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Reduction of Hugoniot-Maslov chains for trajectories of solitary vortices of the “shallow water” equations on the Hill equation. (English. Russian original) Zbl 0978.76512
Theor. Math. Phys. 112, No. 1, 827-843 (1997); translation from Teor. Mat. Fiz. 112, No. 1, 47-66 (1997).
Summary: According to Maslov’s idea, many two-dimensional, quasilinear hyperbolic systems of partial differential equations admit only three types of singularities that are in general position and have the property of “structure self-similarity and stability.” Those are: shock waves, “narrow” solitons, and “square-root” point singularities (solitary vortices). Their propagation is described by an infinite chain of ordinary differential equations (ODE) that generalize the well-known Hugoniot conditions for shock waves. After some reasonable closure of the chain for the case of solitary vortices in the “shallow water” equations, we obtain a nonlinear system of sixteen ODE, which is exactly equivalent to the (linear) Hill equation with a periodic potential. This means that, in some approximations, the trajectory of a solitary vortex can be described by the Hill equation. This result can be used to predict the trajectory of the vortex center if we know its observable part.
76B47 Vortex flows for incompressible inviscid fluids
35Q35 PDEs in connection with fluid mechanics
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