Discrete nonlinear Schrödinger equations with arbitrarily high-order nonlinearities

A class of discrete nonlinear Schrödinger equations with arbitrarily high-order nonlinearities is introduced. These equations are derived from the same Hamiltonian using different Poisson brackets and include as particular cases the saturable discrete nonlinear Schrödinger equation and the Ablowitz-Ladik equation. As a common property, these equations possess three kinds of exact analytical stationary solutions for which the Peierls-Nabarro barrier is zero. Several properties of these solutions, including stability, discrete breathers, and moving solutions, are investigated.

Figure 1

Lowest nonzero eigenvalues for type III solutions [Eq. (23)] for the (solid line) integrable AL equation $(p=0)$, and for the nonintegrable cases of $p=1$ (dashed line) and $p=2$ (dashed-dotted line). For $p=1$ and $p=2$, the eigenvalues are observed to become negative, indicating instability, for certain values of $\beta$.

Figure 2

(a) Stationary solutions of the GAL equation $(\nu =2)$ for different values of $p$ ranging from $p=1$ (lower curve) to $p=8$ (upper curve). Other parameters are $\beta =0.25$, $m=1$. (b)–(d) The time evolution of the solutions corresponding to the cases $p=1,2,3$, respectively.

Figure 3

Time evolution of the position of the maxima of a two-hump solution of the GAL equation with $p=1$, $m=1$, originated from the initial condition in Eq. (72) with the distance between the two humps increased in steps of 1 from $X_0=7$ (lower slopes) to $X_0=10$ (higher slopes). Other parameters are fixed as $\delta =\pi$, $\beta =1.25$.

Figure 4

(a) Time evolution of two solitons of the GAL equation obtained for $p=1$, $m=1$ and initial condition taken as in Eq. (72) with $X_0=4$. Other parameters are fixed as $\delta =\pi$, $\beta =1.25$. (b) Time dependence of the humps amplitude (modulo square of the maximum of the profiles) depicted in the second panel. The minima correspond to the times at which the amplitude moves to the next lattice site. Parameters are the same as for (a). (c) Time evolution of Eq. (31) with $p=1$, $N=100$ and initial condition taken as in Eq. (72) with $X_0=1$, $m=1$, $\beta =1.25$, and $\delta =\pi$. (d) Same as in (c) but for $p=2$ and $X_0=3$.

Figure 5

Top panel: Time evolution of Eq. (31) with $p=1$, $N=100$ and initial condition taken as in Eq. (72) with $X_0=5$, $m=1$, $\beta =0.15$, and $\delta =0$. Bottom panel: Same as for the upper panel but for $p=2$ and $X_0=8$.