Statistical mechanics of a discrete Schrödinger equation with saturable nonlinearity

We study the statistical mechanics of the one-dimensional discrete nonlinear Schrödinger (DNLS) equation with saturable nonlinearity. Our study represents an extension of earlier work [Phys. Rev. Lett. 84, 3740 (2000)] regarding the statistical mechanics of the one-dimensional DNLS equation with a cubic nonlinearity. As in this earlier study, we identify the spontaneous creation of localized excitations with a discontinuity in the partition function. The fact that this phenomenon is retained in the saturable DNLS is nontrivial, since in contrast to the cubic DNLS whose nonlinear character is enhanced as the excitation amplitude increases, the saturable DNLS, in fact, becomes increasingly linear as the excitation amplitude increases. We explore the nonlinear dynamics of this phenomenon by direct numerical simulations.

Figure 1

Parameter space ($a,h$) [for $\nu =1$], where the shaded area is inaccessible. The thick lines represent $\beta =0$ [Eq. (14)] and $\beta =\infty$ [$h=-2a-\nu \ln (1+a)$] and, thereby, bound the regime in which Gibbsian thermodynamics applies. The points are obtained numerically by applying the transfer integral technique to Eq. (8) using $\beta ={10}^{-8}$. The dashed line represents Eq. (15), which corresponds to the cubic DNLS. Initial conditions for direct (microcanonical) numerical calculations (see Figs. 2, 3, 4) are marked by a square, a circle, an up-pointing triangle, a down-pointing triangle, and a star.

Figure 2

Distributions of $A=|{\psi }_n{|}^2$ for four different energies at $a=1.5$ [for $\nu =1$]. The symbols show data obtained by direct numerical solution. Solid lines show the distribution function obtained by solution (using the transfer integral technique) of Eq. (8), while the thin lines show the function $\exp (-A/a)/a.$ The curves have been shifted vertically to facilitate visualization. All data is for $a=1.5$, while the energies are $h=-2.13$ (squares), $h=-0.97$ (circles), $h=-0.76$ (up-pointing triangles), and $h=0.70$ (down-pointing triangles).

Figure 3

Distributions of $A=|{\psi }_n{|}^2$ for $h=2.09$ at $a=1.5$ [for $\nu =1$]. The distribution is obtained by direct numerical solution of Eq. (2).

Figure 4

Spatiotemporal snapshot of the dynamics in the strongly non-Gibbsian regime for $h=2.09$ at $a=1.5$ [for $\nu =1$].