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New conservation laws for the interaction of nonlinear waves. (English) Zbl 0874.35089
Using an analogy between the particle collisions and the resonance wave interactions, the author introduces the problem of invariants of wave interactions. The relation between the integrability and the invariant of wave interactions is considered, too. The existence and physical consequences of a new invariant for the system of Rossby waves is studied. The relation to web geometry enables the author to obtain the following results: the 4-wave interaction in \(1-d\) media can possess no more than one extra invariant and the 3-wave interaction in \(2-d\) media can only have 0,1,2, or infinitely many extra invariants. Finally, one explains the connection between the invariants of wave interactions and the dynamical equations of continuous media.
Reviewer: V.A.Sava (Iaşi)

35Q35 PDEs in connection with fluid mechanics
35Q58 Other completely integrable PDE (MSC2000)
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B65 Rossby waves (MSC2010)
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