Generalized Traveling Wave Research Articles

Stability and uniqueness of generalized traveling front for non-autonomous Fisher–KPP equations with nonlocal diffusion

This paper is concerned with the stability and uniqueness of generalized traveling front of non-autonomous Fisher–KPP equations with nonlocal diffusion. In such non-autonomous nonlocal dispersal equations, both the diffusion kernel and the reaction term generally depend on time. The existence of generalized traveling wave fronts of such equations has been studied in Ducrot and Jin (2022). By using comparison principle and part metric, we show the generalized traveling front is asymptotically stable under well-fitted perturbation and it is also unique.

Constructing Exact and Approximate Diffusion Wave Solutions for a Quasilinear Parabolic Equation with Power Nonlinearities

The paper studies a degenerate nonlinear parabolic equation containing a convective term and a source (reaction) term. It considers the construction of approximate solutions to this equation with a specified law of diffusion wave motion, the existence of these solutions being proved in our previous studies. A stepwise algorithm of the numerical solution with a time-difference scheme is proposed, the second-order difference scheme being used in such problems for the first time. At each step the problem is solved iteratively on the basis of a radial basis function (RBF) collocation method. In order to verify the numerical solution algorithm, two classes of exact generalized traveling wave solutions are proposed, whose construction is reduced to solving a Cauchy problem for second order ordinary differential equations (ODEs) with a singularity at the higher derivative. The theorem of the existence and uniqueness of the analytical solution in the form of a power series is proved for it, and the estimates of the radius of convergence are obtained. The Euler method is used to prove a similar statement concerning the existence of a continuous solution in the non-analytical case. The RBF collocation method is also applied for the approximate solution of the Cauchy problem. The solutions to the Cauchy problem are numerically analyzed, and this has enabled us to reveal and describe some of their properties, including those not previously observed, and to assess the accuracy of the method.

Rogue waves on the general periodic traveling wave background for an extended modified Korteweg–de Vries equation

Under consideration in this paper is rogue waves on the general periodic traveling wave background of an integrable extended modified Korteweg–de Vries equation. The general periodic traveling wave solutions are presented. By means of the Darboux transformation and the nonlinearization of the Lax pair, we present the first‐, second‐ and third‐order rogue waves on the general periodic traveling wave background. Furthermore, the dynamic behaviors of rogue periodic waves are elucidated from the viewpoint of three‐dimensional structures.

Adequate wide-ranging closed-form wave solutions to a nonlinear biological model

The auxiliary equation method and the extended tanh function method are put in use to establish wide-ranging, further general and standard closed-form traveling wave solutions to the time fractional nonlinear biological model. The fractional derivative is considered in the sense of beta-derivative. The ascertained solutions are in the form of exponential, rational, hyperbolic and trigonometric functions with significant physical meaning. The achieved solutions demonstrate that the methods are effective, compatible and efficient mathematical tools. We analyze the attained solutions through sketching graphs for the defined values of the associated parameters to perceive the internal structure by making use the Mathematica software. The developed traveling wave solutions might be helpful to illustrate the internal infrastructure of tangible phenomena associated with the consider model.

Exact Solutions of the KdV Equation with Dual-Power Law Nonlinearity

In this paper, we investigate the KdV equation with dual-power law nonlinearity. As a result, we have obtained general exact travelling wave soliton solutions such as bright soliton solution, dark soliton solution and periodic solution. These solutions have many free parameters in such away that they may be used to simulate many experimental situations. The main contribution in this work is to give the general solution of the obtained equations with different values of parameters $$n$$ .

Periodic travelling waves and rogue waves for the higher-order modified Korteweg-de Vries equation

In this paper, we construct the rogue wave solutions of the higher-order modified Korteweg-de Vries equation on the background of the general periodic travelling wave solutions. By the sub-equation method, we derive the general periodic travelling wave solutions expressed in terms of the Jacobi elliptic functions. By using the Darboux transformation method, we derive one-, two-, and three-fold rogue wave solutions on the background of elliptic functions. We provide some examples of two families of rogue wave solutions for illustration. The method provided in this paper can also be applicable to the construction of rogue wave solutions of other higher-order nonlinear integrable models on the periodic wave background.

