Statistics of wave interactions in nonlinear disordered systems

We study the properties of mode-mode interactions for waves propagating in nonlinear disordered one-dimensional systems. We focus on (i) the localization volume of a mode which defines the number of interacting partner modes, (ii) the overlap integrals which determine the interaction strength, (iii) the average spacing between eigenvalues of interacting modes, which sets a scale for the nonlinearity strength, and (iv) resonance probabilities of interacting modes. Our results are discussed in the light of recent studies on spreading of wave packets in disordered nonlinear systems and are related to the quantum many-body problem in a random chain.

Figure 1

(Color online) Localization length $\xi$ versus normalized eigenvalue for $W=0.5,1,2,4$ (from top to bottom). Inset: zoom for $W=0.5$ around the bandwidth center.

Figure 2

(Color online) Average localization volume $V$ [(r)ed], participation number $P$ [(g)reen)], and localization length $\xi$ [(b)lue] of NMs with eigenvalues near the bandwidth center versus strength of disorder $W$. Dashed lines are estimated asymptotics for $V$ and $\xi$ at small disorder strengths $330/W^2$ and $100/W^2$, respectively.

Figure 3

(Color online) Average integrals $⟨I⟩$ versus strength of disorder $W$ using method I [(r)ed] and method II [(b)lue]. Dashed fitting lines are $3.4\times {10}^{-3}{W}^{3.4}$ (upper line) and $3.84\times {10}^{-5}{W}^{3.4}$ (lower line). Inset: $⟨I⟩$ versus $W$ for $N=40$. For the averaging, 400 disorder realizations were used.

Figure 4

(Color online) PDFs of eigenvalue spacings $s$ for $W=4,10,20$ [(r)ed, (g)reen, and (b)lue]. (o)range curve: PDF for $W=4$ and a short chain with $N=10$. Dashed curve: the Wigner-Dyson distribution with the average spacing $d\approx 0.59$ [see Eq. (18)].

Figure 5

(Color online) Orange (light gray) curve: numerically calculated average spacing $d$ versus strength of disorder $W$ (see Sec. 4). Red (dark gray) curve: width of the spectrum $\Delta$. Dashed curve: the fit of $d$ in the limit of $W\to 0$ by $d=W^2/37$. Dashed-dotted curve: the fit of $d$ in the limit of $W\to \infty$ by $d=W/3-2$.

Figure 6

(Color online) Probability densities $\mathcal{W}\left(R_{\nu ,{\vec{\mu }}_0}\right)$ of resonant NMs in linear-log scale (main figure) and log-log scale (inset). Disorder strength $W=4,6,8,10,15$ [(b)lack, (g)reen, (o)range, (v)iolet, and (r)ed].

Figure 7

(Color online) The constant $C$ as a function of $W$ in linear-logarithmic scale when taking into account contributions from (i) all combinations [(r)ed] (ii) quadruplets [(b)lue], and (iii) triplets [(v)iolet].