Localized Nonlinear Waves in Systems with Time- and Space-Modulated Nonlinearities

Using similarity transformations we construct explicit nontrivial solutions of nonlinear Schrödinger equations with potentials and nonlinearities depending both on time and on the spatial coordinates. We present the general theory and use it to calculate explicitly nontrivial solutions such as periodic (breathers), resonant, or quasiperiodically oscillating solitons. Some implications to the field of matter waves are also discussed.

Figure 1

Solution (16), for $n=1$, ${\omega }_0=2$, and $\epsilon =0.5$. Initial data for Eq. (10) are $\nu (0)=\sqrt{2}$ and ${\nu }_t(0)=0$. (a) Pseudocolor plot of $|\psi (x,t){|}^2$ on the domain $x\in [-3,3]$. (b) Width $\nu (t)$ (upper line) and amplitude $\sqrt{\gamma }$ (lower line) vs $t$.

Figure 2

Width (upper line) and amplitude (lower line) of the solution (16) vs $t$ for $n=1$, ${\omega }_0=2$, and $\epsilon =0.5$. Initial data for Eq. (10) are $\nu (0)=\sqrt{2}$ and ${\nu }_t(0)=-6.125\sqrt{2}$.

Figure 3

Plots of $|\psi (x,t){|}^2$ for breathing solutions (16), for $\epsilon =0$ in Eq. (11), and $\gamma (t)$ given by Eq. (18), corresponding to (a) $n=1$, (b) $n=2$. For both cases, $x\in [-6,6]$, $t\in [0,10]$. The initial data for Eq. (10) are $\nu (0)=\sqrt{2}$ and ${\nu }_t(0)=0$.

Figure 4

Solution (16), for $n=1$ and ${\omega }^2=1+0.5\cos (\sqrt{2}t)$. The initial data for (10) are $\nu (0)=2^{3/4}$ and ${\nu }_t(0)=0$. (a) Pseudocolor plot of $|\psi (x,t){|}^2$. (b) Width $\nu (t)$.

Figure 5

Pseudocolor plots of solutions of Eq. (1). (a) Quasiperiodic solution, with $x\in [-3,3]$, $t\in [0,40]$, for $\epsilon =0.6$, $\nu (0)=(2\sqrt{2}{)}^{1/2}$, and ${\nu }_t(0)=0$. (b) Resonant increasing solution, with $x\in [-6,6]$, $t\in [0,40]$ for $\epsilon =0.3$, $\nu (0)=\sqrt{2}$, and ${\nu }_t(0)=0$.