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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.
MSC:
76B47 Vortex flows for incompressible inviscid fluids
35Q35 PDEs in connection with fluid mechanics
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