HYDRODYNAMIC TYPE SYSTEMS AND THEIR INTEGRABILITY Introduction for Applied Mathematicians

Hydrodynamic type systems are systems of quasilinear equations of the first order. They naturally arise in continuum mechanics but also occur as a result of semi-classical approximations of nonlinear dispersive waves. The mathematical theory of one-dimensional hyperbolic quasilinear equations initiated by B. Riemann in XIX century has been developed into a rich and diverse area of applied mathematics including, e.g., theory of shock waves. Among the classical methods of integration of one-dimensional quasilinear hyperbolic equations are the method of characteristics and the hodograph method, the latter being applicable only to the two-component hydrodynamic systems. A relatively recent breakthrough in the theory of hydrodynamic type systems was made in 1980’s by S. Tsarev who proved Novikov’s hypothesis on the integrability of diagonalisable Hamiltonian systems of hydrodynamic type and formulated the generalised hodograph method. In these notes I will outline some of the basic ideas related to integrability of one-dimensional hydrodynamic type systems. The emphasis will be made on the applicable aspects of the theory.