Universal Nature of the Nonlinear Stage of Modulational Instability

We characterize the nonlinear stage of modulational instability (MI) by studying the longtime asymptotics of the focusing nonlinear Schrödinger (NLS) equation on the infinite line with initial conditions tending to constant values at infinity. Asymptotically in time, the spatial domain divides into three regions: a far left and a far right field, in which the solution is approximately equal to its initial value, and a central region in which the solution has oscillatory behavior described by slow modulations of the periodic traveling wave solutions of the focusing NLS equation. These results demonstrate that the asymptotic stage of MI is universal since the behavior of a large class of perturbations characterized by a continuous spectrum is described by the same asymptotic state.

Figure 1

(Left panel) The spectral $k$ plane, showing the continuous spectrum $\mathrm{\Sigma }$ (the red lines), the regions where $\mathrm{Im}\lambda >0$ (gray) and $\mathrm{Im}\lambda <0$ (white), and a discrete eigenvalue $k_n$ together with its symmetric counterpart. (Right panel) The asymptotic regime for the $xt$ plane, showing the decomposition into two plane wave regions (white) and the modulated elliptic wave region (gray).

Figure 2

The sign of $\mathrm{Im}\theta$ in the complex $k$ plane for various values of the similarity variable $\xi =x/t$ and $q_o=1$: (i) $\xi =-6$, corresponding to $x<-{\xi }_{*}t$; (ii) $\xi =-5.2$, corresponding to $-{\xi }_{*}t<x<0$; (iii) $\xi =3$, corresponding to $0<x<{\xi }_{*}t$; and (iv) $\xi =6.5$, corresponding to $x>{\xi }_{*}t$. Gray, $\mathrm{Im}\theta >0$; white, $\mathrm{Im}\theta <0$.

Figure 3

(Left panels) The asymptotic solution $|q(x,t)|$ (the vertical axis) in the oscillation region as a function of $\xi =x/t$ (the horizontal axis) for $q_o=1$. Top panel, $t=4$; bottom panel, $t=20$. Also shown (the dashed lines) is the time-independent envelope of the solution. (Right panel) Density plot from numerical simulations of Eq. (1) with a small Gaussian perturbation of the constant background. The red lines show the analytically predicted boundaries $x=\pm 4\sqrt{2}{q}_{o}t$.