Abstract
A nontrivially solvable 4-dimensional Hamiltonian system is applied to the problem of wave fronts and to the asymptotic theory of partial differential equations. The Hamilton function we consider is H(x,p)=D(x)|p|. Such Hamiltonians arise when describing the fronts of linear waves generated by a localized source in a basin with a variable depth. We consider two realistic types of bottom shape: 1) the depth of the basin is determined, in the polar coordinates, by the function D(ϱ,φ)=(ϱ2+b)∕(ϱ2+a) and 2) the depth function is D(x,y)=(x2+b)∕(x2+a). As an application, we construct the asymptotic solution to the wave equation with localized initial conditions and asymptotic solutions of the Helmholtz equation with a localized right-hand side.
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