Fundamental Peregrine Solitons of Ultrastrong Amplitude Enhancement through Self-Steepening in Vector Nonlinear Systems

We report the universal emergence of anomalous fundamental Peregrine solitons, which can exhibit an unprecedentedly ultrahigh peak amplitude comparable to any higher-order rogue wave events, in the vector derivative nonlinear Schrödinger system involving the self-steepening effect. We present the exact explicit rational solutions on either a continuous-wave or a periodical-wave background, for a broad range of parameters. We numerically confirm the buildup of anomalous Peregrine solitons from strong initial harmonic perturbations, despite the onset of competing modulation instability. Our results may stimulate the experimental study of such Peregrine soliton anomaly in birefringent crystals or other similar vector systems.

Figure 1

Unusual fundamental PSs in the (a),(b) anomalous dispersion ($s=1$) and (c),(d) normal dispersion ($s=-1$) regimes, normalized to their respective backgrounds (letting $\epsilon =1$). The other parameters are given by (a),(b) ${\omega }_1=-1/4$, ${\omega }_2=-11/4$, $a_1=\sqrt{5}/7$, $a_2=3\sqrt{15}/7$; (c),(d) ${\omega }_1=2$, ${\omega }_2=4$, $a_1=2/7$, $a_2=6\sqrt{3}/7$. The specific PS states in the Manakov system ($\epsilon =0$) are plotted in (e) and (f), with $s=1$, ${\omega }_1=1/2$, ${\omega }_2=-1/2$, $a_1=a_2=1$.

Figure 2

Evolutions of $f_1$ and $f_2$ versus $\delta ={\omega }_1-{\omega }_2$ for $\epsilon =1$, $s=-1$, and $\kappa =6$. The green dashed line gives the values of $a_1$ and $a_2$ specified by Eq. (15), which can be read from the vertical axis that the black arrow points to. In each panel, the insets show the PS structures at $\delta =-2$, with the upper and lower plots corresponding to the red solid circle and blue cross, respectively.

Figure 3

Surface (top) and contour (bottom) plots of fundamental PS solutions on the same periodical wave background, with parameters $\epsilon =1$, $s=-1$, ${\omega }_1=2$, ${\omega }_2=4$, $a_1=1/4$, and $a_2=3\sqrt{3}/4$. (a),(b) $(\mu ,\nu )=(-17/8+\sqrt{3}/8,3\sqrt{5}/8-\sqrt{15}/8)$; (c),(d) $(\mu ,\nu )=(-17/8-\sqrt{3}/8,3\sqrt{5}/8+\sqrt{15}/8)$.

Figure 4

Maps of MI gain ${\gamma }_h$ versus $\mathrm{\Omega }$ and $\delta$ for the (a) DNLS and (b) Manakov systems, under otherwise identical parameter condition, i.e., $\kappa =6$, $a_1=2/7$, and $a_2=6\sqrt{3}/7$. The inset in (a) shows the gain profile ${\gamma }_h$ at $\delta =-2$, and the yellow cross indicates the maximum gain ${\gamma }_h^{\max }\approx 1.30$ at $\mathrm{\Omega }\approx 1.87$.

Figure 5

Numerical recurrence of an eightfold-amplitude PS shown in Figs. 1 and 1 from (a),(b) the analytical solutions (5) and (6) at $\xi =-3$; (c),(d) the initial profiles at $\xi =-1$ but perturbed by low-amplitude harmonic waves; and (e),(f) the plane-wave solutions (3) at $\xi =-1$ perturbed by low-amplitude harmonic waves, respectively, under otherwise identical conditions.