On Maslov's method for constructing combined asymptotics for $h$-pseudodifferential equations

The authors discuss a scheme proposed by V. P. Maslov for constructing combined (with respect to smoothness and a small parameter $h$) asymptotics of the solution of the Cauchy problem for $h$-pseudodifferential equations. The exposition is carried out by means of examples of equations for the oscillations of a crystal lattice and for water waves. The main attention is given to the isolation of the leading term of the asymptotics. A number of estimates are proved for the remainders in formulas for the action of an $h$-pseudodifferential operator on the exponential function, with respect to smoothness and the parameter.
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