Higher dimensional localized and periodic wave dynamics in an integrable (2+1)-dimensional deep water oceanic wave model

In this article, a new integrable (2+1)-dimensional Kundu–Mukherjee–Naskar model which is a variant of the well-known nonlinear Schrödinger equation is investigated. Bright–dark optical solitons along with periodic waves, complexiton, and rational solutions are constructed by employing a generalized traveling wave analysis, Jacobian-elliptic function, Riccati equation, and ansatz approach. Further, the dynamics of these bright/dark optical solitons and complexiton/periodic waves are studied by exploring the importance of arbitrary physical parameters with graphical demonstrations for a clear understanding and stability of the considered model is also investigated. The obtained higher dimensional nonlinear wave solutions of this integrable system shall have useful applications in different physical systems including the dynamics of beam propagation in optical fibers, ion-acoustic waves in magnetized plasma, and deep water oceanic rogue waves.

Rogue waves of the fifth-order Ito equation on the general periodic travelling wave solutions background

In this paper, on the background of general periodic travelling wave solutions, we construct rogue wave solutions of the fifth-order Ito equation. In particular, those solutions cover the known results in the literature. By means of the Darboux transformation, we derive one-, two- and three-fold rogue wave solutions on the periodic travelling wave solutions background. We provide several illustrations of such rogue waves and analyze their generation mechanisms.

Dynamical properties of generalized traveling waves of exactly solvable forced Burgers equations with variable coefficients

The initial value problem for a generalized forced Burgers equation with variable coefficients Ut+(μ˙(t)/μ(t))U+UUx=(1/2μ(t))Uxx−a(t)Ux+b(t)(xU)x−ω2(t)x+f(t),x∈R,t>0, is solved using Cole-Hopf linearization and Wei-Norman Lie algebraic approach for finding the evolution operator of the associated linear diffusion type equation. As a result, solution of the initial value problem is obtained in terms of a corresponding linear second-order inhomogeneous ordinary differential equation and a standard Burgers model. Then, using the translation and Galilean invariance of standard Burgers equation, families of generalized nonlinear waves propagating according to a Newtonian type equation of motion are constructed. The influence of the damping, dilatation and forcing terms on the dynamics of shocks, multi-shocks, triangular and N-shaped generalized traveling waves and rational type solutions with moving singularities is investigated. Finally, exactly solvable models with concrete time-variable coefficients are introduced and dynamical properties of certain particular solutions are discussed.

Generalized traveling waves for time-dependent reaction–diffusion systems

Traveling wave solutions in general time-dependent (including time-periodic) reaction–diffusion equations and systems of equations have attracted great attention in the last two decades. The aim of this paper is to study the propagation phenomenon in a general time-heterogeneous environment. More specifically, we investigate generalized traveling wave solutions for a two-component time-dependent non-cooperative reaction–diffusion system which has applications in epidemiology and ecology. Sufficient conditions on the existence and nonexistence of generalized traveling wave solutions are established. In the susceptible-infectious epidemic model setting, generalized traveling waves describe the spatio-temporal invasion of a disease into a totally susceptible population. In the context of predator–prey systems, the generalized traveling waves describe the spatial invasion of predators introduced into a new environment where the prey population is at its carrying capacity.

Symmetry reductions and new functional separable solutions of nonlinear Klein—Gordon and telegraph type equations

The paper is concerned with different classes of nonlinear Klein—Gordon and telegraph type equations with variable coefficients\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c(x){u_{tt}} + d(x){u_t} = {[a(x){u_x}]_x} + b(x){u_x} + p(x)f(u),$$\\end{document}where f (u) is an arbitrary function. We seek exact solutions to these equations by the direct method of symmetry reductions using the composition of functions u = U (z) with z = φ(x, t). We show that f (u) and any four of the five functional coefficients a(x), b(x), c(x), d(x), and p(x) in such equations can be set arbitrarily, while the remaining coefficient can be expressed in terms of the others. The study investigates the properties and finds some solutions of the overdetermined system of PDEs for φ(x, t). Examples of specific equations with new exact functional separable solutions are given. In addition, the study presents some generalized traveling wave solutions to more complex, nonlinear Klein—Gordon and telegraph type equations with delay.

