An exactly solvable problem of wave fronts and applications to the asymptotic theory

This is the full and extended version of the brief note arXiv:1908.00938. 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 $H(\mathbf x,\mathbf p)=\sqrt{D(\mathbf{x})}|\mathbf{p}|$. 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 \emph{realistic} types of bottom shape: 1) the depth of the basin is determined, in the polar coordinates, by the function $D(\varrho,\varphi)=(\varrho^2+b)/(\varrho^2+a)$ and 2) the depth function is $D(x,y)=(x^2+b)/(x^2+a)$. 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.