Noisy dynamics in long and short Josephson junctions

The study of nonlinear dynamics in long Josephson junctions and the features of a particular kind of junction realized using a graphene layer, are the main topics of this research work. The superconducting state of a Josephson junction is a metastable state, and the switching to the resistive state is directly related to characteristic macroscopic quantities, such as the current the voltage across the junction, and the magnetic field through it. Noise sources can affect the mean lifetime of this superconducting metastable state, so that noise induced effects on the transient dynamics of these systems should be taken into account. The long Josephson junctions are investigated in the sine-Gordon framework, stressing the relations beetwen nonlinear excitations travelling into the medium and switching dynamics towards the resistive state. Nonlinear travelling wave solutions of the sine-Gordon equation are solitons and antisolitons (and their combinations), breathers and plasma waves. The effect of a non-Gaussian noise source is considered, by changing peculiar system parameters, such as the junction length or the frequency and the amplitude of an applied oscillating bias current, and features of the noise sources, such as the amplitude and the statistic of the stochastic signal. Fortran codes are implemented to integrate the nonlinear stochastic differential equations for the order parameter of these systems. Typical noise induced effects, such as the noise enhanced stability and the resonant activation, are evident exploring the mean switching time from the superconducting regime, as a function of the noise amplitude and driving frequency, respectively. Attention is given to the soliton evolution in connection with the escape dynamics from the superconducting metastable state. Moreover, noise induced breathers are detected. Breathers are special solutions of the sine-Gordon equation, composed by a coupled soliton-antisoliton pair, oscillating in an internal frame with a proper frequency. These solutions are highly unstable, and their detection in long Josephon junctions is an open challenge. The possibility to generate only breathers into a junction properly excited is the main focus of the second part of this work. The phenomenon of nonlinear supratransmission in long Josephson junction stimulated by an external signal is analyzed. In correspondence of precise combinations of values of amplitude A and frequency omega of the external sinusoidal pulse, the generation of only breathers emerges. Variations of the pulse durations, both of the applied bias current and of the damping parameter affect the localizations of breathers on a (A,omega) 2D parametric space. The robustness of the generated breathers is tested inserting into the model a thermal noise source to mimic the environmental influence. The last part of this work deals with the characteristics of a Josephson junctions designed suspending over a graphene layer two superconducting electrodes. The resistively and capacitively shunted junction model is used to analyze the dynamics of this system, including the Josephson current affected by the graphene. The mean escape times under the influence of a colored noise source are calculated varying the noise intensity and driving frequency, and setting different values of the mean bias current. Noise enhanced stability characterizes the mean escape times as a function of the noise intensity. Dynamic and stochastic resonant activation effects can be clearly distinguished in different noise amplitude ranges. A complete probability density function analysis shades light on the features and the details of all these noise induced effects. The experimental implications of this work are finally discussed, togheter with its possible future developments.