Wave Solutions In Terms Research Articles

Exact Periodic Wave Solutions To The Melnikov Equation

Exact periodic wave solutions in terms of the Jacobi elliptic functions are obtained to the Melnikov equation by means of the extended mapping method with symbolic computation. The stability of these periodic waves is numerically studied. The results show that the linearly combined waves of cn− and dn− functions can propagate stably. For the blow up solutions, the linearly combined periodic solutions of nd− and sd− functions and of dc− and sc− functions can also propagate stably and others can not do. Under the limit conditions, some new solitary wave solutions and trigonometric periodic wave solutions are got. The method is applicable to a large variety of nonlinear partial differential equations. - PACS: 05.45.Yv, 02.30.Ik, 02.30.Jr

Elliptic equation rational expansion method and new exact travelling solutions for Whitham–Broer–Kaup equations

Abstract Based on a new general ansatz and a general subepuation, a new general algebraic method named elliptic equation rational expansion method is devised for constructing multiple travelling wave solutions in terms of rational special function for nonlinear evolution equations (NEEs). We apply the proposed method to solve Whitham–Broer–Kaup equation and explicitly construct a series of exact solutions which include rational form solitary wave solution, rational form triangular periodic wave solutions and rational wave solutions as special cases. In addition, the links among our proposed method with the method by Fan [Chaos, Solitons & Fractals 2004;20:609], are also clarified generally.

CONSTRUCTING FAMILIES TRAVELING WAVE SOLUTIONS IN TERMS OF SPECIAL FUNCTION FOR THE ASYMMETRIC NIZHNIK–NOVIKOV–VESSELOV EQUATION

By means of a more general ansatz and the computerized symbolic system Maple, a generalized algebraic method to uniformly construct solutions in terms of special function of nonlinear evolution equations (NLEEs) is presented. We not only successfully recover the previously-known traveling wave solutions found by Fan's method, but also obtain some general traveling wave solutions in terms of the special function for the asymmetric Nizhnik–Novikov–Vesselov equation.

Exact periodic wave solutions to the coupled Schrödinger-KdV equation and DS equations

The exact solutions for the coupled non-linear partial differential equations are studied by means of the mapping method proposed recently by the author. Taking the coupled Schrodinger-KdV equation and DS equations as examples, abundant periodic wave solutions in terms of Jacobi elliptic functions are obtained. Under the limit conditions, soliton wave solutions are given.

Duffing's equation and its applications to the Hirota equation

In this Letter, we illustrate a connection between the Duffing's equation and the Hirota equation. By applying the exact solutions of the Duffing's equation, we obtain two periodic wave solutions in terms of Jacobi elliptic functions to the Hirota equation.

Travelling wave solutions in terms of special functions for nonlinear coupled evolution systems

We present a simple and effective transformation that decompose the solution of a coupled system into solving a set of algebraic equations and a first order ordinary differential equation of a certain unknown function. This method not only gives us a clear relation between unknown functions for a nonlinear coupled system, but also easily provides us with a series of new and more general travelling wave solutions in terms of special functions such as hyperbolic, rational, triangular, Weierstrass and Jacobi elliptic double periodic functions.

The reverse bending fatigue strength of drill pipes under the effect of drilling fluids: Helbig, R. and Vogt, G.H. Mannesmann Forschungsber. 1988 (1059), 510–514 (in German)

In this work, the analytic solutions for different types of nonlinear partial differential equations are obtained using the multiple Exp-function method. We consider the stated method for the (3+1)-dimensional generalized shallow water-like (SWL) equation, the (3+1)-dimensional Boiti–Leon- Manna–Pempinelli (BLMP) equation, (3+1)-dimensional generalized variable-coefficient B-type Kadomtsev–Petviashvili (VC B-type KP) equation and the (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada (CDGKS) equation. We obtain multi classes of solutions containing one-soliton, two-soliton, and triple-soliton solutions. All the computations have been performed using the software package Maple. The obtained solutions include three classes of soliton wave solutions in terms of one-wave, two-waves, and three-waves solutions. Then the multiple soliton solutions are presented with more arbitrary autocephalous parameters, in which the one, two, and triple solutions localized in all directions in space. Moreover, the obtained solutions and the exact solutions are shown graphically, highlighting the effects of non-linearity. The different types of obtained solutions of aforementioned nonlinear equations arising in fluid dynamics and nonlinear phenomena.

On Classical Boussinesq Equation

The Classical Boussinesq equation has been taken into consideration for calculating the periodic wave solutions in terms of Jacobian cosine elliptic functions. A relation between the translation speed of the periodic wave and the maximum height of pulse is established. The linear and the solitary wave limits have also been discussed.

Expansion of continuum wave solutions in terms of standing natural functions

It is shown that in the interior region of a finite-range potential a continuum outgoing wave solution may be expressed as an infinite sum involving standing natural functions, which are defined as the residues at the poles, real and complex, of the corresponding standing Green function of the problem. The above representation is valid provided the elastic scattering by the potential is known.