Localized wave and vortical solutions to linear hyperbolic systems and their application to linear shallow water equations. (English) Zbl 1180.35336

Linear hyperbolic systems of differential equations for an m-D vector function are considered. There is a large discussion on the asymptotics of solutions to the associated Cauchy problem with fast-decaying or localized initial data. The aim is to present explicit formulas for the asymptotic solutions of the Cauchy problem. Authors concentrate and prove results for the Cauchy problem with localized initial data for the linearized shallow water equations. A main point of proofs is the possibility of representation of fast decaying functions by means of Maslov canonical operator over a specific Lagrange manifold.


35L45 Initial value problems for first-order hyperbolic systems
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35Q35 PDEs in connection with fluid mechanics
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
47G30 Pseudodifferential operators
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B47 Vortex flows for incompressible inviscid fluids
Full Text: DOI


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