Exact Solutions to Degenerate Conservation Laws

Abstract

We consider large variation solutions to systems of conservation laws, for which the Glimm--Lax theory of decay breaks down. We identify and isolate geometric nonlinearities which are distinct from the usual genuine nonlinearity of each wave field by describing some degenerate systems in which all nonlinearity is geometric and is manifested in the coupling of the different wave families. We then construct exact explicit solutions to these equations and examine properties of these solutions. We find a wide variety of phenomena, depending on the form of the nonlinearity. The most striking of these include strong nonlinear instability of solutions and nontrivial time-periodic solutions. We also find solutions which grow or decay exponentially and oscillating solutions which correspond to rotations byan irrational angle. These oscillating, periodic, and exponential solutions can all appear in a single system with small initial data, demonstrating sensitive dependence on initial conditions.