Non-travelling Wave Solutions Research Articles

The new kink type and non-traveling wave solutions of (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation

In this paper, the new solitary wave solutions of the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation are obtained by Lie group symmetry method and the extended homoclinic test approach. Firstly, the equation can be reduced to (1+1)-dimensional partial differential equation by Lie group symmetry, and corresponding bilinear forms of the equation are given by symmetry functions. Secondly, the extended homoclinic test approach is employed to obtain the new kink type and singular solitary wave solutions. In addition, some new traveling and non-traveling wave solutions with arbitrary functions and oscillating tail are investigated through the special transformations for the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation.

New exact solutions of nontraveling wave and local excitation of dynamic behavior for GGKdV equation

For GGKdV equation, solutions of the compatible KdV equation are obtained by using CKdVE method, and Lie point symmetry group of the equation is also obtained. Further, some new exact non-traveling wave solutions are obtained by using the equivalent transformation method and elliptic function method on solving the corresponding symmetric reduction equation, and local excitation modes of three kinds of solutions under three different groups of parameters are presented. Finally, the integrability in the sense of the CkdVE and the Lie Symmetric are proved, which shows the effectiveness of the organic combination of various kinds of nonlinear analytical methods. This CkdVE method communicated the mathematical relations of different nonlinear models. It is a new bridge between the known and unknown solutions of the nonlinear partial differential equations, and it is a new way to explore complex nonlinear complex phenomena.

Use of optimal subalgebra for the analysis of Lie symmetry, symmetry reductions, invariant solutions and conservation laws of the (3 + 1)-dimensional extended Sakovich equation

This paper investigates the [Formula: see text]-dimensional extended Sakovich equation, which represents an essential nonlinear scientific model in the field of ocean physics. The Lie symmetry analysis has been utilized for extracting the non-traveling wave solutions of the [Formula: see text]-dimensional extended Sakovich equation. These solutions are investigated through infinitesimal generators, which are obtained from Lie’s continuous group of transformations. As there are infinite possibilities for the linear combination of infinitesimal generators, so a one-dimensional optimal system of subalgebra has been established using Olver’s standard approach. Moreover, by considering the optimal system of subalgebra, the extended Sakovich equation is converted into a solvable nonlinear PDE through symmetry reductions. Finally, the conservation laws for the governing equation have been derived using Ibragimov’s generalized theorem and quasi-self-adjointness condition.

A New Composite Technique to Obtain Non-traveling Wave Solutions of the (2+1)-dimensional Extended Variable Coefficients Bogoyavlenskii–Kadomtsev–Petviashvili Equation

A New Composite Technique to Obtain Non-traveling Wave Solutions of the (2+1)-dimensional Extended Variable Coefficients Bogoyavlenskii–Kadomtsev–Petviashvili Equation

Exact Solutions and Non-Traveling Wave Solutions of the (2+1)-Dimensional Boussinesq Equation

By the extended (G′G) method and the improved tanh function method, the exact solutions of the (2+1) dimensional Boussinesq equation are studied. Firstly, with the help of the solutions of the nonlinear ordinary differential equation, we obtain the new traveling wave exact solutions of the equation by the homogeneous equilibrium principle and the extended (G′G) method. Secondly, by constructing the new ansatz solutions and applying the improved tanh function method, many non-traveling wave exact solutions of the equation are given. The solutions mainly include hyperbolic, trigonometric and rational functions, which reflect different types of solutions for nonlinear waves. Finally, we discuss the effects of these solutions on the formation of rogue waves according to the numerical simulation.

Abundant explicit non-traveling wave solutions for the (2+1)-dimensional breaking soliton equation

By combining the generalized variable separation method with the extended homoclinic test approach (EHTA) explicit exact non-traveling wave solutions of the (2+1)-dimensional breaking soliton equation are constructed. With the aid of symbolic computation, a series of new non-traveling wave solutions of the (2+1)-dimensional breaking soliton equation are expressed explicitly. These non-traveling wave solutions are new solutions with three arbitrary functions, which have a more general form than that in the previous literatures. The result obtained here reveals the complex structure of the solutions of the (2+1)-dimensional breaking soliton equation. The previous results obtained in literatures can be regarded as special cases here. When some arbitrary functions included in these solutions are taken as some special functions, exact periodic solitary wave, cross soliton-like wave, periodic cross-kink wave, periodic two-solitary wave are presented.

Study on double-periodic soliton and non-traveling wave solutions of integrable systems with variable coefficients

An improved test function method for finding double-periodic soliton and non-traveling wave solutions of integrable systems with variable coefficients is proposed and applied to solve the variable-coefficient Kadomtsev–Petviashvili equation, which describes waves in ferromagnetic media and matter-wave pulses in Bose–Einstein condensates.

