On the Question of the B\"acklund Transformations and Jordan Generalizations of the Second Painlev\'e Equation

TL;DR

This paper presents a method to derive the second Painlevé equation and its Bäcklund transformations from the nonlinear Schrödinger equation, and introduces Jordan algebra-based generalizations called JP-systems.

Contribution

It introduces a new algorithm to construct Jordan algebra-based integrable generalizations of P2 and their Bäcklund transformations, expanding the scope of Painlevé equations.

Findings

Derived P2 and Bäcklund transformations from NLS deformations.

Constructed matrix generalizations of P2 using Jordan algebras.

Proposed JP-systems as new integrable models.

Abstract

We demonstrate the way to derive the second Painlev\'e equation $P_2$ and its B\"acklund transformations from the deformations of the Nonlinear Schr\"odinger equation (NLS), all the while preserving the strict invariance with respect to the Schlesinger transformations. The proposed algorithm allows for a construction of Jordan algebra-based completely integrable multiple-field generalizations of $P_2$ while also producing the corresponding B\"acklund transformations. We suggest calling such models the JP-systems. For example, a Jordan algebra $J_{_{{\rm Mat}(N,N)}}$ with the Jordan product in the form of a semi-anticommutator is shown to generate an integrable matrix generalization of $P_2$, whereas the $V_{_N}$ algebra produces a different JP-system that serves as a generalization of the Sokolov's form of a vectorial NLS.