We study the dynamics of relaxation and thermalization in an exactly solvable model with the goal of understanding the effects of off-shell processes. The focus is to compare the exact evolution of the distribution function with different approximations to the relaxational dynamics: Boltzmann, non-Markovian and Markovian quantum kinetics. The time evolution of the distribution function is evaluated exactly using two methods: time evolution of an initially prepared density matrix and by solving the Heisenberg equations of motion. There are two different cases that are studied in detail: i) no stable particle states below threshold of the bath and a quasiparticle resonance above it and ii) a stable discrete exact `particle' state below threshold. For the case of quasiparticles in the continuum (resonances) the exact quasiparticle distribution asymptotically tends to a statistical equilibrium distribution that differs from a simple Bose-Einstein form as a result of off-shell processes. In the case ii), the distribution of particles does not thermalize with the bath. We study the kinetics of thermalization and relaxation by deriving a non-Markovian quantum kinetic equation which resums the perturbative series and includes off-shell effects. A Markovian approximation that includes off-shell contributions and the usual Boltzmann equation are obtained from the quantum kinetic equation in the limit of wide separation of time scales upon different coarse-graining assumptions. The relaxational dynamics predicted by the non-Markovian, Markovian and Boltzmann approximations are compared to the exact result of the model. The Boltzmann approach is seen to fail in the case of wide resonances and when threshold and renormalization effects are important.Comment: 49 pages, LaTex, 17 figures (16 eps figures

We study the dynamics of relaxation and thermalization in an exactly solvable model of a particle interacting with a harmonic oscillator bath. Our goal is to understand the effects of non-Markovian processes on the relaxational dynamics and to compare the exact evolution of the distribution function with approximate Markovian and non-Markovian quantum kinetics. There are two different cases that are studied in detail: (i) a quasiparticle (resonance) when the renormalized frequency of the particle is above the frequency threshold of the bath and (ii) a stable renormalized "particle" state below this threshold. The time evolution of the occupation number for the particle is evaluated exactly using different approaches that yield to complementary insights. The exact solution allows us to investigate the concept of the formation time of a quasiparticle and to study the difference between the relaxation of the distribution of bare particles and that of quasiparticles. For the case of quasiparticles, the exact occupation number asymptotically tends to a statistical equilibrium distribution that differs from a simple Bose-Einstein form as a result of off-shell processes whereas in the stable particle case, the distribution of particles does not thermalize with the bath. We derive a non-Markovian quantum kinetic equation which resums the perturbative series and includes off-shell effects. A Markovian approximation that includes off-shell contributions and the usual Boltzmann equation (energy conserving) are obtained from the quantum kinetic equation in the limit of wide separation of time scales upon different coarse-graining assumptions. The relaxational dynamics predicted by the non-Markovian, Markovian, and Boltzmann approximations are compared to the exact result. The Boltzmann approach is seen to fail in the case of wide resonances and when threshold and renormalization effects are important.

The existence regimes and dynamics of soliton molecules in dispersion-managed (DM) optical fibers have been studied. Initially we develop a variational approximation to describe the periodic dynamics of a soliton molecule within each unit cell of the dispersion map. The obtained system of coupled equations for the pulse width and chirp allows to find the parameters of DM soliton molecules for the given dispersion map and pulse energy. Then by means of a scaling transformation and averaging procedure we reduce the original nonlinear Schrödinger equation (NLSE) with piecewise-constant periodic dispersion to its counterpart with constant coefficients and additional parabolic potential. The obtained averaged NLSE with expulsive potential can explain the essential features of solitons and soliton molecules in DM fibers related to their energy loss during propagation. Also, the model of averaged NLSE predicts the instability of the temporal position of the soliton, which may lead to difficulty in holding the pulse in the middle of its time slot. All numerical simulations are performed using the parameters of the existing DM fiber setup and illustrated via pertinent examples.

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