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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\textit{S. Yu. Dobrokhotov}, Theor. Math. Phys. 112, No. 1, 827--843 (1997; Zbl 0978.76512); translation from Teor. Mat. Fiz. 112, No. 1, 47--66 (1997)

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