54 research outputs found

### Destruction of Anderson localization in quantum nonlinear Schr\"odinger lattices

The four-wave interaction in quantum nonlinear Schr\"odinger lattices with
disorder is shown to destroy the Anderson localization of waves, giving rise to
unlimited spreading of the nonlinear field to large distances. Moreover, the
process is not thresholded in the quantum domain, contrary to its "classical"
counterpart, and leads to an accelerated spreading of the subdiffusive type,
with the dispersion $\langle(\Delta n)^2\rangle \sim t^{1/2}$ for
$t\rightarrow+\infty$. The results, presented here, shed new light on the
origin of subdiffusion in systems with a broad distribution of relaxation
times.Comment: 4 pages, no figure

### A topological approximation of the nonlinear Anderson model

We study the phenomena of Anderson localization in the presence of nonlinear
interaction on a lattice. A class of nonlinear Schrodinger models with
arbitrary power nonlinearity is analyzed. We conceive the various regimes of
behavior, depending on the topology of resonance-overlap in phase space,
ranging from a fully developed chaos involving Levy flights to pseudochaotic
dynamics at the onset of delocalization. It is demonstrated that quadratic
nonlinearity plays a dynamically very distinguished role in that it is the only
type of power nonlinearity permitting an abrupt localization-delocalization
transition with unlimited spreading already at the delocalization border. We
describe this localization-delocalization transition as a percolation
transition on a Cayley tree. It is found in vicinity of the criticality that
the spreading of the wave field is subdiffusive in the limit
t\rightarrow+\infty. The second moment grows with time as a powerlaw t^\alpha,
with \alpha = 1/3. Also we find for superquadratic nonlinearity that the analog
pseudochaotic regime at the edge of chaos is self-controlling in that it has
feedback on the topology of the structure on which the transport processes
concentrate. Then the system automatically (without tuning of parameters)
develops its percolation point. We classify this type of behavior in terms of
self-organized criticality dynamics in Hilbert space. For subquadratic
nonlinearities, the behavior is shown to be sensitive to details of definition
of the nonlinear term. A transport model is proposed based on modified
nonlinearity, using the idea of stripes propagating the wave process to large
distances. Theoretical investigations, presented here, are the basis for
consistency analysis of the different localization-delocalization patterns in
systems with many coupled degrees of freedom in association with the asymptotic
properties of the transport.Comment: 20 pages, 2 figures; improved text with revisions; accepted for
publication in Physical Review

### Probabilistic approach to a proliferation and migration dichotomy in the tumor cell invasion

The proliferation and migration dichotomy of the tumor cell invasion is
examined within a two-component continuous time random walk (CTRW) model. The
balance equations for the cancer cells of two phenotypes with random switching
between cell proliferation and migration are derived. The transport of tumor
cells is formulated in terms of the CTRW with an arbitrary waiting time
distribution law, while proliferation is modelled by a logistic growth. The
overall rate of tumor cell invasion for normal diffusion and subdiffusion is
determined.Comment: Accepted for publication as a Regular Article in Physical Review

### Finite-velocity diffusion on a comb

A Cattaneo equation for a comb structure is considered. We present a rigorous
analysis of the obtained fractional diffusion equation, and corresponding
solutions for the probability distribution function are obtained in the form of
the Fox $H$-function and its infinite series. The mean square displacement
along the backbone is obtained as well in terms of the infinite series of the
Fox $H$-function. The obtained solutions describe the transition from normal
diffusion to subdiffusion, which results from the comb geometry.Comment: 7 page

### Heterogeneous diffusion in comb and fractal grid structures

We give an exact analytical results for diffusion with a power-law position
dependent diffusion coefficient along the main channel (backbone) on a comb and
grid comb structures. For the mean square displacement along the backbone of
the comb we obtain behavior $\langle x^2(t)\rangle\sim t^{1/(2-\alpha)}$, where
$\alpha$ is the power-law exponent of the position dependent diffusion
coefficient $D(x)\sim |x|^{\alpha}$. Depending on the value of $\alpha$ we
observe different regimes, from anomalous subdiffusion, superdiffusion, and
hyperdiffusion. For the case of the fractal grid we observe the mean square
displacement, which depends on the fractal dimension of the structure of the
backbones, i.e., $\langle x^2(t)\rangle\sim t^{(1+\nu)/(2-\alpha)}$, where
$0<\nu<1$ is the fractal dimension of the backbones structure. The reduced
probability distribution functions for both cases are obtained by help of the
Fox $H$-functions

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