Non-travelling Wave Research Articles

Application of the [formula omitted]-expansion method for the generalized Fisher’s equation and modified equal width equation

In this work, the exact traveling wave solutions of the generalized Fisher’s equation and modified equal width equation are studied by using the G′G-expansion method. As a result, many solitary wave solutions are derived from the solutions via hyperbolic functions, trigonometric functions and rational functions. When the parameters were taken at special values, the results obtained were compared with the solution via the tanh method established earlier. In fact, many general non-traveling wave solutions are obtained. The efficiency of the method is demonstrated by applying it for a variety of selected equations.

Exact solutions of (3+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq equations

The symmetries and the exact solutions of the (3+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq (INHB) equations, which describe atmospheric gravity waves, are studied in this paper. The calculation on symmetry shows that the equations are invariant under the Galilean transformations, the scaling transformations, and the space—time translations. Three types of symmetry reduction equations and similar solutions for the (3+1)-dimensional INHB equations are proposed. Traveling and non-traveling wave solutions of the INHB equations are demonstrated. The evolutions of the wind velocities in latitudinal, longitudinal, and vertical directions with space—time are demonstrated. The periodicity and the atmosphere viscosity are displayed in the (3+1)-dimensional INHB system.

SOME EXACT BLOWUP SOLUTIONS TO MULTIDIMENSIONAL SCHRÖDINGER MAP EQUATION ON HYPERBOLIC SPACE AND CONE

Exact solutions for the multidimensional Schrödinger map equation (SM for short) on hyperbolic 2-space [Formula: see text] cone are obtained. Consequently, we show the non-traveling wave solution on [Formula: see text] is a finite energy solution on the finite spacial domain. The question of whether a solution of SM can develop a finite time singularity on [Formula: see text] with smooth initial data is not clear. Our result show that blowup can really happen on this initial data. In addition, some exact global smooth solutions are constructed.

Novel loop-like solitons for the generalized Vakhnenko equation

A non-traveling wave solution of a generalized Vakhnenko equation arising from the high-frequent wave motion in a relaxing medium is derived via the extended Riccati mapping method. The solution includes an arbitrary function of an independent variable. Based on the solution, two hyperbolic functions are chosen to construct new solitons. Novel single-loop-like and double-loop-like solitons are found for the equation.

Saddle-Node Heteroclinic Orbit and Exact Nontraveling Wave Solutions for (2+1)D KdV-Burgers Equation

We have undertaken the fact that the periodic solution of (2+1)D KdV-Burgers equation does not exist. The Saddle-node heteroclinic orbit has been obtained. Using the Lie group method, we get two-(1+1)-dimensional PDE, through symmetric reduction; and by the direct integral method, spread F-expansion method, and <svg style="vertical-align:-2.3205pt;width:47.662498px;" id="M1" height="20.6" version="1.1" viewBox="0 0 47.662498 20.6" width="47.662498" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,17.65)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,5.944,17.65)"><path id="x1D43A" d="M713 296l-5 -25q-47 -7 -59 -20t-23 -72l-15 -79q-9 -48 -3 -74q-15 -3 -55 -13t-63 -15t-59.5 -10t-70.5 -5q-149 0 -243 80t-94 220q0 169 127.5 276.5t336.5 107.5q91 0 206 -36l-10 -165l-29 -1q1 85 -47.5 126t-146.5 41q-153 0 -243.5 -97t-90.5 -242&#xA;q0 -122 68.5 -198.5t188.5 -76.5q121 0 139 75l20 86q13 58 -1.5 70.5t-99.5 20.5l5 26h267z" /></g> <g transform="matrix(.012,-0,0,-.012,18.25,9.488)"><path id="x2032" d="M227 744l-123 -338l-31 15l73 368q12 3 41.5 -8t36.5 -20z" /></g> <g transform="matrix(.017,-0,0,-.017,22.412,17.65)"><path id="x2F" d="M368 703l-264 -866h-60l265 866h59z" /></g><g transform="matrix(.017,-0,0,-.017,29.416,17.65)"><use xlink:href="#x1D43A"/></g><g transform="matrix(.017,-0,0,-.017,41.723,17.65)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g> </svg>-expansion method, we obtain exact nontraveling wave solutions, for the (2+1)D KdV Burgers equation, and find out some new strange phenomenons of sympathetic vibration to evolution of nontraveling wave.

