The B\"{a}cklund transforms of Peterson's deformations of quadrics

TL;DR

This paper explores explicit deformations of quadrics using Bäcklund transforms, linking classical differential geometry with integrable systems like the sine-Gordon equation, and proposes a model for solitons of quadrics.

Contribution

It introduces a novel approach to derive Bäcklund transforms of Peterson's deformations of quadrics via connections with sine-Gordon solitons and Bianchi's geometric methods.

Findings

Explicit Bäcklund transforms for Peterson's deformations derived.

Connection established between quadrics deformations and sine-Gordon solitons.

Proposed a new model for solitons of quadrics.

Abstract

In trying to provide explicit deformations of quadrics the starting point of our investigation is to use Bianchi's link between real deformations of totally real regions of real paraboloids and various totally real forms of the sine-Gordon equation coupled with Bianchi's simple observation that the vacuum soliton of these totally real forms of the sine-Gordon equation provides precisely Peterson's deformations of such quadrics in order to derive explicit B\"{a}cklund transforms of Peterson's deformations of quadrics. Based also on Bianchi's approach of the B\"{a}cklund transformation for quadrics via common conjugate systems and in analogy to the solitons of the sine-Gordon equation corresponding at the level of the geometric picture to the solitons of the pseudo-sphere we propose a model for the solitons of quadrics.