Semiclassical approximation for a nonlinear oscillator with dissipation

An $S$--matrix approach is developed for the chaotic dynamics of a nonlinear oscillator with dissipation. The quantum--classical crossover is studied in the framework of the semiclassical expansion for the $S$--matrix. Analytical expressions for the braking time and the $S$--matrix are obtained

Localization-delocalization transition on a separatrix system of nonlinear Schrodinger equation with disorder

Localization-delocalization transition in a discrete Anderson nonlinear Schr\"odinger equation with disorder is shown to be a critical phenomenon $-$ similar to a percolation transition on a disordered lattice, with the nonlinearity parameter thought as the control parameter. In vicinity of the critical point the spreading of the wave field is subdiffusive in the limit $t\rightarrow+\infty$. The second moment grows with time as a powerlaw $\propto t^\alpha$, with $\alpha$ exactly 1/3. This critical spreading finds its significance in some connection with the general problem of transport along separatrices of dynamical systems with many degrees of freedom and is mathematically related with a description in terms fractional derivative equations. Above the delocalization point, with the criticality effects stepping aside, we find that the transport is subdiffusive with $\alpha = 2/5$ consistently with the results from previous investigations. A threshold for unlimited spreading is calculated exactly by mapping the transport problem on a Cayley tree.Comment: 6 pages, 1 figur

Geometrical enhancement of the electric field: Application of fractional calculus in nanoplasmonics

We developed an analytical approach, for a wave propagation in metal-dielectric nanostructures in the quasi-static limit. This consideration establishes a link between fractional geometry of the nanostructure and fractional integro-differentiation. The method is based on fractional calculus and permits to obtain analytical expressions for the electric field enhancement.Comment: Published in EP