Analytic solitary waves of nonintegrable equations

Abstract

A major drawback of most methods to find analytic expressions for solitary waves is the a priori restriction to a given class of expressions. To overcome this difficulty, we present a new method, applicable to a wide class of autonomous equations, which builds as an intermediate information the first order autonomous ODE satisfied by the solitary wave. We discuss its application to the cubic complex one-dimensional Ginzburg-Landau equation, and conclude to the elliptic nature of the yet unknown most general solitary wave.

Introduction to the Painlevé property, test and analysis

“…The first method [20,6,7] implements a classical theorem of Briot and Bouquet, stating that any such solution obeys a first order algebraic ODE F(u , u) = 0 in which the degrees of the polynomial F in u and u are known and obtained from the given N-th order ODE by carefully counting its number of movable poles.…”

“…The singular parts of these M series determine completely the coefficients c k, j in (20). In order to then determine C and a k , one uses the addition formula of the elliptic functions to write (20) as a rational function of ℘(x − a 1 ),℘ (x − a 1 ), see Eq. (13).…”

Nonlinear Differential Equations With Exact Solutions Expressed Via The Weierstrass Function

“…In recent years one can observe a splash of papers where authors presented a lot of different approaches to look for exact solutions of nonlinear differential equations [1,2,3,4,5,6,7,8,9]. There are two reasons to make the study in this direction.…”

Solitary Waves of Nonlinear Nonintegrable Equations

“…In the second class of methods [27], presented in section 7, rather than directly looking for the unknown solution…”

“…together with the knowledge of the singular part operators (27) and (132), first yields the correct values of c 0 and c 1 ,…”