Generation and nonlinear dynamics of X waves of the Schrödinger equation

The generation of finite-energy packets of X waves is analyzed in normally dispersive cubic media by using an X-wave expansion. The three-dimensional nonlinear Schrödinger model is reduced to a one-dimensional equation with anomalous dispersion. Pulse splitting and beam replenishment as observed in experiments with water and Kerr media are explained in terms of a higher-order breathing soliton. The results presented also hold in periodic media and Bose-condensed gases.

Figure 1

Sketch of spatiotemporal spectra of “slash” and “backslash” X waves.

Figure 2

Sketch of the generated spatiotemporal spectrum (symmetrized for $k_{\perp }<0$) when $\overline{\beta }>0$. The straight lines are slash and backslash spectra.

Figure 3

Normalized energy distribution $W$ vs inverse differential velocity $\beta$ for two different values of pump velocity determined by $\overline{\beta }$. (a) Branch $\beta <0$ for $\overline{\beta }=0.01\Delta {\beta }_P$, (b) branch $\overline{\beta }<\beta <2\Delta {\overline{\beta }}_P$ for $\overline{\beta }=0.01\Delta {\beta }_P$, (c) as in (a) with $\overline{\beta }=0.5\Delta {\beta }_P$, and (d) as in (b) with $\overline{\beta }=0.5\Delta {\beta }_P$. All quantities are dimensionless.

Figure 4

Nonlinearity profile induced by the spatial overlap of the fundamental X wave.

Figure 5

Instantaneous frequency corresponding to X-wave self-overlap in Fig. 4.

Figure 6

(Color online) Typical spatiotemporal profile (at $y=0$) of a breathing X wave. Two isosurfaces are displayed: the darkest corresponds to higher intensity.