General Traveling Wave Solutions Research Articles

Consistent travelling waves solutions to the non-linear time fractional Klein–Gordon and Sine-Gordon equations through extended tanh-function approach

In this study, the extended tanh-function method has been used to find further general travelling wave solutions for space-time fractional nonlinear partial differential equations, namely, the time fractional nonlinear Sine-Gordon equation and Klein–Gordon equation. The mentioned equations are useful for explaining a series of experimental seismic data, modelling of strain waves, and structures correlated to fault dynamics and the subduction slab, as well as slow earthquakes, episodic tremor, slow slip events, and tremor migration patterns. This method is used to attain exact solutions to a variety of nonlinear partial differential equations that were spatially temporal fractional. To get analytical solutions of the travelling wave type of certain nonlinear evolution equations, a power series in tanh was first utilized as an ansatz. These nonlinear fractional partial differential equations (NLFPDEs) are solved on a variety of non-rectangular domains. By using the fractional complex transform and the properties of the confirmable derivative, the two equations are reduced to ordinary differential equations. The solutions are sketched in 3D, and contour patterns, including king type, single soliton, double solitons, multiple solitons, bell shape, and other sorts of solutions. The achieved solutions are very much effective for explaining the above-stated phenomena.

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.

General Traveling Wave Solutions of Nonlinear Conformable Fractional Sharma-Tasso-Olever Equations and Discussing the Effects of the Fractional Derivatives

The exact traveling wave solution of the fractional Sharma-Tasso-Olever equation can be obtained by using the function expansion method, but the general traveling wave solution cannot be obtained. After transforming it into the Sharma-Tasso-Olever equation of the integer order by the fractional complex transformation, the general solution of its traveling wave is obtained by a specific function transformation. Through parameter setting, the solution of the kinked solitary wave is found from the general solution of the traveling wave, and it is found that when the two fractional derivatives become smaller synchronically, the waveform becomes more smooth, but the position is basically unchanged. The reason for this phenomenon is that the kink solitary wave reaches equilibrium in the counterclockwise and clockwise rotation, and the stretching phenomenon is accompanied in the process of reaching equilibrium. This is a further development of our previous work, and this kind of detailed causative analysis is rare in previous papers.

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.

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.

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.

Further fresh and general traveling wave solutions to some fractional order nonlinear evolution equations in mathematical physics

PurposeFractional order nonlinear evolution equations (FNLEEs) pertaining to conformable fractional derivative are considered to be revealed for well-furnished analytic solutions due to their importance in the nature of real world. In this article, the autors suggest a productive technique, called the rational fractional (DξαG/G)-expansion method, to unravel the nonlinear space-time fractional potential Kadomtsev–Petviashvili (PKP) equation, the nonlinear space-time fractional Sharma–Tasso–Olver (STO) equation and the nonlinear space-time fractional Kolmogorov–Petrovskii–Piskunov (KPP) equation. A fractional complex transformation technique is used to convert the considered equations into the fractional order ordinary differential equation. Then the method is employed to make available their solutions. The constructed solutions in terms of trigonometric function, hyperbolic function and rational function are claimed to be fresh and further general in closed form. These solutions might play important roles to depict the complex physical phenomena arise in physics, mathematical physics and engineering.Design/methodology/approachThe rational fractional (DξαG/G)-expansion method shows high performance and might be used as a strong tool to unravel any other FNLEEs. This method is of the form U(ξ)=∑i=0nai(DξαG/G)i/∑i=0nbi(DξαG/G)i.FindingsAchieved fresh and further abundant closed form traveling wave solutions to analyze the inner mechanisms of complex phenomenon in nature world which will bear a significant role in the of research and will be recorded in the literature.Originality/valueThe rational fractional (DξαG/G)-expansion method shows high performance and might be used as a strong tool to unravel any other FNLEEs. This method is newly established and productive.

Construction of exact solutions to nonlinear PDEs with delay using solutions of simpler PDEs without delay

We describe new indirect methods for constructing exact solutions of nonlinear PDEs with delay. Features of the proposed methods are illustrated by nonlinear delay reaction-diffusion and wave-type equations with variable coefficients. All the presented equations contain from three to seven arbitrary functions, which depend on the spatial variable and/or unknown quantity. We obtain new generalized traveling-wave solutions and functional separable solutions in explicit or implicit form for equations with constant delay as well as exact solutions with time-varying delay (a special case of which are pantograph-type PDEs with proportional delay). We show that the proposed methods can also be applied to construct exact solutions for nonlinear systems of coupled delay PDEs and higher-order delay PDEs. The considered equations and their exact solutions can be used to formulate test problems to check the adequacy and estimate the accuracy of numerical and approximate analytical methods of solving nonlinear initial-boundary value problems for delay PDEs.

