Darboux transformations via Painlevé analysis

Abstract

The interesting result obtained in this paper involves using the generalized singular manifold method to determine the Darboux transformations for the equations. It allows us to establish an iterative procedure to obtain multisolitonic solutions. This procedure is closely related to the Hirota τ -function method. In this paper, we report how to improve the singular manifold method when the equation has more than one Painlevé branch. The singular manifold method generalized in such a way is applied to a pair of equations in 2 + 1 dimensions

Painlevé Analysis and Singular Manifold Method for a (2 + 1) Dimensional Non-Linear Schrödinger Equation

“…We can consider the Lax pair (3.3) as a system of coupled non linear PDE's ( [3,6]) in ψ + , ψ − , m , u and ω so that the singular manifold method can be applied to the Lax pair itself and truncated expansions for ψ + and ψ − should be added to the expansions (4.1). Such expansions could be written as:…”

Darboux transformation and solutions for an equation in 2+1 dimensions

“…The ± sign of w 0 means that there are two possible Painlevé expansions: The problem of systems with two Painlevé branches has been extensively discussed in [14], [15], [12] and [13]. The suggestion of the author and coworker is that, for this class of systems, it is necessary to consider both branches simultaneously by using two singular manifolds; one for each branch.…”

Miura transformation between two non-linear equations in 2+1 dimensions

“…• The Painlevé Property One of the present authors [10] has studied and discussed the system given by (2.1) from the point of view of Painlevé Analysis. As is well known [7] the Painlevé Property (PP), for practical purposes can be summarized in the statement that all solutions of equation (2.1) could be written in the form:…”

“…In reference [10] a detailed discussion is presented on how to deal with both expansions simultaneously.…”

“…In order to show this, we shall be using singularity analysis of the GLDW that has already been succesfully applied to this equation by one of the present authors [10]. Also using techniques based upon Painlevé Analysis, we shall find the Miura transformation which converts (1.1) into the GLDW.…”