Abstract
Singular solutions with algebraic “square-root” type singularity of two-dimensional equations of shallow-water theory are propagated along the trajectories of the external velocity field on which the field satisfies the Cauchy-Riemann conditions. In other words, the differential of the phase flow is proportional to an orthogonal operator on such a trajectory.
It turns out that, in the linear approximation, this fact is closely related to the effect of “blurring” of solutions of hydrodynamical equations; namely, a singular solution of the Cauchy problem for the linearized shallow-water equations preserves its shape exactly (i.e., is not blurred) if and only if the Cauchy-Riemann conditions are satisfied on the trajectory (of the external field) along which the perturbation is propagated.
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The research was financially supported by the Russian Foundation for Basic Research (under grants nos. 05-01-00968 and 04-01-00682).
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Dobrokhotov, S.Y., Tirozzi, B. & Shafarevich, A.I. Cauchy—Riemann conditions and point singularities of solutions to linearized shallow-water equations. Russ. J. Math. Phys. 14, 217–223 (2007). https://doi.org/10.1134/S1061920807020112
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DOI: https://doi.org/10.1134/S1061920807020112