Autoresonant excitation of dark solitons

Continuouslyphase-locked (autoresonant) dark solitons of the defocusing nonlinear Schrodinger equation are excited and controlled by driving the system by a slowly chirped wavelike perturbation. The theory of these excitations is developed using Whitham's averaged variational principle and compared with numerical simulations. The problem of the threshold for transition to autoresonance in the driven system is studied in detail, focusing on the regime when the weakly nonlinear frequency shift in the problem differs from the typical quadratic dependence on the wave amplitude. The numerical simulations in this regime show a deviation of the autoresonance threshold on the driving amplitude from the usual 3/4 power dependence on the driving frequency chirp rate. The theory of this effect is suggested.

Figure 1

The effective potential $V_{\mathrm{eff}}\left(U\right)$ at two stages of autoresonant excitation: (dotted line) as the driving frequency passes the linear resonance ${\omega }_d={\omega }_r$ and (thick solid line) after the driving frequency shifts by $\Delta {\omega }_d=1.62$. The horizontal line in the latter case shows the location of the quasienergy $A$ at this stage.

Figure 2

Comparison of the variational theory (dashed lines) with the numerical solution of the NLS equation (solid lines). Line 1: $U_0=2,k=3,\varepsilon =4.85\times {10}^{-5}$, and $\alpha =6\times {10}^{-6}$; line 2: $U_0=1,k=1,\varepsilon =6\times {10}^{-5}$, and $\alpha =6\times {10}^{-6}$.

Figure 3

The amplitude $\left|\psi \right(x\left)\right|$ (solid line) and the phase $\text{arg}\left[\psi \right(x\left)\right]$ (dashed line) of the dark soliton at $t=416$; $U_0=1,k=1,\varepsilon =0.005$, and $\alpha =0.003$.

Figure 4

The depth $min\left|\psi \right|$ of the dark soliton versus time. Line 1: $\varepsilon =0.005$, line 2: $\varepsilon =0.0048$, line 3: soliton depth after switching off the drive at $t=400$. $U_0=1,k=1$, and $\alpha =0.003$. The threshold value is ${\varepsilon }_{th}\approx 0.0049$.

Figure 5

The soliton frequency ${\omega }_k$: solid line 1: $\varepsilon =0.005$, solid line 2: $\varepsilon =0.0048$. The dashed line shows the driving frequency ${\omega }_d$. $U_0=1,k=1$, and $\alpha =0.003$.

Figure 6

The threshold ${\varepsilon }_{th}$ versus $k$. The dashed line–Eq. (32); the solid lines–the numerical solution of the NLS equation: line 1: $\alpha ={10}^{-2}$, line 2: $\alpha =6\times {10}^{-6}$. $U_0=2$.

Figure 7

The amplitude $u\left(t\right)$ versus time. Solid line 1: $U_0=2,k=3,\varepsilon =4.85\times {10}^{-5}$, and $\alpha =6\times {10}^{-6}$. The dashed line 4 shows $u=0.0037t^{1/2}$. Solid line 2: $U_0=2,k=1.1,\varepsilon =1.07\times {10}^{-4}$, and $\alpha =6\times {10}^{-6}$. The dashed line 3 shows $u=0.018t^{1/3}$.

Figure 8

The threshold ${\varepsilon }_{th}\left(\alpha \right)$ versus $\alpha$ in the singular region, $U_0=2,k=1.1$. The dashed line represents Eq. (34); the solid line is from the numerical solution of the NLS equation.