The Integration Algorithm of Lax equation for both Generic Lax matrices and Generic Initial Conditions

TL;DR

This paper provides a comprehensive integration formula for Lax equations applicable to any initial Lax matrix, including non-diagonalizable cases, with a rigorous proof ensuring broad applicability in physical models.

Contribution

It introduces a complete general integration formula for Lax equations with arbitrary initial conditions, extending previous work to non-diagonalizable matrices and providing a rigorous mathematical proof.

Findings

Unified solution formula for all Lax matrices

Rigorous proof of the integration method's validity

Applicable to both diagonalizable and non-diagonalizable cases

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.