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The modulation equations of nonlinear geometric optics. (English) Zbl 0867.35061
The initial value problem to a system of integro-differential equations \[ (\partial_tU^i)(t,y)+{1\over 2}\Gamma^i_{ii}\partial_y[U^i(t,y)^2]+\sum_{i\neq p<q\neq i}\Gamma^i_{pq}\partial_y\Biggl[\int^1_0 U^p(t,y-s)U^q(t,s)ds\Biggr]=0 \] \[ U^i(0,y)=U^i_0(y),\quad 1\leq i\leq N \] with \((t,y)\in[0,\infty)\times\mathbb{R}\) and with initial data \(U^i_0\) of period 1 is studied. This system describes the leading order interaction of high frequency waves satisfying a system of \(N\) strictly hyperbolic conservation laws, for which every field is genuinly nonlinear. The constants \(\Gamma^i_{pq}\) are computed from second derivatives of the nonlinearities in the conservation laws. As a main result, the author proves that the initial boundary value problem has a global solution if a certain quadratic form formed with the coefficients \(\Gamma^i_{pq}\) is negative. Also, a BV-estimate is proved for the solution. As a second result, a regularity theorem for the solutions is proved.

MSC:
35L65 Hyperbolic conservation laws
78A05 Geometric optics
35A30 Geometric theory, characteristics, transformations in context of PDEs
35L45 Initial value problems for first-order hyperbolic systems
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