First integrals of the Maxwell–Bloch system

Huang, Kaiyin; Shi, Shaoyun; Li, Wenlei

doi:10.5802/crmath.6

Équations différentielles, Systèmes dynamiques

First integrals of the Maxwell–Bloch system

Huang, Kaiyin ^1,² ; Shi, Shaoyun ^2,³ ; Li, Wenlei ²

¹ School of Mathematics, Sichuan University, Chengdu 610000, P. R. China
² School of Mathematics, Jilin University, Changchun 130012, P. R. China
³ State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun 130012, P. R. China

Comptes Rendus. Mathématique, Tome 358 (2020) no. 1, pp. 3-11.

Résumé
Abstract

Nous étudions les premières intégrales analytiques, rationnelles et $C^{1}$ du système de Maxwell–Bloch

\dot{E} = - κ E + g P, \dot{P} = - γ_{⊥} P + g E ▵, \dot{▵} = - γ_{∥} (▵ - ▵_{0}) - 4 g P E,

où $κ, γ_{⊥}, g, γ_{∥}, ▵_{0}$ sont des paramètres réels. En outre, nous prouvons que ce système est non intégrable rationnel dans le sens de Bogoyavlenskij pour presque toutes les valeurs de paramètres.

We investigate the analytic, rational and $C^{1}$ first integrals of the Maxwell–Bloch system

\dot{E} = - κ E + g P, \dot{P} = - γ_{⊥} P + g E ▵, \dot{▵} = - γ_{∥} (▵ - ▵_{0}) - 4 g P E,

where $κ, γ_{⊥}, g, γ_{∥}, ▵_{0}$ are real parameters. In addition, we prove this system is rationally non-integrable in the sense of Bogoyavlenskij for almost all parameter values.

Reçu le : 2018-05-18
Révisé le : 2019-12-27
Accepté le : 2020-01-13
Publié le : 2020-03-18

DOI : 10.5802/crmath.6

Affiliations des auteurs :

Huang, Kaiyin ^{1, 2} ; Shi, Shaoyun ^{2, 3} ; Li, Wenlei ²

@article{CRMATH_2020__358_1_3_0,
     author = {Huang, Kaiyin and Shi, Shaoyun and Li, Wenlei},
     title = {First integrals of the {Maxwell{\textendash}Bloch} system},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {3--11},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {1},
     year = {2020},
     doi = {10.5802/crmath.6},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/crmath.6/}
}

TY  - JOUR
AU  - Huang, Kaiyin
AU  - Shi, Shaoyun
AU  - Li, Wenlei
TI  - First integrals of the Maxwell–Bloch system
JO  - Comptes Rendus. Mathématique
PY  - 2020
SP  - 3
EP  - 11
VL  - 358
IS  - 1
PB  - Académie des sciences, Paris
UR  - http://www.numdam.org/articles/10.5802/crmath.6/
DO  - 10.5802/crmath.6
LA  - en
ID  - CRMATH_2020__358_1_3_0
ER  -

%0 Journal Article
%A Huang, Kaiyin
%A Shi, Shaoyun
%A Li, Wenlei
%T First integrals of the Maxwell–Bloch system
%J Comptes Rendus. Mathématique
%D 2020
%P 3-11
%V 358
%N 1
%I Académie des sciences, Paris
%U http://www.numdam.org/articles/10.5802/crmath.6/
%R 10.5802/crmath.6
%G en
%F CRMATH_2020__358_1_3_0

Huang, Kaiyin; Shi, Shaoyun; Li, Wenlei. First integrals of the Maxwell–Bloch system. Comptes Rendus. Mathématique, Tome 358 (2020) no. 1, pp. 3-11. doi : 10.5802/crmath.6. http://www.numdam.org/articles/10.5802/crmath.6/

Bibliographie
Cité par

[1] Arecchi, Fortunato T. Chaos and generalized multistability in quantum optics, Phys. Scr., Volume T9 (1985), pp. 85-92 | DOI

[2] Ayoul, Michaël; Zung, Nguyen Tien Galoisian obstructions to non-Hamiltonian integrability, C. R. Math. Acad. Sci. Paris, Volume 348 (2010) no. 23-24, pp. 1323-1326 | DOI | MR | Zbl

