Variational approach to studying solitary waves in the nonlinear Schrödinger equation with complex potentials

We discuss the behavior of solitary wave solutions of the nonlinear Schrödinger equation (NLSE) as they interact with complex potentials, using a four-parameter variational approximation based on a dissipation functional formulation of the dynamics. We concentrate on spatially periodic potentials with the periods of the real and imaginary part being either the same or different. Our results for the time evolution of the collective coordinates of our variational ansatz are in good agreement with direct numerical simulation of the NLSE. We compare our method with a collective coordinate approach of Kominis and give examples where the two methods give qualitatively different answers. In our variational approach, we are able to give analytic results for the small oscillation frequency of the solitary wave oscillating parameters which agree with the numerical solution of the collective coordinate equations. We also verify that instabilities set in when the slope $dp\left(t\right)/dv\left(t\right)$ becomes negative when plotted parametrically as a function of time, where $p\left(t\right)$ is the momentum of the solitary wave and $v\left(t\right)$ the velocity.

Figure 1

Solitary wave in the trapped case. Comparison of the 4 CC theory (black solid lines) with the 2 CC theory (red dashed lines) and numerical simulation (blue open circles), (a) position $q\left(t\right)$, (b) velocity $v\left(t\right)$, (c) amplitude $\beta \left(t\right)$, (d) momentum $p\left(t\right)$, and (e) relative phase ${\varphi }_{\mathrm{relative}}\left(t\right)$. Here $l_0=1, k_0=1, V_0=-0.01, W_0=-0.01, {\beta }_0=0.5, q_0=1, v_0=0.1, p_0=0.0531649, {\varphi }_0=0$.

Figure 2

Traveling solitary wave. Comparison of the 4 CC theory (black solid lines) with the 2 CC theory (red dashed lines) and numerical simulation (blue open circles), (a) relative position $q_{\mathrm{relative}}\left(t\right)$, (b) velocity $v\left(t\right)$, (c) amplitude $\beta \left(t\right)$, (d) momentum $p\left(t\right)$, and (e) relative phase ${\varphi }_{\mathrm{relative}}\left(t\right)$. Here $l_0=1, k_0=1, V_0=-0.01, W_0=-0.01, {\beta }_0=0.5, q_0=1, v_0=0.2, p_0=0.103165, {\varphi }_0=0$.

Figure 3

Moving solitary wave in the quasiperiodic case, (a) relative position $q_{\mathrm{relative}}\left(t\right)$, (b) velocity $v\left(t\right)$, (c) amplitude $\beta \left(t\right)$, (d) momentum $p\left(t\right)$, and (e) relative phase ${\varphi }_{\mathrm{relative}}\left(t\right)$. Here $l_0=\sqrt{2}, k_0=1, V_0=-0.01, W_0=-0.01, {\beta }_0=0.5, q_0=\pi , v_0=0.05, p_0=0.0243222, {\varphi }_0=0$.

Figure 4

Moving solitary wave in the quasiperiodic case. Left: discrete Fourier transform (DFT) of $v\left(t\right)$. Right: DFT of $\beta \left(t\right).$

Figure 5

Moving solitary wave in the quasiperiodic case when $k_0=1,l_0=\sqrt{2}$. Magnification of the turnarounds in the parametric plot of the momentum $p\left(t\right)$ vs velocity $v\left(t\right)$. The slope $dp/dv$ is negative for short pieces of the curve.

Figure 6

Blowup (increasing amplitude) situation, (a) position $q\left(t\right)$, (b) velocity $v\left(t\right)$, (c) amplitude $\beta \left(t\right)$, (d) momentum $p\left(t\right)$, (e) phase $\varphi \left(t\right)$, and (f) magnification of the turnaround in the parametric plot of the momentum $p\left(t\right)$ vs velocity $v\left(t\right)$. Parameters and initial conditions: $l_0=1/3, k_0=1, V_0=-0.01, W_0=-0.01, {\beta }_0=0.5, q_0=\pi , v_0=-0.1, p_0=-0.0457085, {\varphi }_0=0$.

Figure 7

Shifted potential-trapped case: (a) position $q\left(t\right)$, (b) velocity $v\left(t\right)$, (c) amplitude $\beta \left(t\right)$, (d) momentum $p\left(t\right)$, and (e) phase $\varphi \left(t\right)$. Here $l_0=1, k_0=1, V_0=-0.01, W_0=-0.005, {\beta }_0=0.5, q_0=1.3\pi /3, v_0=0, p_0=0.000608946, {\varphi }_0=0, \mathrm{\Delta }=-\pi /3$.

Figure 8

Trapped solitary wave when $\mathrm{\Delta }=-\pi /3,q_0=1.3\pi$. Parametric plot of $p\left(t\right)$ vs $v\left(t\right)$ for $0<t<120$. Blue curve is our 4 CC approximation.

Figure 9

Moving solitary wave in a shifted potential in a blowup situation: (a) position $q\left(t\right)$, (b) velocity $v\left(t\right)$, (c) amplitude $\beta \left(t\right)$, (d) momentum $p\left(t\right)$, and (e) phase $\varphi \left(t\right)$. Here $l_0=1, k_0=1, V_0=-0.01, W_0=-0.005, {\beta }_0=0.5, q_0=2.8\pi /3, v_0=0, p_0=0.000608946, {\varphi }_0=0, \mathrm{\Delta }=-\pi /3$.

Figure 10

Moving solitary wave when $\mathrm{\Delta }=-\pi /3,q_0=2.8\pi /3$. Parametric plot of the momentum $p\left(t\right)$ vs $v\left(t\right)$.