First integrals of the Maxwell–Bloch system

Kaiyin Huang,Wenlei Li,Shaoyun Shi

doi:10.5802/crmath.6

Abstract

First integrals of the Maxwell–Bloch system

Highlights

E = −κE + g P, P = −γ⊥P + g E, ̇ = −γ ( − 0) − 4g PE, où κ, γ⊥, g, γ, 0 sont des paramètres réels
Which describes the interaction between the coupling of the fundamental cavity mode E, the collective atomic polarization P and the population inversion ∆ [1]
As indicated in [1, 7, 8], it can be used to model Type I laser (He-Ne), Type II laser (Ruby and CO2) and Type III laser in the case of γ⊥ ≈ γ κ, γ⊥ γ ≈ κ and 0 large enough, respectively. This system has been analyzed as a dynamical system by many researchers, see for instance [5, 7, 13, 18] and ISSN : 1778-3569 https://comptes- rendus.academie- sciences.fr/mathematique/

Summary

Introduction and statement of the main results

Which describes the interaction between the coupling of the fundamental cavity mode E , the collective atomic polarization P and the population inversion ∆ [1]. The numerical analysis yields that the dynamics of (2) are complex and chaotic, which inspire us to prove both systems are non-integrable. The result shows that system (2) is rationally non-integrable in the sense of Bogoyavlenskij for almost all parameter values. Morales–Ruiz, Ramis, Simó, Baider, Churchill, Rod and Singer have applied the differential Galois theory to the non-integrability of Hamiltonian systems and developed the Morales–Ramis theory, see [3, 6, 19, 20] and references therein. (iii) If a = 0, υ := b = c = 0, it has no global analytic first integrals which are analytic in the parameter υ in a neighborhood of υ = 0. The proof of Theorem 2 and Proposition 3 will be given in Section 3 and Section 4, respectively

Proof of Theorem 1

Proof of Theorem 2

Proof of Proposition 3