Kink Profile Solitary Wave Solutions Research Articles

Qualitative Analysis and Travelling Wave Solutions for the Generalized Pochhammer-Chree Equation with a Dissipation Term

The purpose of this paper is to reveal the influence of dissipation on travelling wave solutions of the generalized Pochhammer–Chree equation with a dissipation term, and provides travelling wave solutions for this equation. Applying the theory of planar dynamical systems, we obtain ten global phase portraits of the dynamic system corresponding to this equation under various parameter conditions. Moreover, we present the relations between the properties of travelling wave solutions and the dissipation coefficient r of this equation. We find that a bounded travelling wave solution appears as a bell profile solitary wave solution or a periodic travelling wave solution when r= 0; a bounded travelling wave solution appears as a kink profile solitary wave solution when |r| > 0 is large; a bounded travelling wave solution appears as a damped oscillatory solution when |r| > 0 is small. Further, by using undetermined coefficient method, we get all possible bell profile solitary wave solutions and approximate damped oscillatory solutions for this equation. Error estimates indicate that the approximate solutions are meaningful.

Solitary wave and periodic wave solutions for the thermally forced gravity waves in atmosphere

By introducing a new transformation, a new direct and unified algebraic method for constructing multiple travelling wave solutions of general nonlinear evolution equations is presented and implemented in a computer algebraic system, which extends Fan's direct algebraic method to the case when r > 4. The solutions of a first-order nonlinear ordinary differential equation with a higher degree nonlinear term and Fan's direct algebraic method of obtaining exact solutions to nonlinear partial differential equations are applied to the combined KdV–mKdV–GKdV equation, which is derived from a simple incompressible non-hydrostatic Boussinesq equation with the influence of thermal forcing and is applied to investigate internal gravity waves in the atmosphere. As a result, by taking advantage of the new first-order nonlinear ordinary differential equation with a fifth-degree nonlinear term and an eighth-degree nonlinear term, periodic wave solutions associated with the Jacobin elliptic function and the bell and kink profile solitary wave solutions are obtained under the effect of thermal forcing. Most importantly, the mechanism of propagation and generation of the periodic waves and the solitary waves is analysed in detail according to the values of the heating parameter, which show that the effect of heating in atmosphere helps to excite westerly or easterly propagating periodic internal gravity waves and internal solitary waves in atmosphere, which are affected by the local excitation structures in atmosphere. In addition, as an illustrative sample, the properties of the solitary wave solution and Jacobin periodic solution are shown by some figures under the consideration of heating interaction.

New explicit and exact traveling wave solutions for two nonlinear evolution equations

In this paper, with the aid of computer symbolic computation system such as Maple, an algebraic method is firstly applied to two nonlinear evolution equations, namely, nonlinear Schrodinger equation and Pochhammer–Chree (PC) equation. As a consequence, some new types of exact traveling wave solutions are obtained, which include bell and kink profile solitary wave solutions, triangular periodic wave solutions, and singular solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.

New Types of Travelling Wave Solutions From (2+1)-DimensionalDavey–Stewartson Equation

In this paper, based on new auxiliary nonlinear ordinarydifferential equation with a sixth-degree nonlinear term, westudy the (2+1)-dimensional Davey–Stewartson equation and newtypes of travelling wave solutions are obtained, which includenew bell and kink profile solitary wave solutions, triangularperiodic wave solutions, and singular solutions. The method usedhere can be also extended to many other nonlinear partial differentialequations.

Construction of a series of travelling wave solutions to nonlinear equations

In this paper, based on new auxiliary ordinary differential equation with a sixth-degree nonlinear term, we study the (1 + 1)-dimensional combined KdV–MKdV equation, Hirota equation and (2 + 1)-dimensional Davey–Stewartson equation. Then, a series of new types of travelling wave solutions are obtained which include new bell and kink profile solitary wave solutions, triangular periodic wave solutions and singular solutions. The method used here can be also extended to many other nonlinear partial differential equations.

New exact travelling waves solutions to the combined KDV-MKDV and generalized Zakharov equations

In this paper, the solutions of a first-order nonlinear ordinary differential equation with a sixth-degree nonlinear term and Fan's direct algebraic method of obtaining exact solution to nonlinear PDEs are applied to two nonlinear evolution equations, the combined KdV-MKdV equation and generalized Zakharov equations. As a result, some special exact travelling wave solutions are obtained which include new bell and kink profile solitary wave solutions, bright and dark solitary wave solutions, triangular periodic wave solutions and singular solutions.

Exact solutions for the cubic–quintic nonlinear Schrödinger equation

Abstract In this paper, the cubic–quintic nonlinear Schrodinger equation is solved through the extended elliptic sub-equation method. As a consequence, many types of exact travelling wave solutions are obtained which including bell and kink profile solitary wave solutions, triangular periodic wave solutions and singular solutions.

Link between travelling waves and first order nonlinear ordinary differential equation with a sixth-degree nonlinear term

Many travelling wave solutions of nonlinear evolution equations can be written as a polynomial in several elementary or special functions which satisfy a first order nonlinear ordinary differential equation with a sixth-degree nonlinear term. From that property, we deduce an algebraic method for constructing those solutions by determining only a finite number of coefficients. Being concise and straightforward, the method is applied to three nonlinear evolution equations. As a result, many exact travelling wave solutions are obtained which include new bell and kink profile solitary wave solutions, triangular periodic wave solutions and singular solutions.

