A Quasi Separable Dissipative Maxwell–Bloch System for Laser Dynamics

Abstract

The Maxwell-Bloch dissipative equations describe laser dynamics. Under a simple condition on the parameters there exist two time dependent first integrals, that allow a nonstandard separation of variables in the equations. That condition has a precise physical meaning. The separated differential equations lead naturally to simple conjectures on the asymptotic behavior of the physical variables.

Nonlocal constants of motion in Lagrangian Dynamics of any order

“…• The dissipative Maxwell-Bloch equations for laser dynamics, taken from Gorni-Residori-Zampieri [9], for which a quite natural family q λ (t) yields a constant of motion (1.2) which, for a special choice of the parameters, turns out to be a genuine first integral N(t, q, q) since the integral term vanishes. The first integral permits some kind of separation of variables.…”

Lagrangian dynamics by nonlocal constants of motion

“…The conjecture is that the nontrivial solutions in the q 1 , q 2 plane converge to a point on the circle with radius r ∞ and center in the origin, as in Figure 2b. 7. Conclusions.…”

“…• hydraulic fluid resistance in Section 5, • the Maxwell-Bloch equations for laser dynamics in Section 6, taken from [5] in the conservative case and from [6] and [7] in the dissipative case.…”

“…where a, b, c ≥ 0, g > 0, k ∈ R are parameters (see Arecchi and Meucci [1] and our paper with Residori [7]). We are going to briefly describe two kinds of nonstandard separation of variables that hold when a = b = c = 0 (conservative case) and when a, b, c > 0 (dissipative) with c = 2a.…”

“…From these separations, we deduced or conjectured some dynamical features which we will not repeat here. All details are in our paper [5] in the conservative case, and in the already cited [7] in the dissipative case.…”