Non-Traveling Wave Solutions for the (1 + 1)-Dimensional Burgers System by Riccati Equation Mapping Approach

Starting from the symbolic computation system Maple and Riccati equation mapping approach and a linear variable separation approach, a new family of non-traveling wave solutions of the (1 + 1)-dimensional Burgers system is derived.

Group invariant solutions for the unsteady MHD flow of a third grade fluid in a porous medium

This work describes the time-dependent flow of an incompressible non-Newtonian fluid over an infinite rigid plate. The flow is induced due to the arbitrary velocity V(t) of the plate. The fluid occupies the porous half space y>0 and is also electrically conducting in the presence of a constant applied magnetic field in the transverse direction to the flow. Analytical solutions of the governing non-linear partial differential equation for the unidirectional flow of a third grade fluid are established using the symmetry approach. We construct three types of analytical solutions by employing the Lie symmetry method and the better solution from the physical point of view is shown to be the non-travelling wave solution. We also present numerical solutions of the governing PDE and the reduced ODE and compare with the analytical results. Finally, the influence of emerging parameters are studied through several graphs with emphasis on the study of the effects of the magnetic field and non-Newtonian fluid parameters.

Non-Traveling Wave Solutions for the (2+1)-Dimensional Breaking Soliton System

In this work, starting from the (G'/G)-expansion method and a variable separation method, a new non-traveling wave general solutions of the (2+1)-dimensional breaking soliton system are derived. By selecting appropriately the arbitrary functions in the solutions, special soliton-structure excitations and evolutions are studied.

Novel Chaotic Behaviors and Evolutions for the Nonlinear (3+1)-Dimensional Burgers System

Based on the non-traveling wave solution and a chemical chaos system, chaotic soliton excitations are established for the nonlinear (3+1)-dimensional Burgers system. The chaotic behavior and chaotic evolution of the system are studied.

Application of rational expansion method for stochastic differential equations

By means of the Hermite transformation, a new general ansätz and the symbolic computation system Maple, we apply the Riccati equation rational expansion method [24] to uniformly construct a series of stochastic non-traveling wave solutions for stochastic differential equations. To illustrate the effectiveness of our method, we take the stochastic mKdV equation as an example, and successfully construct some new and more general solutions. The method can also be applied to solve other nonlinear stochastic differential equation or equations.

Variable-coefficient Jacobi elliptic function expansion method for (2+1)-dimensional Nizhnik–Novikov–Vesselov equations

Abstract In this paper, a variable-coefficient Jacobi elliptic function expansion method is proposed to seek more general exact solutions of nonlinear partial differential equations. Being concise and straightforward, this method is applied to the (2+1)-dimensional Nizhnik–Novikov–Vesselov equations. As a result, many new and more general exact non-travelling wave and coefficient function solutions are obtained including Jacobi elliptic function solutions, soliton-like solutions and trigonometric function solutions. To give more physical insights to the obtained solutions, we present graphically their representative structures by setting the arbitrary functions in the solutions as specific functions.

Exact Solutions of Bose-Einstein Condensate in Linear Magnetic Field and Time-Dependent Laser Field

Nonlinear evolution equations (NLEEs) are frequently used to model a wide variety of phenomena in physics, chemistry, biology and even in social science field. Seeking and constructing exact solutions especially soliton solutions of NLEEs is one of the most important and essential task in nonlinear science. With the help of exact solutions, when they exist, the nonlinear phenomena can be better understood. In the past decades, various sophisticated methods have been created such as the inverse scattering method [1], the Backlund transformations [2], the Darboux transformations [3] and the Hirota bilinear method [4]. Especially, in recent years, with the rapid development of computerized symbolic computation, many direct and effective algebraic methods are proposed such as the homogeneous balance method [5], tanh function method [6], the Jacobi elliptic function expansion method [7], the F-expansion techniques [8], exp-function method [9] and so on. Recently, Wang et al. [10] introduced a new direct method called G ′ G -expansion method for a reliable treatment of the NLEEs. Then this method is generalized and widely used to seek travelling wave solutions, non-travelling wave solutions and coefficient function solutions for many nonlinear physical models [11–14], and even extended to solve NLEE with variable coefficients [15]. The Gross–Pitaevskii equation appears as a relevant model in various areas of physics: nonlinear optics, fluid mechanics, Bose–Einstein condensation (BEC). The realization of BEC of weakly interacting atomic gases stim-

Chaotic Behavior of the Nonlinear (3+1)-Dimensional Burgers System

Based on the non-traveling wave solution and Rossler chaos system, chaotic soliton excitations are established for the nonlinear (3+1)-dimensional burgers system. The chaotic behavior and chaotic evolution of the system are investigated.

