Rogue Waves in the Generalized Derivative Nonlinear Schrödinger Equations

Abstract

General rogue waves are derived for the generalized derivative nonlinear Schrödinger (GDNLS) equations by a bilinear Kadomtsev–Petviashvili (KP) reduction method. These GDNLS equations contain the Kaup–Newell equation, the Chen–Lee–Liu equation and the Gerdjikov–Ivanov equation as special cases. In this bilinear framework, it is shown that rogue waves to all members of these equations are expressed by the same bilinear solution. Compared to previous bilinear KP reduction methods for rogue waves in other integrable equations, an important improvement in our current KP reduction procedure is a new parameterization of internal parameters in rogue waves. Under this new parameterization, the rogue wave expressions through elementary Schur polynomials are much simpler. In addition, the rogue wave with the highest peak amplitude at each order can be obtained by setting all those internal parameters to zero, and this maximum peak amplitude at order N turns out to be $2N+1$ times the background amplitude, independent of the individual GDNLS equation and the background wavenumber. It is also reported that these GDNLS equations can be decomposed into two different bilinear systems which require different KP reductions, but the resulting rogue waves remain the same. Dynamics of rogue waves in the GDNLS equations is also analyzed. It is shown that the wavenumber of the constant background strongly affects the orientation and duration of the rogue wave. In addition, some new rogue patterns are presented.

Appendix: Bilinear derivation of rogue waves in the Kundu–Eckhaus equation

When $a=b$, Eq. (1) becomes the Kundu–Eckhaus equation (Kundu 1984)

$$\begin{aligned} \text {i} \phi _{t} + \phi _{\xi \xi } + \rho |\phi |^2\phi + \text {i}a (|\phi |^2)_{\xi }\phi +\frac{1}{4}a^2|\phi |^4\phi =0. \end{aligned}$$

(90)

Under a gauge transformation

$$\begin{aligned} \phi (\xi ,t)=w(\xi ,t) e^{-\frac{a}{2}\text {i}\int |w(\xi ,t)|^2 d\xi }, \end{aligned}$$

this Kundu–Eckhaus equation reduces to the NLS equation

$$\begin{aligned} \text {i}w_{t} + w_{\xi \xi } + \rho |w|^2w=0, \end{aligned}$$

(91)

whose rogue waves have been derived before (Akhmediev et al. 2009b; Ankiewicz et al. 2010a; Dubard et al. 2010; Kedziora et al. 2011; Guo et al. 2012; Ohta and Yang 2012a; Dubard and Matveev 2013). To directly obtain rogue waves in the Kundu–Eckhaus equation (90) without the use of the above gauge transformation, we can apply a similar bilinear approach as we did for the $a\ne b$ case in the main text of this article. Specifically, through a scaling of $(\phi , \xi , t, a)$ together with a Galilean transformation, we can normalize $\rho =2$ in Eq. (90), and the boundary conditions of its rogue waves can be normalized as

$$\begin{aligned} \phi (\xi , t) \rightarrow e^{\mathrm{i}\left( 2t-\frac{1}{2}a\xi \right) }, \quad (\xi , t) \rightarrow \infty . \end{aligned}$$

(92)

Then, we employ a bilinear variable transformation

$$\begin{aligned} \phi (\xi ,t)=e^{\mathrm{i}\left[ 2t-\frac{1}{2}a[\xi +(\ln f)_\xi ]\right] }\frac{g}{f}, \end{aligned}$$

(93)

where f is a real function, and g a complex function. Under this transformation, the Kundu–Eckhaus equation (90) can be split into the following three bilinear equations,

$$\begin{aligned}&\left( \mathrm{i} D_t+D_\xi ^2\right) g\cdot f=0, \end{aligned}$$

(94)

$$\begin{aligned}&(D_\xi ^2+2)f\cdot f=2|g|^2, \end{aligned}$$

(95)

$$\begin{aligned}&D_\xi D_tf\cdot f=2\mathrm{i}D_\xi g\cdot g^*. \end{aligned}$$

(96)

One can recognize that the first two bilinear equations are the ones for the NLS equation (91) with $\rho =2$ (Ohta and Yang 2012a). It turns out that the (f, g) solutions for rogue waves of the NLS equation also satisfy the third bilinear equation above, and thus, rogue waves for the Kundu–Eckhaus equation (90) are given by (93), where (f, g) are those for the NLS equation (91). The reason for this is that under the same differential and difference relations of $\tau$ functions listed in Eq. (3.7) of Ohta and Yang (2012a), the following three multi-dimensional bilinear equations are satisfied simultaneously,

$$\begin{aligned}&(D_{x_1}D_{x_{-1}}-2)\tau _{n}\cdot \tau _n=-2\tau _{n+1}\tau _{n-1}, \end{aligned}$$

(97)

$$\begin{aligned}&(D_{x_2}-D_{x_1}^2)\tau _{n+1}\cdot \tau _n=0, \end{aligned}$$

(98)

$$\begin{aligned}&D_{x_{-1}}D_{x_2}\tau _{n}\cdot \tau _n=2D_{x_1}\tau _{n-1}\cdot \tau _{n+1}. \end{aligned}$$

(99)

Thus, with the same dimension reduction and complex conjugacy conditions of the NLS equation (Ohta and Yang 2012a), and setting $x_1=\xi$, $x_2=\mathrm{i} t$, these multi-dimensional bilinear equations reduce to (94)–(96), and thus, the (f, g) solutions for rogue waves of the NLS equation (91) are also bilinear solutions for rogue waves of the Kundu–Eckhaus Eq. (90) under the bilinear variable transformation (93).