Ralph Willox — Page 3

arXiv2003

Painleve equations from Darboux chains - Part 1: P3-P5

We show that the Painleve equations P3-P5 can be derived (in a unified way) from a periodic sequence of Darboux transformations for a Schrodinger problem with quadratic eigenvalue dependency. The general problem naturally divides into three different branches, each described by an infinite chain of equations. The Painleve equations are obtained by closing the chain periodically at the lowest nontrivial level(s). The chains provide ``symmetric forms'' for the Painleve equations, from which Hirota bilinear forms and Lax pairs are derived. In this paper (Part 1) we analyze in detail the cases P3-P5, while P6 will be studied in Part 2.

nlin.SI↗

arXiv1994

Complete integrability of a modified vector derivative nonlinear Schroedinger equation

Oblique propagation of magnetohydrodynamic waves in warm plasmas is described by a modified vector derivative nonlinear Schroedinger equation, if charge separation in Poisson's equation and the displacement current in Ampere's law are properly taken into account. This modified equation cannot be reduced to the standard derivative nonlinear Schroedinger equation and hence its possible integrability and related properties need to be established afresh. Indeed, the new equation is shown to be integrable by the existence of a bi--Hamiltonian structure, which yields the recursion operator needed to generate an infinite sequence of conserved densities. Some of these have been found explicitly by symbolic computations based on the symmetry properties of the new equation. Since the new equation includes as a special case the derivative nonlinear Schroedinger equation, the recursion operator for the latter one is now readily available.

solv-int↗