Resonantly Interacting Weakly Nonlinear Hyperbolic Waves. I. A Single Space Variable

We present a systematic asymptotic theory for resonantly interacting weakly nonlinear hyperbolic waves in a single space variable. This theory includes as a special case the theory of nonresonant interacting waves for general hyperbolic systems developed recently by J. Hunter and J. B. Keller, when specialized to a single space variable. However, we are also able to treat the general situation when resonances occur in the hyperbolic system. Such resonances are the typical case when the hyperbolic system has at least three equations and when, for example, small-amplitude periodic initial data are prescribed. In the important physical example of the 3 × 3 system describing compressible fluid flow in a single space variable, the resonant asymptotic theory developed by the authors yields, as limit equations, a pair of inviscid Burgers equations coupled through a linear integral operator with known kernel defined through the initial data for the entropy wave. (In the general case we give many new conditions guaranteeing nonresonance for a given hyperbolic system with prescribed initial data, as well as other new structural conditions which imply that resonance occurs.) A method for treating resonantly interacting waves in several space variables, together with applications, will be developed by the authors elsewhere.