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Cauchy-Riemann conditions and point singularities of solutions to linearized shallow-water equations. (English) Zbl 1119.76008
Summary: 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 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 Cauchy problem for 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.
MSC:
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q35 PDEs in connection with fluid mechanics
35A20 Analyticity in context of PDEs
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76U05 General theory of rotating fluids
Keywords:
blurring effect
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