Extended Tanh-function Method Research Articles

Novel types of interactions between solitons in the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Veselov system

Using the extended tanh-function method (ETM) based on mapping method, the variable separation solutions of the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Veselov (ANNV) system are derived. By further studying, we find that these variable separation solutions, which seem independent, actually depend on each other. Based on the variable separation solution and by selecting appropriate functions, new types of interactions among semi-foldons and special dromions, peakons, foldons constructed by multi-valued functions are investigated. Moreover, The explicit phase shifts for all the local excitations offered by the quantity U have been given, and are applied to these novel interactions in detail.

Exotic localized structures based on variable separation solution of (2+1)-dimensional KdV equation via the extended tanh-function method

In this paper, firstly, we obtain the variable separation solutions of (2 + 1)-dimensional KdV equation by the extended tanh-function method (ETM) based on mapping method. Novel localized coherent structures about multi-valued functions, i.e. special dromion, special peakon and foldon, and the interactions among them, are discussed. The interactions between two special dromions and between two special peakons possess novel property, that is, there exists a multi-valued foldon in the process of their collision, which is different from the reported cases in previous literature. Moreover, the explicit phase shifts for all the local excitations offered by the quantity u have been given, and are applied to these novel interactions in detail.

A Burgers equation-based constructive method for solving nonlinear evolution equations

Inspired by the extended tanh function method and the direct method of symmetry reduction, we present a Burgers equation-based constructive method for solving nonlinear evolution equations (NLEEs). This method allows us to generate the solutions of NLEEs by the solutions of the Burgers equation. Applying this method to the NLEEs, we can obtain soliton-like solutions, multi-soliton-like solutions, formal periodic solutions and rational solutions, etc. We take the Boiti–Leon–Pempinelli equation as an example to illustrate the algorithm.

The modified extended tanh-function method for solving Burgers-type equations

By means of the modified extended tanh-function (METF) method the multiple travelling wave solutions of some different kinds of nonlinear partial differential equations are presented and implemented in a computer algebraic system. Solutions for the nonlinear equations such as one-dimensional Burgers, KDV–Burgers, coupled Burgers and two-dimensional Burgers’ equations are obtained precisely and so the efficiency of the method can be demonstrated.

New Method for Finding a Series of Exact Solutions to Generalized Breaking Soliton Equation**The project supported by National Natural Science Foundation ofChina and the Natural Science Foundation of Shandong Province of China

In this paper, a new generalized extended tanh-functionmethod is presented for constructing soliton-like, period-form solutions ofnonlinear evolution equations (NEEs). This method is more powerful than theextended tanh-function method [Phys. Lett. A 277 (2000) 212] and the modifiedextended tanh-function method [Phys. Lett. A 285 (2001) 355]. Abundant newfamilies of the exact solutions of Bogoyavlenskii's generalized breakingsoliton equation are obtained by using this method and symbolic computationsystem Maple.

Symbolic computation in non-linear evolution equation: application to (3+1)-dimensional Kadomtsev–Petviashvili equation

The tanh method is used to find travelling wave solutions to various wave equations. In this paper, the extended tanh function method is further improved by generalizing the Riccati equation and picking up its new solutions. In order to test the validity of this approach, the (3 + 1)-dimensional Kadomtsev–Petviashvili equation is considered. As a result, the abundant new non-travelling wave solutions are obtained.

Extended tanh-Function Method for Finding Travelling Wave Solutions of Some Nonlinear Partial Differential Equations

In this paper, we find travelling wave solutions of some nonlinear partial differential equations (PDEs) by using the extended tanh-function method. Some illustrative equations are investigated by this method and new travelling wave solutions are found. In addition, the properties of these nonlinear PDEs are shown with some figures.

Symbolic computation in non-linear evolution equation: application to (3+1)-dimensional Kadomtsev–Petviashvili equation

Abstract The tanh method is used to find travelling wave solutions to various wave equations. In this paper, the extended tanh function method is further improved by generalizing the Riccati equation and picking up its new solutions. In order to test the validity of this approach, the (3 + 1)-dimensional Kadomtsev–Petviashvili equation is considered. As a result, the abundant new non-travelling wave solutions are obtained.

A New Modified Extended Tanh-Function Method and Its Application

In this paper, a new modified extended tanh-function method is presented for constructing soliton-like, periodic form solutions of nonlinear evolution equation (NEEs). This method is more powerful than the extended tanh-function method [Phys. Lett. A277 (2000) 212] and the modified extended tanh-function method [Phys. Lett. A285 (2001) 355]. Abundant new solutions of two physically important NEEs are obtained by using this method and symbolic computation system Maple.

Symbolic computation and new families of exact soliton-like solutions of Konopelchenko–Dubrovsky equations

We apply a new modified extended tanh function method [Commun. Theor. Phys. (Beijing, China) 39 (2003) 405] to the (2 + 1)-dimensional Konopelchenko–Dubrovsky equations and obtain abundant new exact solutions of the equation. These solutions contain soliton-like solutions, periodic formal solution and rational function solutions. The method can also be applied to solve more (2 + 1)-dimensional (or (3 + 1)-dimensional) nonlinear partial differential equations.

A New Generalization of Extended Tanh-Function Method for Solving Nonlinear Evolution Equations**The project supported by National Natural Science Foundation of China under Grant No. 1007201 and the National Key Basic Research Development Project Program under Grant No. G1998030600

Making use of a new generalized ansätze and a proper transformation, we generalized the extended tanh-function method. Applying the generalized method with the aid of Maple, we consider some nonlinear evolution equations. As a result, we can successfully recover the previously known solitary wave solutions that had been found by the extended tanh-function method and other more sophisticated methods. More importantly, for some equations, we also obtain other new and more general solutions at the same time. The results include kink-profile solitary-wave solutions, bell-profile solitary-wave solutions, periodic wave solutions, rational solutions, singular solutions and new formal solutions.

On a New Modified Extended Tanh-Function Method**The project supported by the National Key Basic Research Development of China (Grant No. 1998030600) and the National Natural Science Foundation of China (Grant No. 10072013)

In this paper, a new modified extended tanh-function method is presented for constructing multiple soliton-like, periodic form and rational solutions of nonlinear evolution equations (NLEEs). This method is more powerful than the extended tanh-function method [Phys. Lett. A 277 (2000) 212] and the modified extended tanh-function method [Phys. Lett. A 299 (2002) 179]. Abundant new solutions of two physically important NLEEs are obtained by using this method and symbolic computation system Maple.

A new extended tanh-function method and its application to the dispersive long wave equations in (2+1) dimensions

In this paper, a new extended tanh-function method is suggested to constructing multiple exact solutions for nonlinear partial differential equations. The validity and reliability of the method is tested by its application to the dispersive long wave equations in (2+1) dimensions. More new explicit exact solutions are found.

Two New Applications of the Modified Extended Tanh-Function Method

Based on a modified extended tanh-function method and symbolic computation, new exact solutions are found for a soliton breaking equation and coupled kdv system. The obtained solutions include rational, soliton, singular and periodical solutions.