Abstract
Weakly nonlinear wave interactions are resonant or nonresonant. The linearized dispersion relation of the wave motion determines the resonant interactions. Resonant interactions cause significant changes in the wave-field. The evolution of the wave-field is determined using weakly nonlinear asymptotics. Quadratically nonlinear resonant interactions of dispersive waves satisfy the three wave resonance condition. The wave amplitudes solve the three wave resonant interaction equations. The phase velocity of hyperbolic waves is independent of frequency. As a result, hyperbolic waves participate in many resonant interactions. The amplitude of a single hyperbolic wave satisfies the inviscid Burgers equation. Harmonic resonance causes wave-form distortion and shock formation. The amplitudes of several interacting hyperbolic waves solve a system of integro-differential equations. The interaction of three oblique hyperbolic planar waves can generate a countably infinite family of new waves. Weak resonance of nonplanar hyperbolic waves also generates infinitely many new waves.
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