Rational Wave Solutions Research Articles

A new general algebraic method with symbolic computation to construct new travelling wave solution for the (1 + 1)-dimensional dispersive long wave equation

With the aid of symbolic computation, a new algebraic method, named Riccati equation rational expansion (RERE) method, is devised for constructing multiple travelling wave solutions for nonlinear evolution equations (NEEs). Compared with most existing tanh methods for finding travelling wave solutions, the proposed method not only recover the results by most known algebraic methods, but also provides new and more general solutions. With the aid of symbolic computation, we choose (1 + 1)-dimensional dispersive long wave equation (DLWE) to illustrate our method. As a result, we obtain many types of solutions including rational form solitary wave solutions, triangular periodic wave solutions and rational wave solutions. The properties of the new solitary wave solutions are shown by some figures.

Fractional transformation and new solutions to mKdV equation

A fractional transformation is introduced to solve modified KdV (mKdV for short) equation, where this fractional transformation is used to map the solutions of an elliptic equation to another elliptic equation. Thus, more new kinds of solutions are obtained, such as rational periodic wave solutions, rational solitary wave solutions and so on. It is shown that this method is more powerful to give more kinds of solutions.

Modified extended tanh-function method and its applications to nonlinear equations

New exact travelling wave solutions for the generalized shallow water wave equation, the improved Boussinesq equation and the coupled system for the approximate equations for water waves are found using a modified extended tanh-function method. The obtained results include rational, periodic, singular and solitary wave solutions.

Exact travelling wave solutions for the generalized shallow water wave equation

Using homogeneous balance method an auto-Bäcklund transformation for the generalized shallow water wave equation is obtained. Then solitary wave solutions are found. Also, modified extended tanh-function method is applied and new exact travelling wave solutions are obtained. The obtained solutions include rational, periodical, singular and solitary wave solutions.

An application for a generalized KdV equation by the decomposition method

The explicit solutions to a generalized Korteweg–de Vries equation (KdV for short) with initial condition are calculated by using the Adomian decomposition method. Using this approach we obtained for the numerical solutions of initial-value KdV equation. Numerical illustrations on well-known KdV equation with the rational and solitary wave solutions indicate that the decomposition method is efficient and accurate. In addition, an illustration of the self canceling phenomena is also given.

Rationalsolutions of the KP equation through symmetry reductions:coherent structures and other solutions

Some singular rational travelling wave solutions of theKadomtsev-Petviashvili (KP) equation are determined by makinguse of the theory of symmetry reductions. Solutions of KP of theform u(x-c1t,y-c2t), c1, c2 constants, satisfy theequation(-c1ūx̄-c2ūȳ + 6ūūx̄ + ūx̄x̄x̄)x̄ = ūȳȳ,with x̄ = x-c1t, ȳ = y-c2t andū(x̄,ȳ) = u(x-c1t,y-c2t). Some nonclassicalsymmetries of this equation are determined: then, by consideringthe corresponding ordinary differential equations, somesolutions for the KP equation are obtained. But in the case ofthe well known one-dimensional soliton, all the solutions weconstruct in this way are rational functions. Moreover, the KPequation reduces to the previous equation with c1 = c2 = 0 undera more general symmetry group. Thus, we can use the solutionsdescribing coherent structures to construct large families of(x,y)-rational solutions.