Geodesic
On Maslov's method for constructing combined asymptotics for $h$-pseudodifferential equations
Izvestiya. Mathematics, Tome 34 (1990) no. 2, pp. 425-439 
V. G. Danilov; P. N. Zhevandrov. On Maslov's method for constructing combined asymptotics for $h$-pseudodifferential equations. Izvestiya. Mathematics, Tome 34 (1990) no. 2, pp. 425-439. http://geodesic.mathdoc.fr/item/IM2_1990_34_2_a10/
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     author = {V. G. Danilov and P. N. Zhevandrov},
     title = {On {Maslov's} method for constructing combined asymptotics for $h$-pseudodifferential equations},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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. Bibliography: 9 titles

[1] Maslov V. P., Operatornye metody, Nauka, M., 1973 | MR

[2] Maslov V. P., Danilov V. G., “Printsip dvoistvennosti Pontryagina dlya vychisleniya effekta tipa Cherenkova v kristallakh i raznostnykh skhemakh, I”, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 166, 1984, 130–160 | MR | Zbl

[3] Dobrokhotov S. Yu., Zhevandrov P. Ya., “Nestandartnye kharakteristiki i operatornyi metod Maslova v lineinykh zadachakh o neustanovivshikhsya volnakh na vode”, Funkts. analiz i ego prilozh., 19:4 (1985), 43–54 | MR | Zbl

[4] Danilov V. G., “Otsenki dlya kanonicheskikh psevdodifferentsialnykh operatorov s kompleksnoi fazoi”, Dokl. AN SSSR, 244:4, 800–807 | MR

[5] Khermander L., “Integralnye operatory Fure, 1”, Matematika, 16:1 (1972), 17–61 | Zbl

[6] Dobrokhotov S. Yu., “Metody Maslova v linearizovannoi teorii gravitatsionnykh voln na poverkhnosti zhidkosti”, Dokl. AN SSSR, 269:1 (1983), 76–80 | MR | Zbl

[7] Danilov V. G., Dzhemesyuk I. A., “Otsenki dlya integralnykh operatorov Fure s kompleksnoi fazoi”, Matem. zametki, 32:4 (1982), 557–576 | MR | Zbl

[8] Karasev M. V., “Ischislenie pochti proizvodyaschikh uporyadochennykh operatorov”, Tr. MIEM, no. 38, 1974, 73–115 | MR

[9] Egorov Yu. V., Lineinye differentsialnye uravneniya glavnogo tipa, Nauka, M., 1984 | MR