# zbMATH — the first resource for mathematics

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
##### Keywords:
high frequency waves
Full Text:
##### References:
 [1] Cheverry C., Seminaire EDP delecole polytechnique (1995) [2] Cheverry C., Institute de recherche mathematique de Rennes (1996) [3] DOI: 10.1512/iumj.1977.26.26088 · Zbl 0377.35051 · doi:10.1512/iumj.1977.26.26088 [4] DOI: 10.1002/cpa.3160180408 · Zbl 0141.28902 · doi:10.1002/cpa.3160180408 [5] Hunter J., Transactions of the sixtheArmy Conference on Applied Mathematics and Computing 2 pp 527– (1989) [6] DOI: 10.1016/0022-247X(74)90079-1 · Zbl 0281.35058 · doi:10.1016/0022-247X(74)90079-1 [7] DOI: 10.1007/BF02105186 · Zbl 0820.35093 · doi:10.1007/BF02105186 [8] DOI: 10.1002/cpa.3160100406 · Zbl 0081.08803 · doi:10.1002/cpa.3160100406 [9] Majda Rosales-Schonbek, Studies in Applied Mathematics 79 pp 205– (1988) · Zbl 0669.76103 [10] Pego L., Studies in Applied Mathematics 79 pp 263– (1988) [11] Randall E., To appear in Stud.in Appl.Math 79 (1988) [12] DOI: 10.1006/jdeq.1994.1133 · Zbl 0856.35080 · doi:10.1006/jdeq.1994.1133
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.