The Cauchy-Riemann conditions and localized asymptotic solutions of the linearized shallow-water equations

Singular solutions of the two-dimensional shallow-water equations with algebraic singularities of the “square root” type, which have been studied before [1-4], propagate along the trajectories of the external velocity field, over which this field satisfies the Cauchy-Riemann conditions. In other words, the differential of the phase flow on such a trajectory is proportional to an orthogonal operator. It turns out that in the linear approximation this situation is strongly linked with the “spreading” effect of solutions of the hydrodynamic equations (cf. [5,6]); namely, a localized asymptotic solution of the Cauchy problem for the linearized shallowwater equations maintains its form (i.e. does not spread) if and only if the Cauchy-Riemann conditions hold on the trajectory of the outer flow along which the disturbance is propagating.