Rogue waves for a system of coupled derivative nonlinear Schrödinger equations

Rogue waves (RWs) are unexpectedly strong excitations emerging from an otherwise tranquil background. The nonlinear Schrödinger equation (NLSE), a ubiquitous model with wide applications to fluid mechanics, optics, plasmas, etc., exhibits RWs only in the regime of modulation instability (MI) of the background. For a system of multiple waveguides, the governing coupled NLSEs can produce regimes of MI and RWs, even if each component has dispersion and cubic nonlinearity of opposite signs. A similar effect is demonstrated here for a system of coupled derivative NLSEs (DNLSEs) where the special feature is the nonlinear self-steepening of narrow pulses. More precisely, these additional regimes of MI and RWs for coupled DNLSEs depend on the mismatch in group velocities between the components, and the parameters for cubic nonlinearity and self-steepening. RWs considered in this paper differ from those of the NLSEs in terms of the amplification ratio and criteria of existence. Applications to optics and plasma physics are discussed.

Figure 1

The existence region of a RW in the plane of parameters $S$ [the strength of the defocusing cubic nonlinearity, see Eq. (4)] and δ (group-velocity mismatch) for $\gamma =2$ and $\rho =1$. Above the dashed curve $\left(S=R\right)$ and below the solid line $\left(S={\mathrm{\Gamma }}^2/2\right)$, RWs exist. Between the dashed and the solid boundaries, RWs do not exist.

Figure 2

The existence region of a RW in the plane of $S$ [the strength of the defocusing cubic nonlinearity, see Eq. (4)] and Γ (the strength of the self-steepening nonlinearity) for $\delta =1$ and $\rho =1$. Above the dashed curve $\left(S=R\right)$ and below the solid line $\left(S={\mathrm{\Gamma }}^2/2\right)$, RWs exist. Between the dashed and the solid curves, RWs do not exist.

Figure 3

The spectrum of modulation instability gain for $S=0.1,\mathrm{\Gamma }=1,$ and $\delta =0,0.5,1$.

Figure 4

The spectrum of modulation instability gain for $S=1.1,\mathrm{\Gamma }=1,\mathrm{and}\delta =0.5$ and 1.

Figure 5

An example of rogue waves of the “elevation-elevation” type for $\rho =1,\sigma =0.1,\gamma =1,\delta =0,\mathrm{and}{\mathrm{\Omega }}_0=1+1.26i$, (a) $\left|A\right|$ and (b) $\left|B\right|$ versus $x$ and $t$.

Figure 6

An example of rogue waves of the “elevation-four-petal” type for $\rho =1,\sigma =0.1,\gamma =1,\delta =0.5,\mathrm{and}{\mathrm{\Omega }}_0=1.11+1.23i$, (a) $\left|A\right|$ and (b) $\left|B\right|$ versus $x$ and $t$.

Figure 7

A rogue wave with the same input parameters as those of Fig. 6 except for a different complex frequency ${\mathrm{\Omega }}_0=-1.11+0.255i$, (a) $\left|A\right|$ and (b) $\left|B\right|$ versus $x$ and $t$.

Figure 8

An example of rogue waves in the additional regime of existence due to the coupling, for $\rho =1,\sigma =1.1,\gamma =1,\delta =0.5,\mathrm{and}{\mathrm{\Omega }}_0=-0.542+0.243i$: (a) $\left|A\right|$ and (b) $\left|B\right|$ versus $x$ and $t$.

Figure 9

The growth phase of the rogue wave in the plane of $x$ and $t$ ($t$ from $-4$ to 0) is robust against a sufficiently small perturbation of $1\%$ with parameters the same as those of Fig. 5. Top: Exact solution; bottom: evolution in the presence of a $1\%$ noise.

Figure 10

The growth phase of the rogue wave in the plane of $x$ and $t$ ($t$ from $-4$ to 0) is also robust against a sufficiently small perturbation of $1\%$ with parameters the same as those of Fig. 6. Top: exact solution; bottom: evolution in the presence of a $1\%$ noise.

Figure 11

The growth phase of the rogue wave in the plane of $x$ and $t$ ($t$ from $-3$ to 0) is obscured by the modulation instability of the background if the noise is sufficiently strong ($5\%$ here) with parameters the same as those of Fig. 6. Top: exact solution; bottom: evolution in the presence of a $5\%$ noise.