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Hugoniót-Maslov chains for singular vortical solutions to quasilinear hyperbolic systems and typhoon trajectory. (English. Russian original) Zbl 1069.37049

J. Math. Sci., New York 124, No. 5, 5209-5249 (2004); translation from Sovrem. Mat., Fundam. Napravl. 2, 5-44 (2003).
Summary: According to Maslov, many 2D quasilinear systems of PDEs possess only three algebras of singular solutions with properties of structural self-similarity and stability. They are the algebras of shock waves, narrow solitons, and square-root point singularities (solitary vortices). Their propagation is described by infinite chains of ODEs (the Hugoniót-Maslov chains). We consider the Hugoniót-Maslov chain for the square-root point singularities of the shallow water equations. We discuss different related mathematical questions (in particular, unexpected integrability effects) as well as their possible application to the problem of typhoon dynamics.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35L60 First-order nonlinear hyperbolic equations
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
76B47 Vortex flows for incompressible inviscid fluids
86A10 Meteorology and atmospheric physics
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