zbMATH — the first resource for mathematics

Invariants of wave systems and web geometry. (English) Zbl 0913.76008
Zakharov, V. E. (ed.), Nonlinear waves and weak turbulence. Transl. from the Russian. Providence, RI: AMS, American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 182 (36), 1-30 (1998).
The aim is to outline the general picture describing the wave systems which possess extra invariants, and how many and what invariants they can have. Our considerations cover various wave systems in hydrodynamics, plasma physics, astrophysics, nonlinear optics, and solid state physics. We reveal a close relationship of this physical problem about invariants with classical differential geometry, namely with the theory of webs. This connection enables us to apply the mathematical apparatus of differential geometry and to find new wave systems with extra invariants, as well as to obtain a clear understanding of the physical problem.
For the entire collection see [Zbl 0879.00029].

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
53A60 Differential geometry of webs
78A40 Waves and radiation in optics and electromagnetic theory
76X05 Ionized gas flow in electromagnetic fields; plasmic flow