Time-Reversal Generation of Rogue Waves

The formation of extreme localizations in nonlinear dispersive media can be explained and described within the framework of nonlinear evolution equations, such as the nonlinear Schrödinger equation (NLS). Within the class of exact NLS breather solutions on a finite background, which describe the modulational instability of monochromatic wave trains, the hierarchy of rational solutions localized in both time and space is considered to provide appropriate prototypes to model rogue wave dynamics. Here, we use the time-reversal invariance of the NLS to propose and experimentally demonstrate a new approach to constructing strongly nonlinear localized waves focused in both time and space. The potential applications of this time-reversal approach include remote sensing and motivated analogous experimental analysis in other nonlinear dispersive media, such as optics, Bose-Einstein condensates, and plasma, where the wave motion dynamics is governed by the NLS.

Figure 1

(a) First-order doubly localized rational solution (Peregrine breather), which at $X=T=0$ amplifies the amplitude of the carrier by a factor of 3. (b) Second-order doubly localized rational solution (Akhmediev-Peregrine breather), which at $X=T=0$ amplifies the amplitude of the carrier by a factor of 5.

Figure 2

Schematic upper view illustration of the wave flume. The single flap, driven by a hydraulic cylinder, is installed at the right end of the wave flume at Position $x_S$. The wave gauge is placed 9 m from the flap, which location is labeled by Position $x_M$. The gauge is far away from the absorbing beach, displayed at the left end of the flume.

Figure 3

Initial conditions provided by theory Eq. (2) at the position of maximal amplification, i.e., at $X=0$, applied to the wave paddle at Position $x_S$: (a) the Peregrine water wave profile for the carrier parameters $a_0=0.3\mathrm{cm}$ and ${ϵ}_0=0.09$; (b) the Akhmediev-Peregrine water wave profile for the carrier parameters $a_0=0.1\mathrm{cm}$ and ${ϵ}_0=0.03$.

Figure 4

(a) Surface water wave profile of the in amplitude decreased Peregrine breather, measured 9 m from the flap at Position $x_M$. (b) Surface water wave profile of the in amplitude decreased Akhmediev-Peregrine breather, measured 9 m from the flap at Position $x_M$. (c) Time-reversed signal of the time series shown in (a), providing new initial conditions to the flap and reemitted at Position $x_S$. (d) Time-reversed signal of the time series shown in (b), providing new initial conditions to the flap and reemitted at Position $x_S$.

Figure 5

(a) Comparison of the Peregrine surface profile measured 9 m from the paddle at Position $x_M$, starting its evolution from time-reversed initial conditions (blue upper line) with the expected theoretical NLS wave profile at the same position (red bottom line). (b) Comparison of the Akhmediev-Peregrine surface profile measured 9 m from the paddle at Position $x_M$, starting its evolution from time-reversed initial conditions (blue upper line) with the expected theoretical NLS wave profile at the same position (red bottom line).