Dynamical chaos in nonlinear Schrödinger models with subquadratic power nonlinearity

We devise an analytical method to deal with a class of nonlinear Schrödinger lattices with random potential and subquadratic power nonlinearity. An iteration algorithm is proposed based on the multinomial theorem, using Diophantine equations and a mapping procedure onto a Cayley graph. Based on this algorithm, we are able to obtain several hard results pertaining to asymptotic spreading of the nonlinear field beyond a perturbation theory approach. In particular, we show that the spreading process is subdiffusive and has complex microscopic organization involving both long-time trapping phenomena on finite clusters and long-distance jumps along the lattice consistent with Lévy flights. The origin of the flights is associated with the occurrence of degenerate states in the system; the latter are found to be a characteristic of the subquadratic model. The limit of quadratic power nonlinearity is also discussed and shown to result in a delocalization border, above which the field can spread to long distances on a stochastic process and below which it is Anderson localized similarly to a linear field.

Figure 1

Mapping dynamical Eqs. (a) (13) and (b) (30) onto a Cayley tree with coordination number $z=3$.

Figure 2

Mapping dynamical equations for ${\sigma }_k\left(t\right)$ onto a Cayley tree in the (a) first and (b) second orders assuming $s<1$. The dotted oval on the right-hand side encircles a degenerate state.

Figure 3

Replacing (a) disconnected bonds within a degenerate state by (b) a connected (wave transmitting) bond. The fractal dimension of the shared bond is $d_f=q_{i_1,j_1}+2q_{i_2,j_2}=3s/2$. The bond is conducting in the second order, if $d_f\ge 1$, demanding $s\ge \frac{2}{3}$.

Figure 4

Mapping dynamical equations for ${\sigma }_k\left(t\right)$ onto a Cayley tree in the third order. The wave processes giving rise to a degenerate state are inside the dotted oval structure. The reduction of these processes into a single (shared) bond is illustrated in the top right corner. The fractal dimension of this bond is given by the sum of respective fractional indices, i.e., $d_f=q_{i_1,j_1}+2q_{i_2,j_2}+2q_{i_3,j_3}=5s/3$. The bond is conducting in the third order, if $d_f\ge 1$, demanding $s\ge \frac{3}{5}$.

Figure 5

Nearest-neighbor random walk on a percolating system of dephased oscillators (closed circles). The trajectory of the random walk is represented by a piecewise black curve. The oscillators in a regular state (not available for the random walk) are shown as open circles. It is assumed that dephased oscillators are organized in connected infinite (i.e., percolating) clusters lying on a Cayley graph with coordination number $z=3$. The onset of unlimited transport corresponds to a critical concentration of dephased oscillators, which depends on the coordination number $z$ and is given by $p_c=1/(z-1)$. The dispersion of the random walk corresponds to the scaling $\langle {\left(\mathrm{\Delta }n\right)}^2\left(t\right)\rangle \propto t^{1/3}$ for $t\to +\infty$.