Dimensional Soliton Equation Research Articles

Breathers, Soliton and Hybrid Solutions for generalized (2＋1) dimensional Soliton Equation

This article studies lump and its interaction solutions, breather and solitary wave solutions for generalized (2＋1) dimensional Soliton equation by using direct method based on Hirota’s bilinear form. Lump and its interaction solutions are obtained by assuming test function as a combination of positive, trigonometric and exponential functions. Peak and depth values of lump solutions are also determined using extreme points. 3D, 2D and contour plots are given to comprehend the physical features of these solutions.

Integrability of a generalized (2+1)-dimensional soliton equation via Bell polynomials

Integrability of a generalized (2+1)-dimensional soliton equation via Bell polynomials

Multiple soliton, M-lump and interaction solutions to the (3+1)-dimensional soliton equation

One of the most effective ways to understand nonlinear quantum systems is with lump solutions. The objective of this study is to acquire more about the (3+1)-dimensional soliton equation. We successfully present this equation with various solitons and M-lump solutions. We adopt specific parameter values to accentuate the physical features of the provided exact solutions through 3D and contour plots as doing so is of extreme significance. The submitted results indicate the physical qualities of lump-and-lump interaction events in various nonlinear physical processes.

Analytical Solutions of the Nonlinear (2 + 1)-Dimensional Soliton Equation by Using Some Methods

In this work, it has been applied two methods for solving the (2+1)-dimensional soliton equation, namely, the ansatz method and the F-expansion method. These methods are utilized to provide new accurate periodic and soliton solutions to this problem that are more generic. An appropriate transformation can be used to convert this nonlinear system into another nonlinear ordinary differential equation. In mathematical physics, it is demonstrated that the ansatz method and the F-expansion method give a strong mathematical tool for solving a large number of systems of nonlinear partial differential equations.

Algebro-Geometric Solutions of a ( 2 + 1 )-Dimensional Integrable Equation Associated with the Ablowitz-Kaup-Newell-Segur Soliton Hierarchy

The ( 2 + 1 )-dimensional Lax integrable equation is decomposed into solvable ordinary differential equations with the help of known ( 1 + 1 )-dimensional soliton equations associated with the Ablowitz-Kaup-Newell-Segur soliton hierarchy. Then, based on the finite-order expansion of the Lax matrix, a hyperelliptic Riemann surface and Abel-Jacobi coordinates are introduced to straighten out the associated flows, from which the algebro-geometric solutions of the ( 2 + 1 )-dimensional integrable equation are proposed by means of the Riemann θ functions.

Extended hyperbolic function method for the (2 +1)-dimensional nonlinear soliton equation

By employing the extended hyperbolic function method (EHFM), we extract the exact solutions of the (2+1)-dimensional nonlinear soliton equation (SE). A soliton equation is used for investigation of the dynamics of nonlinear waves in plasma physics and fluid dynamics. A various new techniques for finding exact solutions of the (2+1)-dimensional nonlinear SE are satisfactorily acquired with the help of EHFM. The EHFM presents various types of new solutions in the form of dark, singular, periodic, bright solitons and some rational function solutions. In addition, for the physical characterization of the acquired solutions of (2+1)-dimensional SE, some 2-dim and 3-dim plots are drawn. The attained results are novel for the considered equation, and results reveal that the method is concise, direct and competent which can be assembled in other complex phenomena.

An investigation to the nonlinear (2 + 1)-dimensional soliton equation for discovering explicit and periodic wave solutions

In this paper, the double (G'/G, 1/G)-expansion method is employed to establish explicit general solutions of the nonlinear (2 + 1)-dimensional soliton equation. A variety of exact travelling wave solutions are attained involving three functions that are classified into rational, trigonometric and hyperbolic with especial parameters that originate the solitary explicit and new periodic solutions. The used method is the generalization of the (G'/G)-expansion method and it rediscovers all the acquainted solitary wave solutions that are obtained by means of the (G'/G)-expansion method.

New lump, lump-kink, breather waves and other interaction solutions to the (3+1)-dimensional soliton equation

This study investigates the (3+1)-dimensional soliton equation via the Hirota bilinear approach and symbolic computations. We successfully construct some new lump, lump-kink, breather wave, lump periodic, and some other new interaction solutions. All the reported solutions are verified by inserting them into the original equation with the help of the Wolfram Mathematica package. The solution’s visual characteristics are graphically represented in order to shed more light on the results obtained. The findings obtained are useful in understanding the basic nonlinear fluid dynamic scenarios as well as the dynamics of computational physics and engineering sciences in the related nonlinear higher dimensional wave fields.

Time-dependent defects in integrable soliton equations.

We study (1 + 1)-dimensional integrable soliton equations with time-dependent defects located at x = c(t), where c(t) is a function of class C 1. We define the defect condition as a Bäcklund transformation evaluated at x = c(t) in space rather than over the full line. We show that such a defect condition does not spoil the integrability of the system. We also study soliton solutions that can meet the defect for the system. An interesting discovery is that the defect system admits peaked soliton solutions.

