We obtain a complete solution for the mean-field dynamics of the BCS paired state with a large, but finite number of Cooper pairs in the nonadiabatic regime. We show that the problem reduces to a classical integrable Hamiltonian system and derive a complete set of its integrals of motion. The condensate exhibits irregular multi-frequency oscillations ergodically exploring the part of the phase-space allowed by the conservation laws. In the thermodynamic limit however the system can asymptotically reach a steady state.The study of the dynamics of the BCS superconductors has a long history [1]. Early attempts to describe nonstationary superconductivity were based on the timedependent Ginzburg-Landau (TDGL) equation [2,3,4], which reduces the problem to the time evolution of a single collective order parameter ∆(t). The TDGL approach is valid only provided the system quickly reaches an equilibrium with the instantaneous value of ∆(t), i.e. a local equilibrium is established faster than the time scale of the order parameter variation, τ ∆ ≃ 1/∆. This requirement limits the applicability of the TDGL to special situations where pair breaking dominates, e.g. due to a large concentration of magnetic impurities. An alternative to TDGL is the Boltzmann kinetic equation [5,6] for the quasiparticle distribution function coupled to a self-consistent equation for ∆(t). This approach is justified only when external parameters change slowly on the τ ∆ time scale, so that the system can be characterized by a quasiparticle distribution.Is it possible to describe theoretically the dynamics of a BCS paired state in the nonadiabatic regime when external parameters change substantially on the τ ∆ time scale? In particular, an important question is whether, following a sudden perturbation, the condensate reaches a steady state on a τ ∆ time scale or on a much longer quasiparticle energy relaxation time scale τ ǫ . In the nonadiabatic regime both TDGL and the Boltzmann kinetic equations fail and one has to deal with the coupled coherent dynamics of individual Cooper pairs. Recent studies [15,16,17,18] of this outstanding problem were motivated by experiments on fermionic pairing in cold atomic alkali gases [7,8]. The strength of pairing interactions in these systems can be fine tuned rapidly by a magnetic field, making it easier than in metals to access the nonadiabatic regime experimentally.The main result of the present paper is an explicit general solution for the dynamics of the BCS model, which describes a spatially homogenous condensate at times t ≪ τ ǫ . We employ the usual BCS mean-field approximation, which is accurate when the number of Cooper pairs is large [10,11]. It turns out that the mean-field BCS dynamics can be formulated as a nonlinear classical Hamiltonian problem. We obtain the exact solution for all initial conditions and a complete set of integrals of motion for the mean-field BCS dynamics.In this paper we assume that the number of Cooper pairs in the system is arbitrary large, but finite. In this case the typica...

We develop an explicit description of a time-dependent response of fermionic condensates to perturbations. The dynamics of Cooper pairs at times shorter than the energy relaxation time can be described by the BCS model. We obtain a general explicit solution for the dynamics of the BCS model. We also solve a closely related dynamical problem -the central spin model, which describes a localized spin coupled to a "spin bath". Here, we focus on presenting the solution and describing its general properties, but also mention some applications, e.g. to nonstationary pairing in cold Fermi gases and to the issue of electron spin decoherence in quantum dots. A typical dynamics of the BCS and central spin models is quasi-periodic with a large number of frequencies and stable under small perturbations. We show that for certain special initial conditions the number of frequencies decreases and the solution simplifies. In particular, periodic solutions correspond to the ground state and excitations of the BCS model.

Using the n-particle periodic Toda lattice and the relativistic generalization due to Ruijsenaars of the elliptic Calogero-Moser system as examples, we revise the basic properties of the Bäcklund transformations (BT's) from the Hamiltonian point of view. The analogy between BT and Baxter's quantum Q-operator pointed out by Pasquier and Gaudin is exploited to produce a conjugated variable µ for the parameter λ of the BT B λ such that µ belongs to the spectrum of the Lax operator L(λ). As a consequence, the generating function of the composition B λ 1 • . . . • B λn of n BT's gives rise also to another canonical transformation separating variables for the model. For the Toda lattice the dual BT parametrized by µ is introduced.

We present a geometric construction of Bäcklund transformations and discretizations for a large class of algebraic completely integrable systems. To be more precise, we construct families of Bäcklund transformations, which are naturally parametrized by the points on the spectral curve(s) of the system. The key idea is that a point on the curve determines, through the Abel-Jacobi map, a vector on its Jacobian which determines a translation on the corresponding level set of the integrals (the generic level set of an algebraic completely integrable systems has a group structure). Globalizing this construction we find (possibly multi-valued, as is very common for Bäcklund transformations) maps which preserve the integrals of the system, they map solutions to solutions and they are symplectic maps (or, more generally, Poisson maps). We show that these have the spectrality property, a property of Bäcklund transformations that was recently introduced. Moreover, we recover Bäcklund transformations and discretizations which have up to now been constructed by ad-hoc methods, and we find Bäcklund transformations and discretizations for other integrable systems. We also introduce another approach, using pairs of normalizations of eigenvectors of Lax operators and we explain how our two methods are related through the method of separation of variables. Contents1991 Mathematics Subject Classification. 35Q58, 37J35, 58J72, 70H06.

For the integrable case of the discrete self-trapping (DST) model we construct a Bäcklund transformation. The dual Lax matrix and the corresponding dual Bäcklund transformation are also found and studied. The quantum analog of the Bäcklund transformation (Q-operator) is constructed as the trace of a monodromy matrix with an infinite-dimensional auxiliary space. We present the Q-operator as an explicit integral operator as well as describe its action on the monomial basis. As a result we obtain a family of integral equations for multivariable polynomial eigenfunctions of the quantum integrable DST model. These eigenfunctions are special functions of the Heun class which is beyond the hypergeometric class. The found integral equations are new and they shall provide a basis for efficient analytical and numerical studies of such complicated functions.