Spreading for the generalized nonlinear Schrödinger equation with disorder

The dynamics of an initially localized wave packet is studied for the generalized nonlinear Schrödinger equation with a random potential, where the nonlinear term is $\beta {\left|\psi \right|}^p\psi$ and $p$ is arbitrary. Mainly short times for which the numerical calculations can be performed accurately are considered. Long time calculations are presented as well. In particular, the subdiffusive behavior where the average second moment of the wave packet is of the form $⟨m_2⟩\approx t^{\alpha }$ is computed. Contrary to former heuristic arguments, no evidence for any critical behavior as function of $p$ is found. The properties of $\alpha \left(p\right)$ for relatively short times are explored, a scaling property and a maximal value for $p\approx \frac{1}{2}$ are found.

Figure 1

(Color online) (a) $⟨m_2⟩$ for $\beta =1$ and $p=2$ as a function of time. The blue solid curve is the second moment calculated numerically and the green dashed line is the fit which we use in order to find $\alpha$. (b) $\alpha \left(p\right)$ for different values of $\beta$. From top to the bottom: $\beta =4,2,1,0.75,0.5,0.25$ (yellow stars, purple triangles, turquoise asterisks, red or black circles, green squares, and blue diamonds). For $\beta =0.75$ we present two lines: the solid red line and circles are based on $⟨m_2⟩$ and the black circles are based on $⟨\text{ln}{m}_2⟩$. Only the data for $\beta =0.25$, 0.5, 0.75, and 1 where points are connected by lines satisfy $\left(\mathbf{t}\mathbf{1}\right)$ and $\left(\mathbf{t}\mathbf{2}\right)$. The black dashed line is the asymptotic prediction $\alpha =\frac{1}{1+p}$ for long times [12]. The linear case $\left(\beta =0\right)$ is represented by orange solid line for comparison. For all realizations, $W=4$ and maximal localization length is of 6 lattice sites. At the initial time the wave packet populates one site $\left(n=0\right)$.

Figure 2

(Color online) Figure 1b after rescaling by Eq. (4).

Figure 3

(Color online) $m_2\left(t\right)$ for a representative realization as a function of time for $\beta =1$ and $W=4$. From top to bottom: $p=1.5,2,2.5,4,8,0$, (green, red, turquoise, purple, yellow, and blue).