New non-traveling wave solutions for the (2+1)-dimensional variable coefficients Date-Jimbo-Kashiwara-Miwa equation

In the paper, we mainly study the (2+1)-dimensional variable coefficients Date-Jimbo-Kashiwara-Miwa equation (VC-DJKM equation) by combining the extended homoclinic test approach and the generalized variable separation method. Applying this thought and with the aid of symbolic computation, we present thirty six kinds of new exact non-traveling wave solutions of the (2+1)-dimensional VC-DJKM equation including kink-like solution, singular solitary wave-like solution, periodic solitary wave-like solution, kinky breather wave-like solution, and so on. These results all have a like-parabolic tail which reveals the complex structure of solution and maybe give a prediction of physical phenomenon. The generalized variable separation method greatly enriches the types and structures of solutions. Moreover, when some functions in the generalized variable separation form take 0 or 1, it will degenerate into the variable separation form of multiplication or addition. As the special case of VC-DJKM equation, the corresponding results of the (2+1)-dimensional DJKM equation with constant coefficients are also given.

Propagation and interaction between special fractional soliton and soliton molecules in the inhomogeneous fiber

IntroductionFractional nonlinear models have been widely used in the research of nonlinear science. A fractional nonlinear Schrödinger equation with distributed coefficients is considered to describe the propagation of pi-second pulses in inhomogeneous fiber systems. However, soliton molecules based on the fractional nonlinear Schrödinger equation are hardly reported although many fractional soliton structures have been studied. ObjectivesThis paper discusses the propagation and interaction between special fractional soliton and soliton molecules based on analytical solutions of a fractional nonlinear Schrödinger equation. MethodsTwo analytical methods, including the variable-coefficient fractional mapping method and Hirota method with the modified Riemann–Liouville fractional derivative rule, are used to obtain analytical non-travelling wave solutions and multi-soliton approximate solutions. ResultsAnalytical non-travelling wave solutions and multi-soliton approximate solutions are derived. The form conditions of soliton molecules are given, and the dynamical characteristics and interactions between special fractional solitons, multi-solitons and soliton molecules are discussed in the periodic inhomogeneous fiber and the exponential dispersion decreasing fiber. ConclusionAnalytical chirp-free and chirped non-traveling wave solutions and multi-soliton approximate solutions including soliton molecules are obtained. Based on these solutions, dynamical characteristics and interactions between special fractional solitons, multi-solitons and soliton molecules are discussed. These theoretical studies are of great help to understand the propagation of optical pulses in fibers.

New non-traveling wave solutions for (3+1)-dimensional variable coefficients Date-Jimbo-Kashiwara-Miwa equation

<abstract><p>In this paper, we investigate non-traveling wave solutions of the (3+1)-dimensional variable coefficients Date-Jimbo-Kashiwara-Miwa (VC-DJKM) equation, which describes the real physical phenomena owing to the inhomogeneities of media. By combining the extended homoclinic test approach with variable separation method, we obtain abundant new exact non-traveling wave solutions of the (3+1)-dimensional VC-DJKM equation. These results with a parabolic tail or linear tail reveal the complex structure of the solutions for (3+1)-dimensional VC-DJKM equation. Moreover, the tail in these solutions maybe give a prediction of physical phenomenon. When arbitrary functions contained in these non-traveling wave solutions are taken as some special functions, we can get the kink-type solitons, singular solitary wave solutions, and periodic solitary wave solutions, and so on. As the special cases of our work, the corresponding results of (3+1)-dimensional DJKM equation, (2+1)-dimensional DJKM equation, (2+1)-dimensional VC-DJKM equation are also given.</p></abstract>

ABUNDANT NEW NON-TRAVELING WAVE SOLUTIONS FOR THE (3+1)-DIMENSIONAL BOITI-LEON-MANNA-PEMPINELLI EQUATION

<abstract> Seeking exact solutions of higher-dimensional nonlinear partial differential equations has recently received tremendous attention in mathematics and physics. In this paper, we investigate exact solutions of (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation which describes nonlinear wave propagation in incompressible fluid. Firstly, by means of extended homoclinic test approach, we get eight kinds of non-traveling wave solutions of (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Then, combining the improved tanh function method and new ansatz solutions, we obtain abundant new exact non-traveling wave solutions of (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. These results include not only many results obtained in other literatures, but also some new exact non-traveling wave solutions. Moreover, the exact kink wave solutions, periodic solitary wave solutions and singular solitary wave solutions are given when arbitrary functions contained in these solutions are taken as some special functions. </abstract>

Abundant new non-travelling wave solutions for the (3+1)-dimensional potential-YTSF equation

By combining the extended homoclinic test approach (EHTA) with the method of separation of variables explicit exact non-traveling wave solutions of the (3+1)-dimensional potential Yu-Toda-Sasa-Fukuyama (YTSF) equation is constructed. With the aid of symbolic computation, abundant new explicit exact solutions of the (3+1)-dimensional potential-YTSF equation are obtained. The results obtained her e reveal the complex structure of the solutions of the (3+1) dimensional potential-YTSF equation. Many of the previous results obtained in other literature can be seen as special cases here. When some arbitrary functions included in these solutions are taken as some special functions, exact periodic solitary wave, periodic soliton, periodic cross-kink wave, periodic two-solitary wave are presented.