Exact Solutions of (2 + 1)-Dimensional Boiti–Leon–Pempinelle Equation with (G′/G)-Expansion Method

In this paper, we construct exact solutions for the (2 + 1)-dimensional Boiti–Leon–Pempinelle equation by using the (G′/G)-expansion method, and with the help of Maple. As a result, non-travelling wave solutions with three arbitrary functions are obtained including hyperbolic function solutions, trigonometric function solutions, and rational solutions. This method can be applied to other higher-dimensional nonlinear partial differential equations.

New Application of the (G'/G)-Expansion Method to Excite Soliton Structures for Nonlinear Equation

The (G'/G)-expansion method is extended to construct non-travelling wave solutions for highdimensional nonlinear equations and to explore special soliton structure excitations and evolutions. Taking an example, a new series of the non-travelling wave solutions are calculated for the (2+1)- dimensional asymmetrical Nizhnik-Novikov-Veselov system by using the (G'/G)-expansion method. By selecting appropriately the arbitrary functions in the solutions, special soliton-structure excitations and evolutions are studied.

Exact solutions of (2+1) -dimensional Bogoyavlenskii’s breaking soliton equation with symbolic computation

In this paper, the generalized ( G ′ G )-expansion method (Wang et al. (2008) and Zhang et al. (2008) [13] , [14] ) is used to construct exact solutions of the (2+1)-dimensional Bogoyavlenskii’s breaking soliton equation. As a result, non-travelling wave solutions with three arbitrary functions are obtained including hyperbolic function solutions, trigonometric function solutions and rational solutions. If the parameters take different values, some more solutions are derived. It is shown that the generalized ( G ′ G )-expansion method, with the help of symbolic computation, provides an effective and powerful method for solving high-dimensional nonlinear partial differential equations in mathematical physics.

A method for constructing nontraveling wave solutions for (1+1)-dimensional evolution equations

A method is proposed to construct nontraveling wave solution for (1+1)-dimensional evolution equations by extending the linear traveling wave transformation of the (G′/G)-expansion method into a nonlinear transformation. Owing to a built-in arbitrary function included in the solution, abundant solutions can be excited. Taking the Vakhnenko equation as an example, a series of nontraveling wave solutions with variable separation is obtained. Some new solutions are excited, and the known solitary wave solutions are special cases of the nontraveling wave solutions.

New exact traveling and non-traveling wave solutions for (2 + 1)-dimensional Burgers equation

In this paper, some novel solitary wave solutions, including solitary-like wave solution, x-periodic soliton solution, y-periodic soliton solution, doubly periodic solution, rational solution, and new non-traveling wave solution, are obtained for (2 + 1)-dimensional Burgers equation by means of the generalized direct ansätz method and different test functions.

The non-traveling wave solutions and novel fractal soliton for the (2 + 1)-dimensional Broer–Kaup equations with variable coefficients

A method is proposed by extending the linear traveling wave transformation into the nonlinear transformation with the ( G′/ G)-expansion method. The non-traveling wave solutions with variable separation can be constructed for the (2 + 1)-dimensional Broer–Kaup equations with variable coefficients via the method. A novel class of fractal soliton, namely, the cross-like fractal soliton is observed by selecting appropriately the arbitrary functions in the solutions.

Symmetry reductions and new exact non-traveling wave solutions of potential Kadomtsev–Petviashvili equation with p-power

With the aid of Maple symbolic computation and Lie group method, PKPp equation is reduced to some (1+1)-dimensional partial differential equations, in which there are linear PDE with constant coefficients, nonlinear PDE with constant coefficients, and nonlinear PDE with variable coefficients. Using the separation of variables, homoclinic test technique and auxiliary equation methods, we obtain new abundant exact non-traveling solutions with arbitrary functions for the PKPp.