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

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 (generalized) 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.