[3] Baider, Alberto; Churchill, Richard C.; Rod, David L.; Singer, Michael F. On the infinitesimal geometry of integrable systems, Mechanics day (Waterloo, ON, 1992) (Fields Institute Communications), Volume 7, American Mathematical Society, 1992, pp. 5-56 | Zbl

[4] Bogoyavlenskij, Oleg I. Extended integrability and bi-Hamiltonian systems, Commun. Math. Phys., Volume 196 (1998) no. 1, pp. 19-51 | DOI | MR | Zbl

[5] Cândido, Murilo R.; Llibre, Jaume; Novaes, Douglas D. Persistence of periodic solutions for higher order perturbed differential systems via Lyapunov-Schmidt reduction, Nonlinearity, Volume 30 (2017) no. 9, pp. 3560-3586 | DOI | MR | Zbl

[6] Churchill, Richard C.; Rod, David L.; Singer, Michael F. Group-theoretic obstructions to integrability, Ergodic Theory Dyn. Syst., Volume 15 (1995) no. 1, pp. 15-48 | DOI | MR | Zbl

[7] Hacinliyan, Avadis S.; Kuşbeyzi, İlknur; Aybar, Orhan O. Approximate solutions of Maxwell-Bloch equations and possible Lotka Volterra type behavior, Nonlinear Dyn., Volume 62 (2010) no. 1-2, pp. 17-26 | DOI | MR | Zbl

[8] Khanin, Ya. I. Dynamics of Lasers, Soviet Radio, 1975

[9] Kuznetsov, Yuri A. Elements of applied bifurcation theory, Springer, 2013 | Zbl

[10] Li, Weigu; Llibre, Jaume; Zhang, Xiang Local first integrals of differential systems and diffeomorphisms, Z. Angew. Math. Phys., Volume 54 (2003) no. 2, pp. 235-255 | MR | Zbl

[11] Li, Wenlei; Shi, Shaoyun Galoisian obstruction to the integrability of general dynamical systems, Dyn. Syst., Volume 252 (2012) no. 10, pp. 5518-5534 | MR | Zbl

[12] Li, Wenlei; Shi, Shaoyun Corrigendum to “Galoisian obstruction to the integrability of general dynamical systems”, Dyn. Syst., Volume 262 (2017) no. 3, pp. 1253-1256 | Zbl

[13] Liu, Lingling; Aybar, O. Ozgur; Romanovski, Valery G.; Zhang, Weinian OIdentifying weak foci and centers in the Maxwell–Bloch system, J. Math. Anal. Appl., Volume 430 (2015) no. 1, pp. 549-571 | Zbl

[14] Llibre, Jaume; Valls, Clàudia Formal and analytic integrability of the Lorenz system, J. Phys. A, Math. Gen., Volume 38 (2005) no. 12, pp. 2681-2686 | DOI | MR | Zbl

[15] Llibre, Jaume; Valls, Clàudia Global analytic integrability of the Rabinovich system, J. Geom. Phys., Volume 58 (2008) no. 12, pp. 1762-1771 | DOI | MR | Zbl

[16] Llibre, Jaume; Valls, Clàudia Analytic first integrals of the FitzHugh–Nagumo systems, Z. Angew. Math. Phys., Volume 60 (2009) no. 2, pp. 237-245 | DOI | MR | Zbl

[17] Llibre, Jaume; Valls, Clàudia On the $C^{1}$ non-integrability of differential systems via periodic orbits, Eur. J. Appl. Math., Volume 22 (2011) no. 4, pp. 381-391 | MR | Zbl

[18] Lăzureanu, Cristian On the Hamilton-Poisson realizations of the integrable deformations of the Maxwell–Bloch equations, C. R. Math. Acad. Sci. Paris, Volume 355 (2017) no. 5, pp. 596-600 | DOI | MR | Zbl

[19] Morales-Ruiz, Juan J. Differential Galois theory and non-integrability of Hamiltonian systems, Progress in Mathematics, 179, Birkhäuser, 1999 | MR | Zbl

[20] Morales-Ruiz, Juan J.; Ramis, Jean Pierre Galoisian obstructions to integrability of Hamiltonian systems. I. II., Methods Appl. Anal., Volume 8 (2001) no. 1, p. 33-95, 97–111 | MR | Zbl

[21] Poincaré, Henri Sur l’intégration algébrique des équations différentielles du premier ordre et du premier degré, Palermo Rend., Volume 5 (1897), pp. 193-239 | DOI | Zbl

Cité par Sources :