A New Extended Jacobi Elliptic Function Expansion Method and Its Application to the Generalized Shallow Water Wave Equation

With the aid of symbolic computation, a new extended Jacobi elliptic function expansion method is presented by means of a new ansatz, in which periodic solutions of nonlinear evolution equations, which can be expressed as a finite Laurent series of some 12 Jacobi elliptic functions, are very effective to uniformly construct more new exact periodic solutions in terms of Jacobi elliptic function solutions of nonlinear partial differential equations. As an application of the method, we choose the generalized shallow water wave (GSWW) equation to illustrate the method. As a result, we can successfully obtain more new solutions. Of course, more shock wave solutions or solitary wave solutions can be gotten at their limit condition

Further Results about Traveling Wave Exact Solutions of the Drinfeld-Sokolov Equations

We employ the complex method to obtain all meromorphic exact solutions of complex Drinfeld-Sokolov equations (DS system of equations). The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that all constant and simply periodic traveling wave exact solutions of the equations (DS) are solitary wave solutions, the complex method is simpler than other methods and there exist simply periodic solutions vs,3(z) which are not only new but also not degenerated successively by the elliptic function solutions. We believe that this method should play an important role for finding exact solutions in the mathematical physics. For these new traveling wave solutions, we give some computer simulations to illustrate our main results

Seven common errors in finding exact solutions of nonlinear differential equations

We analyze the common errors of the recent papers in which the solitary wave solutions of nonlinear differential equations are presented. Seven common errors are formulated and classified. These errors are illustrated by using multiple examples of the common errors from the recent publications. We show that many popular methods in finding of the exact solutions are equivalent each other. We demonstrate that some authors look for the solitary wave solutions of nonlinear ordinary differential equations and do not take into account the well - known general solutions of these equations. We illustrate several cases when authors present some functions for describing solutions but do not use arbitrary constants. As this fact takes place the redundant solutions of differential equations are found. A few examples of incorrect solutions by some authors are presented. Several other errors in finding the exact solutions of nonlinear differential equations are also discussed.Comment: 42 page

On the nonlinear dynamics of the traveling-wave solutions of the Serre system

We numerically study nonlinear phenomena related to the dynamics of traveling wave solutions of the Serre equations including the stability, the persistence, the interactions and the breaking of solitary waves. The numerical method utilizes a high-order finite-element method with smooth, periodic splines in space and explicit Runge-Kutta methods in time. Other forms of solutions such as cnoidal waves and dispersive shock waves are also considered. The differences between solutions of the Serre equations and the Euler equations are also studied.Comment: 28 pages, 20 figures, 3 tables, 33 references. Other author's papers can be downloaded at http://www.denys-dutykh.com

All exact traveling wave solutions of the combined KdV-mKdV equation

In this article, we employ the complex method to obtain all meromorphic solutions of complex combined Korteweg-de Vries-modified Korteweg-de Vries equation (KdV-mKdV equation) at first, then we find all exact traveling wave solutions of the combined KdV-mKdV equation. The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that all rational and simply periodic exact traveling wave solutions of the combined KdV-mKdV equation are solitary wave solutions, the complex method is simpler than other methods, and there exist some rational solutions wr,2(z)wr,2(z) and simply periodic solutions ws,2(z)ws,2(z) such that they are not only new but also not degenerated successively by the elliptic function solutions. We believe that this method should play an important role in finding exact solutions in mathematical physics. We also give some computer simulations to illustrate our main results

On the evolution of scattering data under perturbations of the Toda lattice

We present the results of an analytical and numerical study of the long-time behavior for certain Fermi-Pasta-Ulam (FPU) lattices viewed as perturbations of the completely integrable Toda lattice. Our main tools are the direct and inverse scattering transforms for doubly-infinite Jacobi matrices, which are well-known to linearize the Toda flow. We focus in particular on the evolution of the associated scattering data under the perturbed vs. the unperturbed equations. We find that the eigenvalues present initially in the scattering data converge to new, slightly perturbed eigenvalues under the perturbed dynamics of the lattice equation. To these eigenvalues correspond solitary waves that emerge from the solitons in the initial data. We also find that new eigenvalues emerge from the continuous spectrum as the lattice system is let to evolve under the perturbed dynamics.Comment: 27 pages, 17 figures. Revised Introduction and Discussion section