Separation of variables in PDEs using nonlinear transformations: Applications to reaction–diffusion type equations

The paper describes a new approach to constructing exact solutions of nonlinear partial differential equations that employs separation of variables using special (nonlinear integral) transformations and the splitting principle. To illustrate its effectiveness, the method is applied to nonlinear reaction–diffusion type equations that involve variable coefficients and arbitrary functions. New exact functional separable solutions as well as generalized traveling wave solutions are obtained.

New exact solutions for Chan-Hilliard equation by new sub-equation method

In this present study, new sub-equation method is used to construct exact travelling wave solutions of the Cahn-Hilliard equation (CH). As a consequence, many new general exact travelling wave solutions are obtained including generalized hyperbolic (GH) function, generalized trigonometric (GT) function, exponential function and rational function solutions. Compared with the most existing similar new sub-equation method, the proposed method gives new and more general exact travelling wave solutions. More importantly, the method with the help of symbolic computation prepares a powerful mathematical tool for solving a great many nonlinear evolution equations in mathematical physics.

Periodic Travelling Waves of the Modified KdV Equation and Rogue Waves on the Periodic Background

We address the most general periodic travelling wave of the modified Korteweg-de Vries (mKdV) equation written as a rational function of Jacobian elliptic functions. By applying an algebraic method which relates the periodic travelling waves and the squared periodic eigenfunctions of the Lax operators, we characterize explicitly the location of eigenvalues in the periodic spectral problem away from the imaginary axis. We show that Darboux transformations with the periodic eigenfunctions remain in the class of the same periodic travelling waves of the mKdV equation. In a general setting, there are exactly three symmetric pairs of eigenvalues away from the imaginary axis, and we give a new representation of the second non-periodic solution to the Lax equations for the same eigenvalues. We show that Darboux transformations with the non-periodic solutions to the Lax equations produce rogue waves on the periodic background, which are either brought from infinity by propagating algebraic solitons or formed in a finite region of the time-space plane.

Some applications of the modified (G′/G2)-expansion method in mathematical physics

Abstract This study proposes a modified ( G ′ / G 2 ) -expansion method to seek new more general exact traveling wave solutions to nonlinear evolution equations (NLEEs). To demonstrate the validity of this method, two interesting NLEEs are considered: the (2 + 1)-dimensional typical breaking soliton equation and (1 + 1)-dimensional classical Boussinesq system of equations. Consequently, we construct some new exact solutions involving the free parameters of the considered equations, which are categorized into three unique forms including rational, periodic, and hyperbolic functions. Thus, this study illustrates the effectiveness and the simplicity of the proposed method with the aid of symbolically computational software such as MATHEMATICA.

Traveling wave solutions in closed form for some nonlinear fractional evolution equations related to conformable fractional derivative

Fractional order nonlinear evolution equations involving conformable fractional derivative are formulated and revealed for attractive solutions to depict the physical phenomena of nonlinear mechanisms in the real world. The core aim of this article is to explore further new general exact traveling wave solutions of nonlinear fractional evolution equations, namely, the space time fractional (2+1)-dimensional dispersive long wave equations, the (3+1)-dimensional space time fractional mKdV-ZK equation and the space time fractional modified regularized long-wave equation. The mentioned equations are firstly turned into the fractional order ordinary differential equations with the aid of a suitable composite transformation and then hunted their solutions by means of recently established fractional generalized ($D_\xi^\alpha G/G$)-expansion method. This productive method successfully generates many new and general closed form traveling wave solutions in accurate, reliable and efficient way in terms of hyperbolic, trigonometric and rational. The obtained results might play important roles for describing the complex phenomena related to science and engineering and also be newly recorded in the literature for their high acceptance. The suggested method will draw the attention to the researchers to establish further new solutions to any other nonlinear evolution equations.

