Dispersive nonlinear geometric optics

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Abstract
We construct infinitely accurate approximate solutions to systems of hyperbolic partial differential equations which model short wavelength dispersive nonlinear phenomena. The principal themes are the following. (1) The natural framework for the study of dispersion is wavelength ε solutions of systems of partial differential operators in ε∂. The natural ε-characteristic equation and ε-eikonal equations are not homogeneous. This corresponds exactly to the fact that the speeds of propagation, which are called group velocities, depend on the length of the wave number. (2) The basic dynamic equations are expressed in terms of the operator ε∂t. As a result growth or decay tends to occur at the catastrophic rate ect/ε. The analysis is limited to conservative or nearly conservative models. (3) If a phase φ(x)/ε satisfies the natural ε-eikonal equation, the natural harmonic phases, nφ(x)/ε, generally do not. One needs to impose a coherence hypothesis for the harmonics. (4) In typical examples the set of harmonics which are eikonal is finite. The fact that high harmonics are not eikonal suppresses the wave steepening which is characteristic of quasilinear wave equations. It also explains why a variety of monochromatic models are appropriate in nonlinear settings where harmonics would normally be expected to appear. (5) We study wavelength ε solutions of nonlinear equations in ε∂ for times O(1). For a given system, there is a critical exponent p so that for amplitudes O(εp), one has simultaneously smooth existence for t=O(1), and nonlinear behavior in the principal term of the approximate solutions. This is the amplitude for which the time scale of nonlinear interaction is O(1). (6) The approximate solutions have residual each of whose derivatives is O(εn) for all n>0. In addition, we prove that there are exact solutions of the partial differential equations, that is with zero residual, so that the difference between the exact solution and the approximate solutions is infinitely small. This is a stability result for the approximate solutions.

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