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 $H(\mathbf{x},\mathbf{p})=\sqrt{D\left(\mathbf{x}\right)}\left|\mathbf{p}\right|$. 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 $D(\varrho ,\phi )=({\varrho }^2+b)∕({\varrho }^2+a)$ 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.

非平凡的四维哈密顿系统被应用到波前问题和偏微分方程的渐近理论。我们考虑的汉密尔顿函数是$H（\mathbf{X}，\mathbf{p}）=\sqrt{d（\mathbf{X}）}\left|\mathbf{p}\right|$。当描述由深度可变的盆地中的局部震源产生的线性波的波前时，就会出现这种哈密顿量。我们考虑底部形状的两种现实类型：1）盆的深度在极坐标中由函数确定$d（\varrho ，\phi ）=（{\varrho }^2+b）∕（{\varrho }^2+\mathrm{一种}）$ 2）深度函数是 $d（X，ÿ）=（X^2+b）∕（X^2+\mathrm{一种}）$。作为应用，我们构造了具有局部初始条件的波动方程的渐近解和具有局部右侧的亥姆霍兹方程的渐近解。