Reduction of hugoniot-maslov chains for trajectories of solitary vortices of the “shallow water” equations to the hill equation |
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Authors: | S Yu Dobrokhotov |
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Institution: | (1) Institute of Mechanical Problems, Russian Academy of Sciences, Moscow, Russia |
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Abstract: | 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.
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 1, pp. 47–66. |
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