Three Dimensional Reductions of Four-Dimensional Quasilinear Systems

TL;DR

This paper demonstrates that certain integrable four-dimensional second-order equations have infinitely many three-dimensional reductions, conservation laws, and commuting flows, linking them to dispersionless limits of nonlocal KdV and NLS equations.

Contribution

It introduces a new class of four-dimensional integrable equations and explores their reductions, conservation laws, and connections to nonlocal soliton equations.

Findings

Existence of infinitely many 3D hydrodynamic reductions

Presence of infinite conservation laws and commuting flows

Connections to dispersionless limits of nonlocal KdV and NLS equations

Abstract

In this paper we show that integrable four dimensional linearly degenerate equations of second order possess infinitely many three dimensional hydrodynamic reductions. Furthermore, they are equipped infinitely many conservation laws and higher commuting flows. We show that the dispersionless limits of nonlocal KdV and nonlocal NLS equations (the so-called Breaking Soliton equations introduced by O.I. Bogoyavlenski) are one and two component reductions (respectively) of one of these four dimensional linearly degenerate equations.