Baseband modulation instability as the origin of rogue waves

We study the existence and properties of rogue-wave solutions in different nonlinear wave evolution models that are commonly used in optics and hydrodynamics. In particular, we consider the Fokas-Lenells equation, the defocusing vector nonlinear Schrödinger equation, and the long-wave–short-wave resonance equation. We show that rogue-wave solutions in all of these models exist in the subset of parameters where modulation instability is present if and only if the unstable sideband spectrum also contains cw or zero-frequency perturbations as a limiting case (baseband instability). We numerically confirm that rogue waves may only be excited from a weakly perturbed cw whenever the baseband instability is present. Conversely, modulation instability leads to nonlinear periodic oscillations.

Figure 1

Existence domains of rogue waves in the plane $(\kappa ,\omega )$ with $a=1$ in the focusing regime $\left(\sigma =1\right)$ and defocusing regime $\left(\sigma =-1\right)$. The red dotted line denotes $\omega =1/\kappa +1/{\kappa }^3$; the green dashed line denotes $\omega =1/\kappa$; the black solid line denotes $\omega =1/\kappa -1/{\kappa }^3$.

Figure 2

Typical rogue soliton states. Top, focusing regime $\sigma =1,\kappa =0.5$ and $a=1,\omega =0$. Bottom, defocusing regime $\sigma =-1,\kappa =0.5$ and $a=1,\omega =4$.

Figure 3

Maps of logarithmic MI gain $\left(10{log}_{10}G\right)$ in the focusing $\left(\sigma =1\right)$ FLE (1). Top, MI on the $(\Omega ,\kappa )$ plane, calculated for the case of $a=1,\omega =1$. Bottom, MI on the $(\Omega ,\omega )$ plane, calculated for the case of $a=2,\kappa =0.5$.

Figure 4

Maps of logarithmic MI gain $\left(10{log}_{10}G\right)$ in the defocusing $\left(\sigma =-1\right)$ FLE (1). MI on the $(\Omega ,\kappa )$ plane, calculated for the case of $a=2,\omega =1$.

Figure 5

Color plot of ${\left|\psi (\xi ,\tau )\right|}^2$ from the numerical solution of the focusing FLE (1) in the baseband MI regime. The initial condition is a plane wave perturbed by a random noise seed with parameters: $a=2,\omega =1,\kappa =1.15$, and $\sigma =-1$.

Figure 6

Rogue-wave envelope distributions $|{\psi }^{\left(1\right)}(\tau ,\xi )|$ and $|{\psi }^{\left(2\right)}(\tau ,\xi )|$ of expression (6). Here, $a_1=3,a_2=1$, and $\omega =1.k=2.36954+1.1972i$ and $\lambda =-1.69162-1.79721i$.

Figure 7

Rogue-wave existence condition. (a) $(\omega ,a_1)$ plane with $a_2=3$. (b) $(a_2,a_1)$ plane with $\omega =4$.

Figure 8

Maps of MI gain $2G$ of the VNLSE (5). (a) MI on the $(\Omega ,\omega )$ plane, calculated for the cases of $a_1=3,a_2=3$, and ${\omega }_1=-{\omega }_2=\omega$. Dotted (green) curves represent the analytical marginal stability condition $\Omega =2\omega ,{\Omega }^2=max\{4{\omega }^2-8a^2,0\}$. (b) MI on the $(\Omega ,a_1)$ plane, calculated for the cases of $a_2=3,{\omega }_1=-{\omega }_2=4$.

Figure 9

Color plot of (a) $|{\psi }^{\left(1\right)}(\tau ,\xi )|$ and (b) $|{\psi }^{\left(2\right)}(\tau ,\xi )|$ from the numerical solution of the defocusing VNLSE. The initial condition is a plane wave perturbed by weak random noise. Parameters: $a_1=2,a_2=1$, and $\omega =1$. A rogue wave is highlighted by a surrounding box.

Figure 10

Color plot of (a) $|{\psi }^{\left(1\right)}(\tau ,\xi )|$ and (b) $|{\psi }^{\left(2\right)}(\tau ,\xi )|$ from the numerical solution of the defocusing VNLSE. The initial condition is a plane wave perturbed by weak random noise. Parameters: $a_1=2,a_2=1$, and $\omega =3$. No rogue waves are generated in this case.

Figure 11

Rogue-wave envelope distributions $|{\psi }^{\left(S\right)}(\tau ,\xi )|$ and $|{\psi }^{\left(L\right)}(\tau ,\xi )|$ corresponding to expression (8). Here, $a=1,\omega =0$, and $b=0.5$.

Figure 12

Maps of MI gain $2G$ of the LWSW Eq. (8). MI on the $(\Omega ,\omega )$ plane, calculated for the case of $a=1$. Dashed (green) curves represent the analytical marginal stability condition ${\Omega }^6[\frac{1}{16}{\Omega }^6-\frac{1}{2}{\Omega }^4{\omega }^2-\omega (9a^2-{\omega }^3){\Omega }^2+4a^2{\omega }^3-27a^4]=0$.

Figure 13

Color plot of (a) $|{\psi }^{\left(S\right)}(\tau ,\xi )|$ and (b) $|{\psi }^{\left(L\right)}(\tau ,\xi )|$ from the numerical solution of the LWSW equation. The initial condition is a plane wave perturbed by weak seed. Parameters: $a=1,b=0.5$, and $\omega =0$. A rogue wave is highlighted by a surrounding box.

Figure 14

Color plot of (a) $|{\psi }^{\left(S\right)}(\tau ,\xi )|$ and (b) $|{\psi }^{\left(L\right)}(\tau ,\xi )|$ from the numerical solution of the LWSW equation. The initial condition is a plane wave perturbed by weak random noise. Parameters: $a=1,b=0.5$, and $\omega =2.5$. No rogue wave is generated in this case.