A Study on Lump and Interaction Solutions to a (3 + 1)‐Dimensional Soliton Equation

Based on bilinear formulation of a (3 + 1)‐dimensional soliton equation, lump solution and related interaction solutions are investigated. The lump solutions of the soliton equation are classified into three cases with nonsingularity conditions being given. The interaction solutions between lump and a stripe soliton are obtained in eight cases, which have interesting fusing and fission behaviors with changing time. The interaction solutions of the soliton equation between a lump and a resonant pair of stripe solitons are also given, and we find that the lump just exist for a finite period during the interaction process.

The solitary wave, rogue wave and periodic solutions for the ( $$3+1$$ 3 + 1 )-dimensional soliton equation

Based on the Hirota’s bilinear form, the ( $$3+1$$ )-dimensional soliton equation is investigated. The solitary wave, rogue wave and periodic solutions are obtained. Their interaction behaviors are shown in some three-dimensional plots and corresponding contour plots by setting different values of the parameters.

Construction of lump soliton and mixed lump stripe solutions of (3+1)-dimensional soliton equation

Abstract In this letter, we apply two different ansatzs for constructing the lump soliton and mixed lump strip solutions of (3+1)-dimensional soliton equation, which is associating with the Hirota bilinear form. These lump soliton solutions rationally localized in all directions in the space. The solutions of interactions between a lump and a stripe are shown by graphic illustration of some special solutions which would give us a better understanding on the evolution of solutions of waves.

Solitary wave solutions of (2 + 1)-dimensional soliton equation arising in mathematical physics

In this article, new solitary wave solutions of (2+1)−dimensional soliton equation are constructed. This equation is similar to integrable Zakharov equation in plasma physics. The sine-Gordon expansion method and (G′/G)-expansion method are utilized for a reliable treatment of (2+1)-dimensional soliton equation. Many novel solutions such as rational, hyperbolic and complex function solutions are characterized with some free parameters to the problem studied.

New periodic wave solutions of (3+1)-dimensional soliton equation

In this paper, associating with the Hirota bilinear form, the three-wave method, which is applied to construct some periodic wave solutions of (3+1)-dimensional soliton equation, is a powerful approach to obtain periodic solutions for many non-linear evolution equations in the integrable systems theory.

Elastic collision of mobile solitons of a (3 + 1)-dimensional soliton equation

The multiple exp-function method is a new approach to obtain multiple-wave solutions of nonlinear partial differential equations (NLPDEs). By this method, one can obtain multi-soliton solutions of NLPDEs. Hence, in this paper, using symbolic computation, we apply the multiple exp-function method to construct the exact multiple-wave solutions of a (3 + 1)-dimensional soliton equation. Based on this application, we obtain mobile single-wave, double-wave and multi-wave solutions for this equation. In addition, we employ the straightforward and algebraic Hirota bilinearization method to construct the multi-soliton solutions of NLPDEs, and we reveal the remarkable property of soliton–soliton collision through this approach. Further, we investigate the one- and two-soliton solutions of a (3 + 1)-dimensional soliton equation using the Hirota’s method. We explore the particle-like behavior or elastic interaction of solitons, which has potential application in optical communication systems and switching devices.

Traveling wave solutions of a (2 + 1)-dimensional Zakharov-like equation by the first integral method and the tanh method

In this paper, the first integral method and a variety of solitary waves methods are employed for constructing the new exact traveling wave solutions of a (2+1)-dimensional Zakharov-like soliton equation. Moreover, this equation will be studied by a variety of solitary waves methods and ansatze.

Applications of extended F-expansion and projective Ricatti equation methods to (2+1)-dimensional soliton equations

The (2+1)-dimensional Maccari and nonlinear Schrödinger equations are reduced to a nonlinear ordinary differential equation (ODE) by using a simple transformation, various solutions of the nonlinear ODE are obtained by using extended F-expansion and projective Ricatti equation methods. With the aid of solutions of the nonlinear ODE more explicit traveling wave solutions expressed by the hyperbolic functions, trigonometric functions and rational functions are found out. It is shown that these methods provides a powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

New Periodic and Soliton Solutions of (2 + 1)-Dimensional Soliton Equation

By using the sine-cosine method proposed recently, we give the exact periodic and soliton solutions of the (2 + 1)-dimensional soliton equation in this paper. Many new families of exact traveling wave solutions of the (2 + 1)-dimensional soliton equation are successfully obtained. The computation for the method appears to be easier and faster by general mathematical software.

He's variational method for a (2 + 1)-dimensional soliton equation

In this Letter, a (2 + 1)-dimensional soliton equation is studied by He’s variational approach. The solitary solutions are obtained using the Ritz method.

Periodic Wave Solutions to a (3+1)-Dimensional Soliton Equation

Two classes of periodic wave solutions to the (3+1)-dimensional soliton equation are derived by employing the Hirota bilinear method and theta function identities. These solutions are expressed in terms of Riemann theta functions of genus one and can be converted into an elliptic function format, both their long wave limit and extremum value are discussed in detail.