On the relationship between the classical Dicke-Jaynes-Cummings-Gaudin model and the nonlinear Schrödinger equation

Abstract

In this paper, the relationship between the classical Dicke-Jaynes-Cummings-Gaudin (DJCG) model and the nonlinear Schrödinger (NLS) equation is studied. It is shown that the classical DJCG model is equivalent to a stationary NLS equation. Moreover, the standard NLS equation can be solved by the classical DJCG model and a suitably chosen higher order flow. Further, it is also shown that classical DJCG model can be transformed into the classical Gaudin spin model in an external magnetic field through a deformation of Lax matrix. Finally, the separated variables are constructed on the common level sets of Casimir functions and the generalized action-angle coordinates are introduced via the Hamilton-Jacobi equation.

Action-angle variables for the Lie–Poisson Hamiltonian systems associated with the Hirota–Satsuma modified Boussinesq equation

“…The separation of variables for finite-dimensional integrable systems is important for constructing action-angle variables. A series of literature studies shows research on finite-dimensional integrable systems associated with hyperelliptic spectral curves (see, e.g., Kuznetsov [7]; Babelon and Talon [8]; Kalnins et al; [9]; Eilbeck et al; [10]; Harnad and Winternitz [11]; Ragnisco [12]; Kulish et al; [13]; Qiao [14]; Zeng [15]; Zhou [16]; Zeng and Lin [17]; Cao et al; [18]; Derkachev [19]; Du and Geng [20]; Du and Yang [21]). However, the study on integrable systems associated with non-hyperelliptic spectral curves is much more complicated (see, e.g., Sklyanin [22]; Adams et al; [23]; Buchstaber et al; [24]; Dickey [25]; Derkachov and Valinevich [26]).…”

Action-angle variables for the Lie–Poisson Hamiltonian systems associated with Boussinesq equation

“…We use the method in [17] to construct separation variables and use the Hamilton-Jacobi theory for the generating functions of conserved integrals [15,16] to introduce the action-angle variables.…”

“…Separation of variables and construction of action-angle variables for finite-dimensional integrable systems generated by 2 × 2 Lax matrices associated with hyperelliptic spectral curves have been studied widely and deeply in the past decades [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. However, the systems generated by 3 × 3 Lax matrices, which are related to non-hyperelliptic spectral curves are much more complicated.…”