Stokes phenomena and the monodromy deformation problem for the non-linear Schrodinger equation

Abstract

Following Flaschka and Newell (1979) the authors have formulated the inverse problem for Painleve IV, with the help of similarity variables. The Painleve IV arises as the eliminant of the two second-order differential equations originating from the non-linear Schrodinger equation. They have obtained the asymptotic expansions near the singularities at 0 and infinity of the complex eigenvalue plane. The corresponding analysis then displays Stokes phenomena. The monodromy matrices connecting the solution Yj in the sector Sj to that in Sj+1 are fixed in structure by the imposition of certain conditions. They then show that a deformation keeping the monodromy data fixed leads to the non-linear Schrodinger equation. At this point they may mention that, while Flaschka and Newell did not make any absolute determination of the Stokes parameter, the approach yields the values of the Stokes parameter in an explicit way, which in turn can determine the matrix connecting the solutions near 0 and infinity . Such a realisation was not possible in the approach of Flashchka and Newell. Lastly they show that the integral equation originating from the analyticity and asymptotic nature leads to the similarity solution previously determined by Boiti and Pempinelli (1979).