Lax Representation and Darboux Solutions of the Classical Painlevé Second Equation

Irfan Mahmood,Wen-Xiu Ma,Muhammad Waseem

doi:10.1155/2021/8851043

Abstract

In this article, we present Darboux solutions of the classical Painlevé second equation. We reexpress the classical Painlevé second Lax pair in new setting introducing gauge transformations to yield its Darboux expression in additive form. The new linear system of that equation carries similar structure as other integrable systems possess in the AKNS scheme. Finally, we generalize the Darboux transformation of the classical Painlevé second equation to the N -th form in terms of Wranskian.

Highlights

The six classical Painlevé equations first were introduced by Paul Painlevé and his colleagues while classifying nonlinear second-order ordinary differential equations with respect to their solutions [1]
These were well entertained in mathematics because of their connection to nonlinear partial differential equation, where they appear as ordinary differential equation reduction of higher dimensional integrable that further pursued to establish the Painlevé test for these partial differential equations [2, 3]
Result shows that the classical Painlevé second Darboux transformation in the additive structure where as if we pursue with the old Lax pair, the Darboux expression will be in product form which mostly happened for noncommutative case not recommended in the classical framework

Summary

Introduction

The six classical Painlevé equations first were introduced by Paul Painlevé and his colleagues while classifying nonlinear second-order ordinary differential equations with respect to their solutions [1]. Various properties of these equations have been studied from the mathematical point of such as their integrability through the Painlevé test, their zero-curvature representation is a compatibility condition of associated set of linear systems [6,7,8] as well as in the framework of hamiltonian formalism [9] These equation got considerable attention from the physical point of view because the number of problems in applied mathematics and in physics is made to be integrable in terms of their solutions, Painlevé transcendents. We will present nontrivial Darboux solutions of that equation introducing gauge transformation that brings its Lax pair (11) to a new form as similar as we have for many integrable systems in the AKNS scheme. By iteration, the N -fold Darboux transformation will be generalized to determinantal form in terms of Wronskian

Gauge Transformation and Nontrivial Darboux Solutions "

N-Fold Darboux Solutions in Terms of Wronskians

Conclusion