The extended first kind elliptic sub-equation method and its application to the generalized reaction Duffing model

In this Letter, an extended first kind elliptic sub-equation method and its algorithm are proposed by studying a first order nonlinear ordinary differential equation with a sixth-degree nonlinear term. Being concise and straightforward, the method is applied to the generalized reaction Duffing model. As a consequence, more types of exact travelling wave solutions are obtained which include new bell and kink profile solitary wave solutions, triangular periodic wave solutions and singular solutions. The method can be also extended to many other nonlinear wave equations.

Lienard Equation and Exact Solutions for Some Soliton-Producing Nonlinear Equations**The project supported by National Natural Science Foundation of China under Grant No. 10371023 and the Science Foundation of the Education Committee of Shanghai under Grant No. 03GK15

In this paper, we first consider exact solutions for Lienard equation with nonlinear terms of any order. Then, explicit exact bell and kink profile solitary-wave solutions for many nonlinear evolution equations are obtained by means of results of the Lienard equation and proper deductions, which transform original partial differential equations into the Lienard one. These nonlinear equations include compound KdV, compound KdV–Burgers, generalized Boussinesq, generalized KP and Ginzburg–Landau equation. Some new solitary-wave solutions are found.

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.

New exact solutions for some nonlinear differential equations using symbolic computation

Based on computerized symbolic computation and a new general ansätze, a generalized tanh-function method for constructing multiple travelling wave solutions of nonlinear evolution equations (NEEs) is presented and implemented in a computer algebraic system. Applying the generalized method, with the aid of Maple, we consider some NEEs with physics interests. 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 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. The properties of the new soliton solutions for WBK equations are shown by some figures.

Generalized extended tanh-function method and its application to (1+1)-dimensional dispersive long wave equation

Making use of a new generalized ansatzes, we present the generalized extended tanh-function method for constructing the exact solutions of nonlinear partial differential equations (NPDEs) in a unified way. Applying the generalized method, with the aid of Maple, we consider the Wu–Zhang equation (which describes (1+1)-dimensional dispersive long wave). As a result, we can successfully obtain the solitary wave solutions that can be found by the extended tanh-function method and the modified extended tanh-function method. More importantly, for the equation, 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 other new formal solutions. As an illustrative sample, the properties of some soliton solutions for Wu–Zhang equation are shown by some figures.

EXACT TRAVELING WAVE SOLUTIONS FOR SOME NONLINEAR EVOLUTION EQUATIONS WITH NONLINEAR TERMS OF ANY ORDER

In this paper, we improved the tanh method by means of a proper transformation and general ansätz. Using the improved method, with the aid of Mathematica™, we consider some nonlinear evolution equations with nonlinear terms of any order. As a result, rich explicit exact traveling wave solutions for these equations, which contain kink profile solitary wave solutions, bell profile solitary wave solutions, rational solutions, periodic solutions, and combined formal solutions, are obtained.

Exact solutions for a new class of nonlinear evolution equations with nonlinear term of any order

In this paper, we consider a new class of nonlinear partial differential equations with nonlinear term of any order, u tt + a 1 u xx + a 2 u+ a 3 u p + a 4 u 2 p−1 =0, which contains some particular important equations. We give a new kind of transformation and a new generalized ansätze to treat this class of equations. As a result, many explicit exact solutions, which contain new kink-profile solitary-wave solutions, bell-profile solitary-wave solutions, periodic wave solutions and combined formal solitary-wave solutions, are obtained by the extended method. In addition, we also can derive rational solutions for this class of equations.

Explicit solitary-wave solutions to generalized Pochhammer-Chree equations

For the solitary-wave solution u(ξ) = u(x − vt + ¾0) to the generalized Pochhammer-Chree equation (PC equation) $$u_{tt} - u_{ttxx} + ru_{xxt} - (a_1 u + a_2 u^2 + a_3 u^3 )_{xx} = 0, r,a_i = consts(r \ne 0)$$ (I) the formula\(\int_{ - \infty }^{ + \infty } {[u'(\xi )]^2 d\xi = } \frac{1}{{12r\nu }}(C_ + - C_ - )^3 [3a_3 (C_ + - C_ - ) + 2a_2 ], C_ \pm = \mathop {\lim }\limits_{\xi \to \pm \infty } u(\xi )\), is established, by which it is shown that the generalized PC equation (I) does not have bell profile solitary-wave solutions but may have kink profile solitary-wave solutions. However a special generalized PC equation $$u_{tt} - u_{ttxx} - (a_1 u + a_2 u^2 + a_3 u^3 )_{xx} = 0, a_i = consts$$ (II) may have not only bell profile solitary-wave solutions, but also kink profile solitary-wave solutions whose asymptotic values satisfy 3a3(C++C−)+2a2=0. Furthermore all expected solitary-wave solutions are given. Finally some explicit bell profile solitary-wave solutions to another generalized PC equation $$u_{tt} - u_{ttxx} - (a_1 u + a_3 u^3 + a_5 u^5 )_{xx} = 0, a_i = consts$$ (III) are proposed.