Geodesic
The geometric series method for constructing exact solutions to nonlinear evolution equations
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 7, pp. 1113-1125 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

It is proved that, for the majority of integrable evolution equations, the perturbation series constructed based on the exponential solution of the linearized problem is geometric or becomes geometric as a result of changing the variable in the equation or after a transformation of the series. Using this property, a method for constructing exact solutions to a wide class of nonintegrable equations is proposed; this method is based on the requirement for the perturbation series to be geometric and on the imposition of constraints on the values of the coefficients and parameters of the equation under which the sum of the series is the solution to be found. The effectiveness of using the diagonal Padé approximants the minimal order of which is determined by the order of the pole of the solution to the equation is demonstrated. 
@article{ZVMMF_2017_57_7_a3,
     author = {A. V. Bochkarev and A. I. Zemlyanukhin},
     title = {The geometric series method for constructing exact solutions to nonlinear evolution equations},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1113--1125},
     year = {2017},
     volume = {57},
     number = {7},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_7_a3/}
}
TY  - JOUR
AU  - A. V. Bochkarev
AU  - A. I. Zemlyanukhin
TI  - The geometric series method for constructing exact solutions to nonlinear evolution equations
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2017
SP  - 1113
EP  - 1125
VL  - 57
IS  - 7
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_7_a3/
LA  - ru
ID  - ZVMMF_2017_57_7_a3
ER  - 
%0 Journal Article
%A A. V. Bochkarev
%A A. I. Zemlyanukhin
%T The geometric series method for constructing exact solutions to nonlinear evolution equations
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2017
%P 1113-1125
%V 57
%N 7
%U http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_7_a3/
%G ru
%F ZVMMF_2017_57_7_a3
A. V. Bochkarev; A. I. Zemlyanukhin. The geometric series method for constructing exact solutions to nonlinear evolution equations. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 7, pp. 1113-1125. http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_7_a3/

[1] Koul Dzh., Metody vozmuschenii v prikladnoi matematike, Mir, M., 1972

[2] Korpel A., “Solitary wave formation through nonlinear coupling of finite exponential waves”, Phys. Lett., 68A (1978), 179–181 | DOI | MR

[3] Lambert F., Musette M., “Solitary waves, padeons and solitons”, Lectures Notes Math., 1071, 1984, 197–212 | DOI | MR | Zbl

[4] Hereman W., Banerjee P. P., Korpel A., Assanto G., van Immerzeele A., Meerpoel A., “Exact solitary wave solutions of non-linear evolution and wave equations using a direct algebraic method”, J. Phys. A: Math. Gen., 19 (1986), 607–628 | DOI | MR | Zbl

[5] Baikov V. A., Khusnutdinova K. R., “Formal linearization and exact solutions of some nonlinear partial differential equations”, Nonlinear Mathematical Physics, 3:1–2 (1996), 139–146 | DOI | MR | Zbl

[6] Kont R., Myuzett M., Metod Penleve i ego prilozheniya, In-t kompyuter. issled., Regulyar. i khaotich. dinamika, M.–Izhevsk, 2011

[7] Beiker Dzh., Greivs-Morris P., Approksimatsii Pade, Mir, M., 1986

[8] Sazonov S. V., “Opticheskie solitony v sredakh iz dvukhurovnevykh atomov”, Nauch.-tekhn. vestnik inf. tekhnologii, mekhaniki i optiki, 2013, no. 5(87), 1–22

[9] Shabat A. B., Adler V. E., Marikhin V. G. i dr., Entsiklopediya integriruemykh sistem. ver. 0039/M, ITF, 2009

[10] Borisov A. B., Zykov S. A., Pavlov M. V., “Uravnenie Tsitseiki i razmnozhenie nelineinykh integriruemykh uravnenii”, TMF, 131:1 (2002), 126–134 | DOI | Zbl

[11] Conte R., Musette M., “Link between solitary waves and projective Riccati equations”, J. Phys. A: Math. Gen., 25 (1992), 5609–5623 | DOI | MR | Zbl

[12] Meshkov A. G., Sokolov V. V., “Integriruemye evolyutsionnye uravneniya s postoyannoi separantoi”, Ufimsk. matem. zhurn., 4:3 (2012), 104–154 | Zbl

[13] Ibragimov N. Kh., Gruppy preobrazovanii v matematicheskoi fizike, Nauka, M., 1983

[14] Banerjee P. P., Daoud F., Hereman W., “A straightforward method for finding implicit solitary wave solutions of nonlinear evolution and wave equations”, J. Phys. A: Math. Gen., 23 (1990), 521–536 | DOI | MR | Zbl

[15] Kazeminia M., Tolou P., Mahmoudi J., Khatami I., Khatami I., “Solitary and Periodic Solutions of BBMB Equation via Exp-Function Method”, Adv. Studies Theor. Phys., 3:12 (2009), 461–471 | Zbl

[16] Kudryashov N. A., Metody nelineinoi matematicheskoi fiziki, Izd. Dom “Intellekt”, Dolgoprudnyi, 2010

[17] Gandarias M. L., Bruzon M. S., “Simmetriinyi analiz i tochnye resheniya dlya nekotorykh uravnenii Ostrovskogo”, TMF, 168:1 (2011), 49–64 | DOI

[18] Kudryashov N. A., Sinelshchikov D. I., Demina M. V., “Exact solutions of the generalized Bretherton equation”, Phys. Lett. A, 375 (2011), 1074–1079 | DOI | MR | Zbl

[19] Baldwin D., Goktas U., Hereman W., Hong L., Martino R. S., Miller J. C., “Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs”, J. Symb. Comp., 37 (2004), 669–705 | DOI | MR | Zbl