Nonlinear waves in disordered chains: Probing the limits of chaos and spreading

We probe the limits of nonlinear wave spreading in disordered chains which are known to localize linear waves. We particularly extend recent studies on the regimes of strong and weak chaos during subdiffusive spreading of wave packets [Europhys. Lett. 91, 30001 (2010)] and consider strong disorder, which favors Anderson localization. We probe the limit of infinite disorder strength and study Fröhlich-Spencer-Wayne models. We find that the assumption of chaotic wave packet dynamics and its impact on spreading is in accord with all studied cases. Spreading appears to be asymptotic, without any observable slowing down. We also consider chains with spatially inhomogeneous nonlinearity, which give further support to our findings and conclusions.

Figure 1

DNLS, $W=4$: Evolution of (a) $\langle \log {}_{10}{m}_2(t)\rangle$, (b) $\langle \log {}_{10}P(t)\rangle$, (c) $\langle \zeta (t)\rangle$, (d) finite-difference derivative ${\alpha }_m(t)$ for the smoothed $m_2$ data of panel (a), (e) finite-difference derivative ${\alpha }_P(t)$ for the smoothed $P$ data of panel (b), and (f) $\langle S_V(t)\rangle$ for the spreading of wave packets with initial width $L=21$ and $\beta =0.012,0.04,0.18,0.72,3.6,8.4$ [(bl) black; (m) magenta; (r) red; (b) blue; (g) green; (br) brown]. In panels (a), (b), (d), and (e) straight lines correspond to theoretically predicted power laws $m_2~t^{\alpha }$, $P~t^{\alpha /2}$ with $\alpha =1/3$ (dashed lines) and $\alpha =1/2$ (dash-dotted lines). Error bars in panels (a) and (b) denote representative one-standard-deviation errors.

Figure 2

DNLS, $W=4$: Time evolution of average norm density distributions $\langle z_l\rangle$ in real space for (a) $\beta =0.04$, (b) $\beta =0.72,$ and (c) $\beta =3.6$. The color scales shown on top of panels (a)–(c) are used for coloring each lattice site according to its $\log {}_{10}\langle z_l\rangle$ value.

Figure 3

KG, $W=4$: Evolution of (a) $\langle \log {}_{10}{m}_2(t)\rangle$, (b) $\langle \zeta (t)\rangle$, (c) ${\alpha }_m(t)$, and (d) $\langle H_V(t)\rangle$ vs $\log {}_{10}t$ for the spreading of initially compact wave packets of width $L=21$ with $E=0.01,0.04,0.2,0.75,3$ [(bl) black; (m) magenta; (r) red; (b) blue; (g) green]. In panels (a) and (c) straight lines correspond to the theoretically predicted power laws $m_2~t^{\alpha }$ with $\alpha =1/3$ (dashed lines) and $\alpha =1/2$ (dotted lines). Error bars in panel (a) denote representative one-standard-deviation errors.

Figure 4

DNLS, $W=15$: Evolution of (a) $\langle \log {}_{10}{m}_2(t)\rangle$, (b) $\langle \zeta (t)\rangle$, (c) ${\alpha }_m(t)$, and (d) $\langle S_L(t)\rangle$ vs $\log {}_{10}t$ for the spreading of initially compact wave packets of width $L=10$ with $\beta =0.5,9,30$ [(m) magenta; (g) green; (br) brown]. Mean values are averaged quantities over 100 disorder realizations. In panels (a) and (c) straight lines correspond to theoretically predicted weak chaos behavior $m_2~t^{1/3}$.

Figure 5

DNLS, $W=40$: Evolution of (a) $\langle \log {}_{10}{m}_2(t)\rangle$, (b) $\langle \zeta (t)\rangle$, (c) ${\alpha }_m(t)$, and (d) $\langle S_L(t)\rangle$ vs $\log {}_{10}t$ for the spreading of initially compact wave packets of width $L=10$ with $\beta =1,25,100$ [(m) magenta; (g) green; (br) brown]. Mean values are averaged quantities over 100 disorder realizations. In panels (a) and (c) straight lines correspond to theoretically predicted weak chaos behavior $m_2~t^{1/3}$.

Figure 6

FSW: Evolution of (a) $\langle \log {}_{10}{m}_2(t)\rangle$ [(r) red curve] and $\langle \log {}_{10}P(t)\rangle$ [(b) blue curve], (b) $\langle \zeta (t)\rangle$, (c) ${\alpha }_m(t)$ [(r) red curve] and ${\alpha }_P(t)$ [(b) blue curve], and (d) $\langle H_L(t)\rangle$ vs $\log {}_{10}t$ for initially compact wave packets of width $L=21$ with $E=0.05$.

Figure 7

Evolution of (a) $\langle \log {}_{10}{m}_2(t)\rangle$, (b) $\langle \log {}_{10}P(t)\rangle$, (c) $\langle \zeta (t)\rangle$, (d) ${\alpha }_m(t)$, (e) ${\alpha }_P(t)$, and (f) $\langle H_V(t)\rangle$ vs $\log {}_{10}t$ for the spreading of initially compact wave packets of width $L=21$ with $W=4$ and $E=0.02$ in the KG [(m) magenta curves], the NLN KG [(g) green curves] and the LNL KG [(r) red curves] models. In panels (a), (b), (d), and (e) straight lines correspond to theoretically predicted power laws $m_2~t^{1/3}$, $P~t^{1/6}$ of the weak chaos regime. Error bars in panels (a) and (b) denote representative one-standard-deviation errors.