First integrals of the Maxwell-Bloch system
type: Item
instance of: scholarly article
title: First integrals of the Maxwell-Bloch system
zbMATH Open document ID: 1452.34022
author: Kaiyin Huang
author: Shaoyun Shi
author: Wenlei Li
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2020-08-03T00:00:00Z
review text: Consider the three-dimensional polynomial system \[ \frac{dx}{dt}= -ax+y, \quad \frac{dy}{dt}=-by+xz, \quad \frac{dz}{dt}=c(z-\delta_0)y^2-4xy ,\tag{1} \] which is equivalent to the Maxwell-Bloch system in the laser dynamics. The authors prove that system (1) does not possess any rational first integral in the case \(c \neq 0 \) and \( 2\sqrt{(a-b)^2+4\delta_0}/c\) is not an odd number. In the case \(a=b=c=0\), system (1) has two functionally independent polynomial first integrals \[\Phi_1\equiv 2x+z, \quad \Phi_2\equiv 4y+z^2.\] In the case \(a \neq 0, b=c=0\), the unique global analytic first integrals of (1) are of the form \(\Phi(4y^2+z^2)\), where \(\Phi\) is analytic.
reviewed by: Klaus R. Schneider
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34A05
Mathematics Subject Classification ID: 78A60
zbMATH DE Number: 7226753
zbMATH Keywords: first integrals
zbMATH Keywords: Maxwell-Bloch system
zbMATH Keywords: laser dynamics
MaRDI profile type: Publication
full work available at URL: crmath.6
OpenAlex ID: W3012794148
cites work: Galoisian obstructions to non-Hamiltonian integrability
cites work: Q4870162
cites work: Extended integrability and bi-Hamiltonian systems
cites work: Persistence of periodic solutions for higher order perturbed differential systems via Lyapunov–Schmidt reduction
cites work: Group-theoretic obstructions to integrability
cites work: Approximate solutions of Maxwell Bloch equations and possible Lotka-Volterra type behavior
cites work: Elements of applied bifurcation theory
cites work: Local first integrals of differential systems and diffeomorphisms
cites work: Galoisian obstruction to the integrability of general dynamical systems
cites work: Corrigendum to: ``Galoisian obstruction to the integrability of general dynamical systems''
cites work: Identifying weak foci and centers in the Maxwell-Bloch system
cites work: Formal and analytic integrability of the Lorenz system
cites work: Global analytic integrability of the Rabinovich system
cites work: Analytic first integrals of the FitzHugh-Nagumo systems
cites work: On theC1non-integrability of differential systems via periodic orbits
cites work: On the Hamilton-Poisson realizations of the integrable deformations of the Maxwell-Bloch equations
cites work: Differential Galois theory and non-integrability of Hamiltonian systems
cites work: Galoisian obstructions to integrability of Hamiltonian systems. I.
DOI: 10.5802/CRMATH.6
description: scientific article; zbMATH DE number 7226753