Modified method of simplest equation for obtaining exact solutions of nonlinear partial differential equations: past and present

Simple Equations methodology (SEsM) for searching of multisolitons and other exact solutions of nonlinear partial differential equations

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 relationship used by Hirota \cite{hirota} and the relationship used in the previous version of the methodology; (iii) transformation of the solution that contains as particular case the possibility of use of the Painleve expansion; (iv) more than one balance equation. The discussed version of the methodology allows: (i) obtaining multi-soliton solutions of …

Solitary wave solutions of nonlinear partial differential equations based on the simplest equation for the function $1/\cosh^n$

The method of simplest equation is applied for obtaining exact solitary traveling-wave solutions of nonlinear partial differential equations that contain monomials of odd and even grade with respect to participating derivatives. The used simplest equation is $f_ξ^2 = n^2(f^2 -f^{(2n+2)/n})$. The developed methodology is illustrated on two examples of classes of nonlinear partial differential equations that contain: (i) only monomials of odd grade with respect to participating derivatives; (ii) only monomials of even grade with respect to participating derivatives. The obtained solitary wave so…

Solitary wave solutions of several nonlinear PDEs modeling shallow water waves

We apply the version of the method of simplest equation called modified method of simplest equation for obtaining exact traveling wave solutions of a class of equations that contain as particular case a nonlinear PDE that models shallow water waves in viscous fluid (Topper-Kawahara equation). As simplest equation we use a version of the Riccati equation. We obtain two exact traveling wave solutions of equations from the studied class of equations and discuss the question of imposing boundary conditions on one of these solutions.

Painlevé超越的所有一次双有理变换的截断

A truncation for obtaining all the first degree birational transformations of the Painlevé transcendents

双有理变换是使常微分方程保持不变，只改变其参数的变换。我们首先回顾一致的截断，它使我们能够获得主潘勒韦方程P6的冈本一级双有理变换。然后，我们通过增加一个初步的步骤，这是找到所有的Riccati子方程的考虑Pn之前，执行截断。我们详细讨论了我们的方法的主要新颖之处，以最简单的潘勒韦方程为例，P2。最后，我们将此方法应用于P5，得到了它的两个不等价的一次双有理变换。

Integrability of differential equations with fluid mechanics application: from Painleve property to the method of simplest equation

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 this equation are single poles. The importance of this property can be seen from the Ablowitz-Ramani-Segur conhecture that states that a nonlinear PDE is solvable by inverse scattering transformation only if each nonlinear ODE obtained by exact reduction of this PDE has the Painleve property. The Painleve property motivated motivated much research on obtaining …

Simple equations method (SEsM) and some of its numerous particular cases

We discuss a new version of a method for obtaining exact solutions of nonlinear partial differential equations. We call this method the Simple Equations Method (SEsM). The method is based on representation of the searched solution as function of solutions of one or several simple equations. We show that SEsM contains as particular case the Modified Method of Simplest Equation, G'/G - method, Exp-function method, Tanh-method and the method of Fourier series for obtaining exact and approximate solutions of linear differential equations. These methods are only a small part of the large amount of …

Solitary wave solutions for nonlinear partial differential equations containing monomials of odd and even grades with respect to participating derivatives

We apply the method of simplest equation for obtaining exact solitary traveling-wave solutions of nonlinear partial differential equations that contain monomials of odd and even grade with respect to participating derivatives. We consider first the general case of presence of monomials of the both (odd and even) grades and then turn to the two particular cases of nonlinear equations that contain only monomials of odd grade or only monomials of even grade. The methodology is illustrated by numerous examples.

On a class of second-order PDEs admitting partner symmetries

Recently we have demonstrated how to use partner symmetries for obtaining noninvariant solutions of heavenly equations of Plebanski that govern heavenly gravitational metrics. In this paper, we present a class of scalar second-order PDEs with four variables, that possess partner symmetries and contain only second derivatives of the unknown. We present a general form of such a PDE together with recursion relations between partner symmetries. This general PDE is transformed to several simplest canonical forms containing the two heavenly equations of Plebanski among them and two other nonlinear e…

Dressing for a vector modified KdV hierarchy

We present the hierarchy and soliton solutions associated to a multi-component generalisation of the modified Korteweg-de Vries equation. A recursive formula for obtaining the Lax operators associated to the higher flows of the hierarchy is provided. Using the method of rational dressing and the symmetries of the Lax operators we obtain the one-soliton solution. We also derive the general rank one-breather solution and express it in terms of determinants. We present the simplest case of the one-breather solution, which is parametrised by two constant unit vectors normal to each other. Finally,…