A method for constructing exact solutions of nonlinear delay PDEs

We present a new method for constructing exact solutions of nonlinear delay PDEs using special solutions of simpler auxiliary PDEs without delay. The application of the method is demonstrated on nonlinear reaction–diffusion and wave-type equations with delay that include variable coefficients and arbitrary functions. New generalized traveling-wave solutions and functional separable solutions in implicit form are obtained. Results of the paper can be used to formulate test problems intended to verify the adequacy and evaluate the accuracy of numerical and approximate analytical methods of solving the corresponding nonlinear initial-boundary value problems with delay.

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.

Comment on “the solitons and periodic travelling wave solutions for Dodd-Bullough-Mikhailov and Tzitzeica-Dodd-Bullough equations in quantum field theory, Optik 168 (2018) 807-816”

In the recently published article, S.Saha Ray investigated Dodd-Bullough-Mikhailov (DBM) and Tzitzeica-Dodd-Bullough (TDB) equations via his new extended auxiliary equation method. And he claimed that new types of Jacobi elliptic function solutions had been obtained and meanwhile a new family of explicit travelling wave solutions were derived. In this comment, we show that all of exact traveling solutions presented by S.Saha Ray do not satisfy the new extended auxiliary equation method and therefore are incorrect. In addition, general traveling wave solutions of the two differential equation are presented by making full use of certain results published previously.

Exact wave solutions to the simplified modified Camassa-Holm equation in mathematical physics

In this article, we consider the exact solutions to the simplified modified Camassa-Holm (SMCH) equation which has many potential applications in mathematical physics and engineering sciences. We examine the exact travelling wave solutions by means of the modified simple equation (MSE) method by making use of travelling transformation. The attained solutions are in the form of trigonometric and hyperbolic functions. We demonstrate that the method is more general, straightforward and powerful and can be used to examine more general travelling wave solutions of various kinds of fractional nonlinear differential equations arising in mathematical physics and better than other method. Finally, we show the graphical representation and discuss the physical significance of the obtained solutions for its definite values of the involved parameters through depicting 3D and 2D figures in order to know the physical phenomena.

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.

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.

Construction of functional separable solutions in implicit form for non-linear Klein–Gordon type equations with variable coefficients

The paper deals with non-linear Klein–Gordon type equations c(x)utt=[a(x)f(u)ux]x+b(x)g(u).The direct method for constructing functional separable solutions in implicit form to non-linear PDEs is used. This effective method is based on the representation of solutions in the form ∫h(u)du=ξ(x)ω(t)+η(x),where the functions h(u), ξ(x), η(x), and ω(t) are determined further by analyzing the resulting functional-differential equations. Examples of specific Klein–Gordon type equations and their exact solutions are given. The main attention is paid to non-linear equations of a fairly general form, which contain several arbitrary functions dependent on the unknown u and /or the spatial variable x (it is important to note that exact solutions of non-linear PDEs, that contain arbitrary functions and therefore have significant generality, are of great practical interest for testing various numerical and approximate analytical methods for solving corresponding initial–boundary value problems). Many new generalized traveling-wave solutions and functional separable solutions (in closed form) are described. Solutions of several Klein–Gordon equations with delay are also given.

Construction of exact solutions in implicit form for PDEs: New functional separable solutions of non-linear reaction–diffusion equations with variable coefficients

The paper deals with different classes of non-linear reaction–diffusion equations with variable coefficients c(x)ut=[a(x)f(u)ux]x+b(x)g(u),that admit exact solutions. The direct method for constructing functional separable solutions to these and more complex non-linear equations of mathematical physics is described. The method is based on the representation of solutions in implicit form ∫h(u)du=ξ(x)ω(t)+η(x),where the functions h(u), ξ(x), η(x), and ω(t) are determined further by analyzing the resulting functional-differential equations. Examples of specific reaction–diffusion type equations and their exact solutions are given. The main attention is paid to non-linear equations of a fairly general form, which contain several arbitrary functions dependent on the unknown u and /or the spatial variable x (it is important to note that exact solutions of non-linear PDEs, that contain arbitrary functions and therefore have significant generality, are of great practical interest for testing various numerical and approximate analytical methods for solving corresponding initial–boundary value problems). Many new generalized traveling-wave solutions and functional separable solutions are described.

Abundant wave solutions of the Boussinesq equation and the (2+1)-dimensional extended shallow water wave equation

In this article, we establish the exact wave solutions of the Boussinesq equation and the (2 + 1)-dimensional extended shallow water wave equation by applying the new generalized (G'/G) -expansion method. When the condition of the fluid is such that the horizontal length scale is much greater than the vertical length scale, the shallow water equations are mostly suitable. In Ocean engineering, Boussinesq-type equations are commonly used in computer simulations for the model of water waves in shallow seas and harbors. We explained the new generalized (G'/G) -expansion method to seek further general traveling wave solutions of the above mentioned equations. The traveling wave solutions attained by this method are exposed in terms of hyperbolic, trigonometric and rational functions. The shape of the obtained solutions are bell shaped soliton, kink soliton, singular kink soliton, singular soliton, singular periodic solution and compaction. This method is very influential mathematical tool for extracting exact solutions of NLEEs which frequently arise in mathematical physics, engineering sciences and many scientific real world application fields.