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Nonlinear classical oscillators can be excited to high energies by a weak driving field provided the drive frequency is properly chirped. This process is known as autoresonance (AR). We find that for a large class of oscillators, it is sufficient to consider only the first harmonic of the motion when studying AR, even when the dynamics is highly nonlinear. The first harmonic approximation is also used to relate AR in an asymmetric potential to AR in a "frequency equivalent" symmetric potential and to study the autoresonance breakdown phenomenon.

We present an analysis of the power fluctuations in the statistical steady state of a passively mode-locked laser. We use statistical light-mode theory to map this problem to that of fluctuations in a reference equilibrium statistical physics problem and, in this way, study the fluctuations nonperturbatively. The power fluctuations, being noncritical, are Gaussian and proportional in amplitude to the inverse square root of the number of degrees of freedom. We calculate explicit analytic expressions for the covariance matrix of the overall pulse and cw power variables, providing complete information on the single-time power distribution in the laser, and derive a set of fluctuation-dissipation relations between them and the susceptibilities of the steady-state quantities.

We study the multipulse formation in passive mode locking in the framework of the statistical light-mode dynamics theory. It is a many-body theory that treats the complex many-mode laser system by statistical mechanics. We give a detailed theory and experimental verification for the important case of multiple-pulse formation in the laser cavity. We follow and extend our former work on the subject. We give a detailed analysis with a rigorous calculation of the partition function, the free energy, and the order parameter in the coarse-graining method within the mean-field theory that is exact in the light-mode system. The outcome is a comprehensive picture of multipulse formation and annihilation, pulse after pulse, in an almost quantized manner, as the noise ("temperature") or the light power is varied. We obtain the phase diagram of the system, showing a series of first-order phase transitions, each belonging to a different number of pulses. We also study the hysteresis behavior, typical for such thermodynamic systems. We elaborate on the role of the saturable absorber structure in determining the multipulse formation. The theoretical results are compared to experimental measurements that we obtained with mode-locked fiber lasers, and we find an excellent agreement.

Let the interface between two immiscible fluids in a Hele-Shaw cell have, at t = 0, a wedge shape. As a wedge is scale-free, the fluid relaxation dynamics are self-similar. We find the dynamic exponent of this self-similar flow and show that the interface shape is given by the solution of an unusual inverse problem of potential theory. We solve this problem analytically for an almost flat wedge, and numerically otherwise. The wedge solution is useful for analysis of pinch-off singularities.

We present a rigorous statistical-mechanics theory of nonlinear many mode laser systems. An important example is the passively mode-locked laser that promotes pulse operation when a saturable absorber is placed in the cavity. It was shown by Phys. Rev. Lett. 89, 103901 (2002)] that pulse formation is a first-order phase transition of spontaneous ordering of modes in an effective "thermodynamic" system, in which intracavity noise level is the effective temperature. In this paper we present a rigorous solution of a model of passive mode locking. We show that the thermodynamics depends on a single parameter, and calculate exactly the mode-locking point. We find the phase diagram and calculate statistical quantities, including the dependence of the intracavity power on the gain saturation function, and finite size corrections near the transition point. We show that the thermodynamics is independent of the gain saturation mechanism and that it is correctly reproduced by a mean field calculation. The outcome is a new solvable statistical mechanics system with an unstable self-interaction accompanied by a natural global power constraint, and an exact description of an important many mode laser system.