Discontinuous solutions of one-dimensional hamiltonian systems

Abstract

Systems of evolutionary equations in a class having many physical applications as models for nonlinear waves are investigated from the standpoint of hamiltonian theory. Although in most applications these equations are otherwise accountable as hyperbolic systems, they are not necessarily so and the present account bypasses this more familiar approach, gaining several novel advantages. Conservation laws are related through hamiltonian structure to symmetries of the evolutionary system. Then, on the basis of the established facts about conservation laws, various types of discontinuous solutions are classified systematically; and all possible shocks or other discontinuities are shown to have variational characterizations inherited from the hamiltonian description of continuously differentiable solutions. Focusing the variational properties established in general, simple algebraic representations are found for discontinuities of limitingly small strength. Finally, two examples are treated: the system of equations modelling long waves in liquid-filled elastic tubes, applying also to long water waves in horizontal channels of arbitrary cross-section; and the equations of one-dimensional gas dynamics.