Nonlinear Talbot effect of rogue waves

Akhmediev and Kuznetsov-Ma breathers are rogue wave solutions of the nonlinear Schrödinger equation (NLSE). Talbot effect (TE) is an image recurrence phenomenon in the diffraction of light waves. We report the nonlinear TE of rogue waves in a cubic medium. It is different from the linear TE, in that the wave propagates in a NL medium and is an eigenmode of NLSE. Periodic rogue waves impinging on a NL medium exhibit recurrent behavior, but only at the TE length and at the half-TE length with a $\pi$-phase shift; the fractional TE is absent. The NL TE is the result of the NL interference of the lobes of rogue wave breathers. This interaction is related to the transverse period and intensity of breathers, in that the bigger the period and the higher the intensity, the shorter the TE length.

Figure 1

(a) AB with $q=1/4$. (b) Peregrine soliton. (c) KMB with $q=3/4$. Top panels present the intensity distributions; bottom panels present the view from the above.

Figure 2

Propagation of ${\psi }_0$ (a) and Peregrine soliton (b).

Figure 3

(a) Intensity of the solution in Eq. (6) for $k=1.2$. (b) Periods $D_x$ and $D_z$ versus $k$. Insets show intensities of the solution with $k=3$, 5, 7, and 9, respectively. The dot on the $D_z$ curve corresponds to (a).

Figure 4

(a) NL TE carpet of the AB with $q=1/4$. (b) Intensity profiles at certain distances. (c), (d) Same as (a) and (b) but for the linear TE. (e) Spatial period $D_x$, the maximum intensity of the wave, FWMH of one peak intensity, and NL TE length $z_T$, all as functions of $q$. The green star $*$ corresponds to (a).