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Calculation of integrals of the Hugoniot-Maslov chain for singular vortical solutions of the shallow-water equation. (English) Zbl 1178.37070
Theor. Math. Phys. 139, No. 1, 500-512 (2004); translation from Teor. Mat. Fiz. 139, No. 1, 62-76 (2004).
Summary: We discuss the problems of the Hugoniot-Maslov chain integrability for singular vortical solutions of the shallow-water equations on the \(\beta\) plane. We show that the complex variables used to derive the chain automatically give most of the integrals of the complete and the truncated chains. We also study how some of these integrals are related to the Lagrangian invariant (potential vorticity). We discuss how to choose solutions of the chain that can be used to describe the actual trajectories of tropical cyclones.
MSC:
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q35 PDEs in connection with fluid mechanics
86A10 Meteorology and atmospheric physics
35Q53 KdV equations (Korteweg-de Vries equations)
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