Differential and other reductions of the self-dual conformal structure equations

TL;DR

This work develops a dispersionless integrable framework for the self-dual conformal structure (SDCS) in signature $(2,2)$, centered on a Lax-pair formulation, a three-component hierarchy, and a dressing scheme. It introduces and analyzes several reductions, notably interpolating reductions and Type B (BKP-related) reductions, showing how they yield solvable SDCS equations and relate to the base SDCS system via Miura-type maps. A key insight is that Type B reductions satisfy the same third-order SDCS equations and can be connected to the SDCS system through explicit potential relations, with dressing data and wave-function ansatz guiding solution construction. The approach also explores nonlocal SDCS equations arising from reflection-type symmetries and explicit 3D reductions via the waterbag ansatz, broadening the geometric interpretation and solution-building toolkit for self-dual conformal structures.

Abstract

The dispersionless integrable system we consider here was introduced to the literature rather recently, it is connected with the general local form of self-dual conformal structure (SDCS) for the signature (2,2). In integrability framework this system possesses a rich structure of reductions, including differential reductions. We will discuss several characteristic reductions for this system, using the Lax pair, hierarchy structure and the dressing scheme. We use reductions to construct solutions for the SDCS equations. One of our goals is to present type B SDCS system and consider its relations with the SDCS system.