Tunneling Mediated by $2\mathrm{D}+1$ Conical Waves in a 1D Lattice

The propagation of $2\mathrm{D}+1$ wave packets in 1D band gap systems shows that the interplay of periodicity and nonlinearity leads to the spontaneous formation of fast and slow conical localized waves. Such nonlinear tunneling has features that differ on the two edges of the band gap and it is characterized by the competition of bullets and nonlinear $X$ waves.

Figure 1

Spatiotemporal dispersion curves $k_z=\omega /v$ of LW in the frequency plane $k_y$, $\omega$. Outside the forbidden band (region between the two parabolas) the LW lightline ($v=1$, blue curve) divides the regions of subluminal LW ($v<1$, green thin curves) and superluminal LW ($v>1$, thick red curves).

Figure 2

Excitation close to LB ($\overline{\omega }=-0.99$). (a) Transmission vs input fluence $E$; (b) Output pulse delay $\tau$ vs $E$. The delay is calculated as the position of the first peak in the transmitted $y$-integrated pulse profile. (c),(d) Output spectrum of $u_{+}$ at (c) low fluence $E=1$, (d) high fluence $E=120$ (inset: zoom around $k_y\cong 1$). The red dashed line denotes the forbidden band, the blue thick line is the LW lightline. (e),(f) corresponding output spatio-temporal profile for (c) $E=1$; (d) $E=120$. The vertical dashed line gives the peak of $|u_{+}{|}^2$ at $E=1$.

Figure 3

(a),(b) As in Fig. 2 for $\overline{\omega }=0.99$ (LB). Output spectra (c) $E=40$, (d) $E=64$, (e) $E=90$ (zoom around LW lightline); Corresponding intensity ($t$, $y$) profiles: (f) $E=40$, (g) $E=64$, (h) $E=90$. The dashed line in (e),(f),(g) gives the reference instant corresponding to peak of $|u_{+}{|}^2$ at $E=40$.

Figure 4

Level plot of spatial intensity $|u_{+}{|}^2+|u_{-}{|}^2$ for UB excitation ($\overline{\omega }=0.99$), at two instants: (a) $t=40$, both the fast conical LW and the GB are trapped in the lattice; (b) $t=46$, the fast LW has left the lattice and only the slow GB remains.