The integration algorithm of Lax equation for both generic Lax matrices and generic initial conditions

Wissam Chemissany,Pietro Fré,Alexander S Sorin

doi:10.1016/j.nuclphysb.2010.01.009

Abstract

Several physical applications of Lax equation require its general solution for generic Lax matrices and generic not necessarily diagonalizable initial conditions. In the present paper we complete the analysis started in [arXiv:0903.3771] on the integration of Lax equations with both generic Lax operators and generic initial conditions. We present a complete general integration formula holding true for any (diagonalizable or non-diagonalizable) initial Lax matrix and give an original rigorous mathematical proof of its validity relying on no previously published results.

Highlights

IntroductionLax equation appears in a variety of physical-mathematical problems which turn out to constitute integrable dynamical systems
An integration algorithm for generic matrix Lax equation was originally derived in the mathematical literature in [1], [2] and it was applied in the context of supergravity to cosmic billiards in [3], [4], [5] and to black-holes in [6], [7], [8] on the basis of their 3D description pioneered in [9]
In the appendix of that paper it was conjectured a general formula, which was verified for some nontrivial cases, that provides the integration of Lax equation for completely generic initial data

Summary

Introduction

Lax equation appears in a variety of physical-mathematical problems which turn out to constitute integrable dynamical systems. An integration algorithm for generic matrix Lax equation was originally derived in the mathematical literature in [1], [2] and it was applied in the context of supergravity to cosmic billiards in [3], [4], [5] and to black-holes in [6], [7], [8] on the basis of their 3D description pioneered in [9] The latter application to the case of black-holes revealed that the integration algorithm of [1], [2] did not cover the case of non-diagonalizable initial conditions. Following [8] we first present the integration formula and provide the mathematical proof that it satisfies Lax equation

Integration formula

The theorem and its proof

The Generalized Linear System

Conclusions