# General rogue wave solutions to the discrete nonlinear Schrödinger equation

Yasuhiro Ohta
Department of Mathematics, Kobe University
Rokko, Kobe 657-8501, Japan
Bao-Feng Feng
School of Mathematical and Statistical Sciences
The University of Texas Rio Grande Valley

###### Abstract

In the present paper, we attempt to construct both the general rogue wave
solutions to the fully discrete nonlinear Schrödinger (fd-NLS) equation via
the KP-Toda reduction method. First, we deduce the general breather solution
of the fd-NLS equation starting from a pair of bilinear equations. We then
derive the general rogue wave solution by taking a limit to the breather
solution.

##  1 Introduction

Rogue waves (RWs) or freak waves are spontaneously excited local nonlinear
waves with large amplitudes which appear from nowhere and disappear with no
trace [1]. The simplest form of such waves were firstly discovered Peregrine
in the nonlinear Schrödinger (NLS) equation [2], and their higher order forms
were found 20 years later in [3, 4, 5, 6, 7, 8, 9]. Such extreme wave have
been observed in various different contexts such as oceanography [10],
hydrodynamic [11, 12], Bose-Einstein condensate [13], plasma [14] and
nonlinear optic [11, 15, 16].

Motivated by these physical applications, rogue wave solutions have been found
in many other nonlinear wave equations such as the derivative Schrödinger
(NLS) equation [17, 18, 19, 20, 21], the Manakov system [22, 23, 24], Davey-
Stewartson I and II equation [25, 26], the three-wave equation [27], the
Boussinesq equation [28], the Yajima-Oikawa equation [29, 30],.

On the other hand, the study of rogue waves in discrete integrable systems is
much less. As far as we are aware, only the rogue waves in semi-discrete NLS
equation, or the so-called Ablowitz-Ladik equation have been reported in the
literature [31, 32]. In the present paper, we attempt to construct the rogue
wave solutions to the fully discrete nonlinear Schrödinger (fd-NLS) equation,
which was originally discovered by Ablowitz and Ladik [33, 34] and
rediscovered by Hirota and Ohta through Hirota’s bilinear approach [35]. It
was also discussed in [36, 37]. In a recent paper by Hirose et al., the
integrable discretization of the local induction equation which is gauge
equivalent to the NLS equation was presented in [38].

The remainder of the present paper is organized as follows. In section 2, we
will construct general breather solution to the fd-NLS equation. We derive
general rogue wave solutions to the fd-NLS equation by taking the limit to the
general breather solution in section 3. The paper is concluded in section 4.

##  2 Breather solution for fully discrete NLS equation

Based on the bilinear formulation, we are able to derive the breather solution
of the fully discrete NLS equation, which is given by the following theorem.

######  Theorem 1.

The fully discrete NLS equation,

$\left\\{\begin{array}[]{l}\mathrm{i}(q_{k}^{t+1}-q_{k}^{t})=(q_{k+1}^{t}+q_{k-1}^{t+1})(1+\epsilon|q_{k}^{t}|^{2})\Gamma_{k}^{t},\\\\[5.0pt]
\displaystyle\Gamma_{k+1}^{t}=\frac{1+\epsilon|q_{k}^{t}|^{2}}{1+\epsilon|q_{k}^{t+1}|^{2}}\Gamma_{k}^{t},\end{array}\right.$
(1)

where $\epsilon=\pm 1$, admits the $N$-breather solution,

$q_{k}^{t}=\frac{g_{k}^{t}}{f_{k}^{t}}\frac{1-r}{2\sqrt{|r|}}e^{\mathrm{i}k\theta}\left(\frac{1-c}{1-\bar{c}}\frac{1+r\bar{c}}{1+rc}\right)^{t},\quad\Gamma_{k}^{t}=\frac{4rR}{(1+r)^{2}}\frac{f_{k-1}^{t+1}f_{k}^{t}}{f_{k}^{t+1}f_{k-1}^{t}},$
(2)

where $\bar{\ }$ means complex conjugate, $r$ is a real constant whose sign
coincides with $\epsilon$, $\epsilon=\hbox{sign}\,r$, $c$ is a complex
constant, $R$ and $\theta$ are determined by

$\frac{1}{\mathrm{i}}\frac{1+c}{1-\bar{c}}\frac{1-r\bar{c}}{1+rc}=Re^{\mathrm{i}\theta},$
(3)

and $f_{k}^{t}$ and $g_{k}^{t}$ are given by

$f_{k}^{t}=\tau_{k}^{t}(0),\quad g_{k}^{t}=\tau_{k}^{t}(1),$ (4)

with

$\tau_{k}^{t}(n)=\det_{1\leq i,j\leq N}\left(A_{ij}^{(n)}(k,t)\right),$ (5)
$\displaystyle
A_{ij}^{(n)}=\frac{1}{1-rp_{i}\bar{p}_{j}}\left(a_{i}\varphi_{n}(p_{i})\overline{a_{j}\varphi_{-n}(p_{j})}+b_{i}\varphi_{n}(-p_{i})\overline{b_{j}\varphi_{-n}(-p_{j})}\right)$
$\displaystyle\qquad-\frac{1}{1+rp_{i}\bar{p}_{j}}\left(a_{i}\varphi_{n}(p_{i})\overline{b_{j}\varphi_{-n}(-p_{j})}+b_{i}\varphi_{n}(-p_{i})\overline{a_{j}\varphi_{-n}(p_{j})}\right),$
(6)
$\varphi_{n}(p)=\left(\frac{1-rp}{1+p}\right)^{n}\left(\frac{1+p}{1+rp}\right)^{k}\left(\frac{1+p}{1+rp}\frac{1+p/\bar{c}}{1+rcp}\right)^{t},$
(7)

where $p_{i}$, $a_{i}$, $b_{i}$ are complex constants.

