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On an application of conservation laws in asymptotic problems for equations with operatorvalued symbol. (Russian) Zbl 0731.35062
The following equations (elasto-dynamics) are considered $$(h\ll 1)$$ $h^ 2\nabla^ 2u_ i+u_{izz}+(c^ 2-1)h\frac{\partial}{\partial x_ i}(h\nabla u+u_{3z})-h^ 2\mu^ 2u_{itt}=0,\quad i=1,2,$ $h^ 2\nabla^ 2u_ 3+u_{3zz}+(c^ 2-1)\frac{\partial}{\partial z}(h\nabla u+u_{3z})-h^ 2\mu^ 2u_{3tt}=0,\quad z<-H(x);$ $h^ 2\nabla^ 2\phi +\phi_{zz}=0,\quad -H(x)<z<0;$ together with a system of boundary value conditions. The solution is sought in the form $\left( \begin{matrix} \phi \\ U\end{matrix} \right)=\exp (\frac{iS(x,t)}{h})\sigma (x,t)\left( \begin{matrix} \psi \\ V\end{matrix} \right)\exp (\int^{t}_{0}Md\tau),$ where $$U=(u_ 1,u_ 2,u_ 3)$$. Explicit formulas are indicated for $$\psi$$ and V. Under a normalizing assumption on the energy function, it is shown that $$M\equiv 0$$.

##### MSC:
 35L55 Higher-order hyperbolic systems 35C05 Solutions to PDEs in closed form 35Q72 Other PDE from mechanics (MSC2000) 74B05 Classical linear elasticity