Double-wave solutions and Lie symmetry analysis to the (2 + 1)-dimensional coupled Burgers equations

This paper investigates the (2 + 1)-dimensional coupled Burgers equations (CBEs) which is an important nonlinear physical model. In this respect, by making use of the generalized unified method (GUM), a series of double-wave solutions of the (2 + 1)-dimensional coupled Burgers equations are derived. The Lie symmetry technique (LST) is also utilized for the symmetry reductions of the (2 + 1)-dimensional coupled Burgers equations and extracting a non-traveling wave solution. Through some figures, we discussed the wave structures of the double-wave solutions of the CBEs for different values of parameters in these solutions.

Exact traveling and non-traveling wave solutions of the time fractional reaction–diffusion equation

In this study, solutions to the nonlinear time-dependent fractional reaction–diffusion equation with conformal fractional derivative is considered. In the first part of the manuscript, we reduce the fractional equation to a traditional differential equation using the fractional complex transformation. Two cases where the exact solution exists are then presented. One of these solutions is used to model experimental data showing anomalous diffusion in freestanding graphene. In the second part, we study the traveling solutions and present two cases: Canonical-like transformation method and complete discrimination system for polynomial method.

EXPLICIT AND EXACT NON-TRAVELING WAVE SOLUTIONS OF (3+1)-DIMENSIONAL GENERALIZED SHALLOW WATER EQUATION

In this paper, a (3+1)-dimensional generalized shallow water equation is considered. New exact solutions in forms of the hyperbolic functions and the trigonometric functions are obtained based on an extended $(G'/G)$-expansion method and the variable separation method, which contain traveling wave solutions and non-traveling wave solutions. The particular localized excitations and the interactions between two solitary waves for these obtained exact solutions are shown in some three-dimensional graphics.

Some New Non-Travelling Wave Solutions of the Fisher Equation with Nonlinear Auxiliary Equation

We have generated many new non-travelling wave solutions by executing the new extended generalized and improved (G'/G)-Expansion Method. Here the nonlinear ordinary differential equation with many new and real parameters has been used as an auxiliary equation. We have investigated the Fisher equation to show the advantages and effectiveness of this method. The obtained non-travelling solutions are expressed through the hyperbolic functions, trigonometric functions and rational functional forms. Results showing that the method is concise, direct and highly effective to study nonlinear evolution equations those are in mathematical physics and engineering.

Explicit and exact nontraveling wave solutions of the (3+1)-dimensional potential Yu–Toda–Sasa–Fukuyama equation

With the aid of the symbolic computation, we present an improved ( G ′ / G ) -expansion method, which can be applied to seek more types of exact solutions for certain nonlinear evolution equations. In illustration, we choose the (3 + 1)-dimensional potential Yu–Toda–Sasa–Fukuyama equation to demonstrate the validity and advantages of the method. As a result, abundant explicit and exact nontraveling wave solutions are obtained including two solitary waves solutions, nontraveling wave solutions and dromion soliton solutions. Some particular localized excitations and the interactions between two solitary waves are researched. The method can be also applied to other nonlinear partial differential equations.

New Explicit Exact Solutions of the One-Dimensional Parabolic-Parabolic Keller-Segel Model

One-dimensional parabolic-parabolic Keller-Segel (PP-KS) model of chemotaxis is considered. By using the generalized tanh function method, <i>G'</i>(/G)-expansion method and variable-separating method, plenty of new explicit exact solutions, including travelling wave solutions and non-travelling wave solutions, are obtained for the PP-KS model. Compared to the existing results, more new exact solutions are derived and the obtained solutions all have explicit expressions.

New non-traveling wave solutions for the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation

The ( G ′ ∕ G ) -expansion approach is an efficient and well-developed approach to solve nonlinear partial differential equations. In this paper, the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation is investigated by using this approach, which describes the (2+1)-dimensional interaction of the Riemann wave propagated along the y -axis with a long wave propagated along the x -axis and can be considered as a model for the incompressible fluid. With the aid of symbolic computation, a family of exact solutions are obtained in forms of the hyperbolic functions and the trigonometric functions. When the parameters are selected special values, non-traveling wave solutions are also presented, and these gained solutions have abundant structures. The figures corresponding to these solutions are illustrated to show the particular localized excitations and the interactions between two solitary waves.

Symmetry Reductions and New Exact Non-Traveling Wave Solutions of b -Family Equations

Symmetry Reductions and New Exact Non-Traveling Wave Solutions of b -Family Equations