Geodesic
Approximate double-periodic solutions in $(1+1)$-dimensional $\varphi ^4$-theory
Teoretičeskaâ i matematičeskaâ fizika, Tome 116 (1998) no. 2, pp. 182-192 
S. Yu. Vernov; O. A. Khrustalev. Approximate double-periodic solutions in $(1+1)$-dimensional $\varphi ^4$-theory. Teoretičeskaâ i matematičeskaâ fizika, Tome 116 (1998) no. 2, pp. 182-192. http://geodesic.mathdoc.fr/item/TMF_1998_116_2_a1/
@article{TMF_1998_116_2_a1,
     author = {S. Yu. Vernov and O. A. Khrustalev},
     title = {Approximate double-periodic solutions in $(1+1)$-dimensional $\varphi ^4$-theory},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {182--192},
     year = {1998},
     volume = {116},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1998_116_2_a1/}
}
TY  - JOUR
AU  - S. Yu. Vernov
AU  - O. A. Khrustalev
TI  - Approximate double-periodic solutions in $(1+1)$-dimensional $\varphi ^4$-theory
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 1998
SP  - 182
EP  - 192
VL  - 116
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_1998_116_2_a1/
LA  - ru
ID  - TMF_1998_116_2_a1
ER  - 
%0 Journal Article
%A S. Yu. Vernov
%A O. A. Khrustalev
%T Approximate double-periodic solutions in $(1+1)$-dimensional $\varphi ^4$-theory
%J Teoretičeskaâ i matematičeskaâ fizika
%D 1998
%P 182-192
%V 116
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_1998_116_2_a1/
%G ru
%F TMF_1998_116_2_a1

Voir la notice de l'article provenant de la source Math-Net.Ru

Double-periodic solutions of the Euler–Lagrange equation for the $(1+1)$-dimensional scalar $\varphi^4$-theory are considered. The nonlinear term is assumed to be small, and the Poincarй method is used to seek asymptotic solutions in the standing-wave form. The principal resonance problem, which arises for zero mass, is resolved if the leading-order term is taken in the form of a Jacobi elliptic function.

[1] J. M. Colon, C. R Acad. Sci. Paris. Ser. A, 294 (1982), 127 | MR

[2] H. Brezis, J. M. Colon, L. Nirenberg, Commun. Pure Appl. Math., 33 (1980), 667 | DOI | MR | Zbl

[3] P. Rabinovitz, Commun. Pure Appl. Math., 30 (1977), 31

[4] H. Brezis, Bull. Amer. Math. Soc., 8 (1983), 409 | DOI | MR | Zbl

[5] C. E. Wayne, Commun. Math. Phys., 127 (1990), 479 | DOI | MR | Zbl

[6] G. Duffing, Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung, Vieweg, Braunschweig, 1918 | Zbl

[7] G. Beitman, A. Erdeii, Vysshie transtsendentnye funktsii (Ellipticheskie i avtomorfnye funktsii, funktsii Lame i Mate), Nauka, M., 1967 | MR

[8] S. Aubry, J. Chem. Phys., 64 (1976), 3392 | DOI

[9] D. F. Kurdgelaidze, ZhETF, 36 (1959), 842 | MR | Zbl

[10] P. K. Silaev, O. A. Khrustalev, TMF (to appear)

[11] A. Puankare, Novye metody nebesnoi mekhaniki. Izbrannye trudy, Nauka, M., 1971–1974

[12] N. N. Bogolyubov, Yu. A. Mitropolskii, Asimptoticheskie metody v teorii nelineinykh kolebanii, Nauka, M., 1974 | MR

[13] A. Naife, Metody vozmuschenii, Mir, M., 1976 | MR