Response of exact solutions of the nonlinear Schrödinger equation to small perturbations in a class of complex external potentials having supersymmetry and parity-time symmetry

We discuss the effect of small perturbation on nodeless solutions of the nonlinear Schrödinger equation in 1 + 1 dimensions in an external complex potential derivable from a parity-time symmetric superpotential that was considered earlier (Kevrekidis et al 2015 Phys. Rev. E 92 042901). In particular, we consider the nonlinear partial differential equation , where represents the complex potential. Here, we study the perturbations as a function of b and m using a variational approximation based on a dissipation functional formalism. We compare the result of this variational approach with direct numerical simulation of the equations. We find that the variational approximation works quite well at small and moderate values of the parameter product bm which controls the strength of the imaginary part of the potential. We also show that the dissipation functional formalism is equivalent to the generalized traveling wave method for this type of dissipation.

Stability and uniqueness of generalized traveling waves of lattice Fisher-KPP equations in heterogeneous media

In this paper, we investigate the stability and uniqueness of generalized traveling wave solutions of lattice Fisher-KPP equations with general time and space dependence. We first show the existence, uniqueness, and stability of strictly positive entire solutions of such equations. Next, we show the stability and uniqueness of generalized traveling waves connecting the unique strictly positive entire solution and the trivial solution zero. Applying the general stability and uniqueness theorem, we then prove the existence, stability and uniqueness of periodic traveling wave solutions of lattice Fisher-KPP equations in time and space periodic media, and the existence, stability and uniqueness of generalized traveling wave solutions of lattice Fisher-KPP equations in time heterogeneous media. The general stability result established in this paper implies that the generalized traveling waves obtained in many cases are asymptotically stable under well-fitted perturbation.

Abundant general solitary wave solutions to the family of KdV type equations

This work explores the construction of more general exact traveling wave solutions of some nonlinear evolution equations (NLEEs) through the application of the (G′/G, 1/G)-expansion method. This method is allied to the widely used (G′/G)-method initiated by Wang et al. and can be considered as an extension of the (G′/G)-expansion method. For effectiveness, the method is applied to the family of KdV type equations. Abundant general form solitary wave solutions as well as periodic solutions are successfully obtained through this method. Moreover, in the obtained wider set of solutions, if we set special values of the parameters, some previously known solutions are revived. The approach of this method is simple and elegantly standard. Having been computerized it is also powerful, reliable and effective.

Traveling wave solutions of the cylindrical nonlinear Maxwell's equations

Study on propagation of cylindrical electromagnetic waves in various inhomogeneous and nonlinear media is of fundamental importance, which can be described by the cylindrical nonlinear Maxwell's equations. In recent years, finding exact solutions for these equations has emerged as a popular research topic. The exact solutions play an irreplaceable role in understanding and predicting physical phenomena, and developing numerical calculation methods, and so on. However, due to the extreme complexity of nonlinear partial differential equations, exact solutions of the cylindrical Maxwell's equations were only able to be obtained in a nonlinear and nondispersive medium whose dielectric function is an exponential function in previous researches. Actually, there is no general method at present which can exactly solve arbitrary cylindrical nonlinear Maxwell's equations. Therefore, finding physically admissible solutions meeting certain particular condition for the cylindrical nonlinear Maxwell's equations might be feasible. In this paper, we discuss the traveling wave solutions which are very important in electromagnetic theory, especially in solitary wave theory. To our knowledge, research on obtaining traveling wave solutions of the cylindrical nonlinear Maxwell's equations is still lacking. The main conclusions in this paper are listed as follows. Firstly, we introduce the cylindrical nonlinear Maxwell's equations mentioned in some previous publications, which can describe cylindrical electromagnetic waves propagation in inhomogeneous nonlinear and nondispersive media. In this paper, we focus on the nondispersive media with arbitrary nonlinearity and power-law inhomogeneity. Secondly, we point out that the electric field component E of the model has no plane traveling wave solutions E=g(r-kt), after theoretical analysis and study. Then generalized traveling wave solutions in form of E=g(lnr-kt) for the electric field component are obtained by finding correct variable substitution and solving second-order nonlinear ordinary differential equation.Finally, we provide two examples to show the physical meanings of our generalized traveling wave solutions. We find that the transmitting speeds of vibrations vary with different points of the electric field. Actually, the transmitting speed of the vibration of a certain point closer to the cylinder center is lower. As a result, we observed a physical phenomenon similar to that of self-steepening. Our work can be used to analyze the electromagnetic properties of ferroelectric materials and new materials. Theoretically, it can also provide an approach to studying the cylindrical nonlinear Maxwell's equations.