Related papers: New developments of the methodology of the Modifie…
We discuss a version the methodology for obtaining exact solutions of nonlinear partial differential equations based on the possibility for use of: (i) more than one simplest equation; (ii) relationship that contains as particular cases the…
We present a short review of the evolution of the methodology of the Method of simplest equation for obtaining exact particular solutions of nonlinear partial differential equations (NPDEs) and the recent extension of a version of this…
We discuss the last version as well as applications of a method for obtaining exact solutions of nonlinear partial differential equations. As this version is based on more than one simple equation we call it Simple Equations Method (SEsM).…
In these lectures we discuss how the Painleve equations can be written in terms of entire functions, and then in the Hirota bilinear (or multilinear) form. Hirota's method, which has been so useful in soliton theory, is reviewed and…
We discuss the relation between the modified method of simplest equation and the exp-function method. First on the basis of our experience from the application of the method of simplest equation we generalize the exp-function ansatz. Then…
We discuss the application of a variant of the method of simplest equation for obtaining exact traveling wave solutions of a class of nonlinear partial differential equations containing polynomial nonlinearities. As simplest equation we use…
We give an elementary introduction to Hirota's direct method of constructing multisoliton solutions to integrable nonlinear evolution equations. We discuss in detail how this works for equations in the Korteweg-de Vries class. We also show…
A multiple exp-function method to exact multiple wave solutions of nonlinear partial differential equations is proposed. The method is oriented towards ease of use and capability of computer algebra systems, and provides a direct and…
It is well recognized that new types of exact travelling wave solutions to nonlinear partial differential equations can be obtained by modifications of the methods which are in hand. In this study, we extend the class of auxiliary equations…
The Painleve expansion for the second Painleve equation (PII) and fourth Painleve equation (PIV) have two branches. The singular manifold method therefore requires two singular manifolds. The double singular manifold method is used to…
This paper discusses two equations with the conditional Painleve property. The usefulness of the singular manifold method as a tool for determining the non-classical symmetries that reduce the equations to ordinary differential equations…
The modification of simplest equation method to look for exact solutions of nonlinear partial differential equations is presented. Using this method we obtain exact solutions of generalized Korteweg-de Vries equation with cubic source and…
Bilinearization of a given nonlinear partial differential equation is very important not only to find soliton solutions but also to obtain other solutions such as the complexitons, positons, negatons, and lump solutions. In this work we…
In this paper, we present a systematic procedure to derive discrete analogues of integrable PDEs via Hirota's bilinear method. This approach is mainly based on the compatibility between an integrable system and its B\"acklund…
Hirota's bilinear method ("direct method") has been very effective in constructing soliton solutions to many integrable equations. The construction of one- and two-soliton solutions is possible even for non-integrable bilinear equations,…
We present a brief overview of integrability of nonlinear ordinary and partial differential equations with a focus on the Painleve property: an ODE of second order has the Painleve property if the only movable singularities connected to…
The modified method of simplest equation is applied to the extended Korteweg - de Vries equation and to generalized Camassa - Holm equation. Exact traveling wave solutions of these two nonlinear partial differential equations are obtained.…
We show that we can also apply the Hirota method to some non-integrable equations. For this purpose, we consider the extensions of the Kadomtsev-Petviashvili (KP) and the Boussinesq (Bo) equations. We present several solutions of these…
We investigate existence of solitonic solutions for higher-order partial differential equations with polynomial nonlinearities. Using the Hirota method we obtain classification for higher-order integrable systems of equations.
This short review is an introduction to a great variety of methods, the collection of which is called the Painlev\'e analysis, intended at producing all kinds of exact (as opposed to perturbative) results on nonlinear equations, whether…