Rogue-wave pattern transition induced by relative frequency

We revisit a rogue wave in a two-mode nonlinear fiber whose dynamics is described by two-component coupled nonlinear Schrödinger equations. The relative frequency between two modes can induce different rogue wave patterns transition. In particular, we find a four-petaled flower structure rogue wave can exist in the two-mode coupled system, which possesses an asymmetric spectrum distribution. Furthermore, spectrum analysis is performed on these different type rogue waves, and the spectrum relations between them are discussed. We demonstrate qualitatively that different modulation instability gain distribution can induce different rogue wave excitation patterns. These results would deepen our understanding of rogue wave dynamics in complex systems.

Figure 1

The process for which an eye-shaped RW transits to be a four-petaled flower structure RW in the component $|{\psi }_2|$ with varying the relative frequency. (a) With $k=0.999$, (b) $k=1.01$, and (c) $k=1.1$. The other parameters are $s_1=1,s_2=1,A_3=3$.

Figure 2

The process for which a four-petaled flower structure RW transits to be an anti-eye-shaped RW in the component ${\psi }_2$ with varying the relative frequency. (a) With $k=1.3$, (b) $k=1.5$ (c) $k=1.8$. The other parameters are $s_1=s_2=1,A_3=3$.

Figure 3

(a) The spectrum evolution $\sqrt{|F_1(\omega ,z)|}$ of the RW in component ${\psi }_1$. (b) The spectrum evolution $\sqrt{|F_2({\omega }^{\prime },z)|}$ (where ${\omega }^{\prime }=\omega +1.1$) of the RW with the four-petaled flower structure in Fig. 1. (c) The spectrum evolution $\sqrt{|F_1(\omega ,z)|}$ of the RW in component ${\psi }_1$. (d) The spectrum evolution $\sqrt{|F_2({\omega }^{\prime },z)|}$ (where ${\omega }^{\prime }=\omega +1.8$) of the RW with anti-eye structure in Fig. 2.

Figure 4

The modulational instability spectrum of the coupled system with relative wave vector $k=0$ (a), $k=1.1$ (b), and $k=1.8$ (c).