approximate solutions of nonlinear equations. It is important to keep in mind that the suggested methods can perform better in general as they need less computational effort than the alternative methods, while keeping a high level of numerical precision. The actual and estimated outcomes are demonstrated in graphs and tables to be quite similar, demonstrating the usefulness of the proposed approaches. Additionally, several simulations are used to show the physical behaviors of the found solutions with regard to fractional order. The article’s results possess complimentary properties that relate to the symmetry of partial differential equations.

Optical soliton solutions for Lakshmanan-Porsezian-Daniel equation with parabolic law nonlinearity by trial function method

<abstract><p>In this paper, the trial function method is used to address the Lakshmanan-Porsezian-Daniel (LPD) equation with parabolic law nonlinearity. Implementing the traveling wave hypothesis reduces the LPD equation to an ordinary differential equation (ODE). From this ODE, many exact solutions, such as kink solitary wave solutions, bell shaped solitary wave solutions, triangular function solutions, periodic function solutions, singular solutions and Jacobian elliptic function solutions, are retrieved. Among them, some solutions are new. By suitable choice of parameters, we also draw 3D surface and 2D graphs of density, contour and level curves of some precise solutions for intuitive investigation.</p></abstract>

Bright, dark, periodic and kink solitary wave solutions of evolutionary Zoomeron equation

The modified auxiliary equation (MAE) approach and the generalized projective Riccati equation (GPRE) method are used for the first time to solve Zoomeron problem which provided different types of exact traveling wave solutions, including some new dynamical behaviors. The governing model covers unique examples of solitons with distinct properties that emerge in a variety of physical situations, including laser physics, fluid dynamics and nonlinear optics. The achieved exact traveling wave solutions include solitary wave, periodic wave, bright, dark peakon, and kink-type wave solutions. Earned results are given as hyperbolic and trigonometric functions. Moreover, the dynamical features of obtained results are demonstrated through interesting plots.

Optical solitons and traveling wave solutions for the higher-order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms

This work mainly concerned with the optical solitons and other traveling wave solutions for the higher-order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms. With the assistance of the second-order and triple-order polynomial complete discriminant system method, we construct a series of optical soliton solutions which include bright soliton solutions, singular soliton solutions, dark soliton solutions, periodic solutions, kink solitary wave solutions, anti-kink solitary wave solutions and some other traveling wave solutions. What is more, numerical simulations are also given to visualize the mechanism of the higher-order nonlinear Schrödinger system by selecting some suitable parameters of optical soliton solutions. Finally, we give the comparison between the obtained works in this paper and other’s results.

Two component model of microtubules – subsonic and supersonic solitary waves

This work represents a contribution to modelling nonlinear dynamics of microtubules, the basic components of the eukaryotic cytoskeleton. Their dynamics can be explained in terms of kink and antikink solitary waves. Special attention was paid to the stability of solitonic solutions of differential equations describing the dynamics of microtubules. It is shown that subsonic solitons are stable, while supersonic ones are not.

A new study of soliton solutions for the variable coefficients Caudrey–Dodd–Gibbon equation

We present a paper on the different methods for finding solutions to the Caudrey–Dodd–Gibbon equation with variable coefficients, which has wide applications in quantum mechanics and nonlinear optics. We have studied the equation analytically by using the unified method and the modified Kudryashov method. With the aid of symbolic computation and several types of auxiliary equations, we have obtained soliton solutions and other solutions. Then, we assign different values to the parameters, generating two-dimensional and three-dimensional graphics of the solutions, and discuss the interaction of several groups of solion solutions to form periodic wave solutions and kink solitary wave solutions.

Some advanced chirped pulses for generalized mixed nonlinear Schrödinger dynamical equation

In an optical fibre, we investigate the propagation properties of nonlinear periodic waves (PW) for a generalized mixed nonlinear Schrödinger (GMNLS) equation. The Jacobi elliptic (JE) solutions will be used to find the nonlinear chirp. The chirp varies with two intensity-dependent chirping terms are included in the linear section of the pulse chirp. The presence of the newly discovered periodic waves will be discussed in terms of fibre parameter conditions. The long-wave limit generates a wide range of solitary pulse forms, including topological, non topological, dark, kink, hyperbolic and periodic solitary waves (SW). Our findings show that for periodic and solitary waves, a nonlinear chirp obtains. Finally, under finite perturbations, the stability of these nonlinearly chirped solutions will be quantitatively investigated.

Solitary Solutions to the Fractional Generalized Hirota–Satsuma-Coupled KdV Equations

In this paper, the complete discrimination system method is used to the fractional generalized Hirota–Satsuma-coupled KdV equations, which depicts the dispersive long wave widely applied in fluid mechanics. First, by the wave transformation, the fractional generalized Hirota–Satsuma-coupled KdV equation can be transformed into ordinary differential equations. Second, a series of new solutions are constructed by the discrimination system method of the fourth order polynomial, these solutions include kink solitary wave solutions, Jacobian elliptic function double periodic solutions, trigonometric function solution, solitary wave solutions, two-period solutions of elliptic function, and implicit solutions formula solutions. Finally, the comparison between our results and others’ works is also given.