Resonantly Interacting,Weakly Nonlinear Hyperbolic Waves. II. Several Space Variables

A uniformly valid asymptotic theory of resonantly interacting high-frequency waves for nonlinear hyperbolic systems in several space dimensions is developed. When applied to the equations of two-dimensional compressible fluid flow, this theory both predicts the geometric location of the new sound waves produced from the resonant interaction of sound waves and vorticity waves as well as yielding a simplified system which governs the evolution of the amplitudes. In this important special case, this system is two Burgers equations coupled by a linear integral operator with known kernel given by the vortex strength of the shear wave. Several inherently multidimensional assumptions for the phases are needed in this theory, and theoretical examples are given which delineate these assumptions. Furthermore, explicit necessary and sufficient conditions for the validity of the earlier noninteracting wave theory of Hunter and Keller are derived; these explicit conditions indicate that generally waves resonate and interact in several dimensions.