Integrable discretization of hodograph-type systems, hyperelliptic integrals and Whitham equations

TL;DR

This paper develops discrete analogues of key concepts in semi-Hamiltonian hydrodynamic systems, including hodograph equations and Whitham equations, preserving integrability in a discrete geometric framework.

Contribution

It introduces a novel discrete geometric approach to semi-Hamiltonian systems, extending continuous integrable structures to the discrete setting.

Findings

Discrete hodograph equations are formulated.

Hyperelliptic integrals are discretized.

Discrete Whitham equations are constructed.

Abstract

Based on the well-established theory of discrete conjugate nets in discrete differential geometry, we propose and examine discrete analogues of important objects and notions in the theory of semi-Hamiltonian systems of hydrodynamic type. In particular, we present discrete counterparts of (generalised) hodograph equations, hyperelliptic integrals and associated cycles, characteristic speeds of Whitham type and (implicitly) the corresponding Whitham equations. By construction, the intimate relationship with integrable system theory is maintained in the discrete setting.