Parametric autoresonant excitation of the nonlinear Schrödinger equation

Parametric excitation of autoresonant solutions of the nonlinear Schrodinger (NLS) equation by a chirped frequency traveling wave is discussed. Fully nonlinear theory of the process is developed based on Whitham's averaged variational principle and its predictions verified in numerical simulations. The weakly nonlinear limit of the theory is used to find the threshold on the amplitude of the driving wave for entering the autoresonant regime. It is shown that above the threshold, a flat (spatially independent) NLS solution can be fully converted into a traveling wave. A simplified, few spatial harmonics expansion approach is also developed for studying this nonlinear mode conversion process, allowing interpretation as autoresonant interaction within triads of spatial harmonics.

Figure 1

(a) The autoresonant growth of the energy $H$ in time (line 1) for $\mu =0.00225$ just above the threshold and the small energy jump after passage through the linear resonance (line 2) for $\mu =0.00223$. The numerical threshold value is ${\mu }_{th}=0.00224$. (b) The evolution of the phase mismatch $\mathrm{\Phi }$ for $\mu =0.00225$; in all cases $\sigma =1$, $\alpha ={10}^{-3}$, $k=6$, and $U_0=1$.

Figure 2

The theoretical (solid lines) and numerical (squares) thresholds ${\mu }_{th}$ vs $k$ for $\sigma =1$ (a) and $\sigma =-1$ (b), $\left|\alpha \right|={10}^{-3}$, and $U_0=1$.

Figure 3

Autoresonant mode conversion at different times for the example above the threshold $(\mu =0.00225)$ in Fig. 1. (a) $t=0$, (b) $t=2000$, and (c) $t=4000$.

Figure 4

Autoresonant dynamics in $\sigma =-1$ case. ${\left|\psi \right|}_{min}$ and ${\left|\psi \right|}_{max}$ vs time. Solid red lines show numerical simulations, circles and squares represent results from the quasi-equilibrium theory. The parameters are $k=3$, $\alpha =-5\times {10}^{-4}$, and $U_0=1$.

Figure 5

Autoresonant dynamics in $\sigma =+1$ case. ${\left|\psi \right|}_{min}$ and ${\left|\psi \right|}_{max}$ versus time. Solid red lines show numerical simulations, circles and squares represent results from the quasi-equilibrium theory. Other parameters are as in Fig. 4.

Figure 6

The amplitudes (a, b) and the relative phases (c, d) of $a_0,a_{-},a_{+}$ harmonics in the $\sigma =1$ example ($U_0=1$, $k=3$, $\mu =0.002$, and $\alpha =-0.0005$). The results of full simulations are shown by red dashed lines.

Figure 7

The amplitudes (a, b) and the relative phases (c, d) of $b_{+},b_{-}$ harmonics in the example in Fig. 6.

Figure 8

The amplitudes (a b) and the relative phases (b, c) of $a_0,a_{+},a_{-}$ harmonics in $\sigma =-1$ example ($U_0=1$, $k=3$, $\mu =0.002$, and $\alpha =-0.0005$). The results of full simulations are shown by red dashed line.