###### Proof.

By using the Gramian technique, we can directly prove that the $\tau$ function
of discrete KP/Toda hierarchy,

$\tau_{n}(k,K,l,L)=\det_{1\leq i,j\leq N}\left(m_{ij}^{(n)}(k,K,l,L)\right),$

satisfies the discrete bilinear equations,

$\left\\{\begin{array}[]{l}(1-aA)\tau_{n}(k+1,K+1,l,L)\tau_{n}(k,K,l,L)\\\\[5.0pt]
\quad-\tau_{n}(k+1,K,l,L)\tau_{n}(k,K+1,l,L)\\\\[5.0pt]
\quad+aA\tau_{n+1}(k+1,K,l,L)\tau_{n-1}(k,K+1,l,L)=0,\\\\[5.0pt]
A(a-b)(1-aB)\tau_{n+1}(k+1,K,l+1,L+1)\tau_{n}(k,K+1,l,L)\\\\[5.0pt]
\quad-a(A-B)(1-Ab)\tau_{n+1}(k+1,K,l,L)\tau_{n}(k,K+1,l+1,L+1)\\\\[5.0pt]
\quad+Ab(1-aA)\tau_{n+1}(k,K,l+1,L+1)\tau_{n}(k+1,K+1,l,L)\\\\[5.0pt] \quad-
aB(1-aA)\tau_{n+1}(k+1,K+1,l,L)\tau_{n}(k,K,l+1,L+1)=0,\end{array}\right.$ (8)

if the matrix element $m_{ij}^{(n)}$ satisfies

$\displaystyle
m_{ij}^{(n+1)}(k,K,l,L)-m_{ij}^{(n)}(k,K,l,L)=(-1)^{n+1}\phi_{i}^{(n)}(k,K,l,L)\psi_{j}^{(-n-1)}(k,K,l,L),$
$\displaystyle
m_{ij}^{(n)}(k+1,K,l,L)-m_{ij}^{(n)}(k,K,l,L)=(-1)^{n}a\phi_{i}^{(n)}(k+1,K,l,L)\psi_{j}^{(-n)}(k,K,l,L),$
$\displaystyle
m_{ij}^{(n)}(k,K+1,l,L)-m_{ij}^{(n)}(k,K,l,L)=(-1)^{n}A\phi_{i}^{(n-1)}(k,K+1,l,L)\psi_{j}^{(-n-1)}(k,K,l,L),$
$\displaystyle
m_{ij}^{(n)}(k,K,l+1,L)-m_{ij}^{(n)}(k,K,l,L)=(-1)^{n}b\phi_{i}^{(n)}(k,K,l+1,L)\psi_{j}^{(-n)}(k,K,l,L),$
$\displaystyle
m_{ij}^{(n)}(k,K,l,L+1)-m_{ij}^{(n)}(k,K,l,L)=(-1)^{n}B\phi_{i}^{(n-1)}(k,K,l,L+1)\psi_{j}^{(-n-1)}(k,K,l,L),$

where $\phi_{i}^{(n)}$ and $\psi_{j}^{(n)}$ are arbitrary functions satisfying
the linear dispersion relations,

$\displaystyle\phi_{i}^{(n)}(k,K,l,L)-\phi_{i}^{(n)}(k-1,K,l,L)=a\phi_{i}^{(n+1)}(k,K,l,L),$
$\displaystyle\phi_{i}^{(n)}(k,K,l,L)-\phi_{i}^{(n)}(k,K-1,l,L)=A\phi_{i}^{(n-1)}(k,K,l,L),$
$\displaystyle\phi_{i}^{(n)}(k,K,l,L)-\phi_{i}^{(n)}(k,K,l-1,L)=b\phi_{i}^{(n+1)}(k,K,l,L),$
$\displaystyle\phi_{i}^{(n)}(k,K,l,L)-\phi_{i}^{(n)}(k,K,l,L-1)=B\phi_{i}^{(n-1)}(k,K,l,L),$
$\displaystyle\psi_{j}^{(n)}(k+1,K,l,L)-\psi_{j}^{(n)}(k,K,l,L)=a\psi_{j}^{(n+1)}(k,K,l,L),$
$\displaystyle\psi_{j}^{(n)}(k,K+1,l,L)-\psi_{j}^{(n)}(k,K,l,L)=A\psi_{j}^{(n-1)}(k,K,l,L),$
$\displaystyle\psi_{j}^{(n)}(k,K,l+1,L)-\psi_{j}^{(n)}(k,K,l,L)=b\psi_{j}^{(n+1)}(k,K,l,L),$
$\displaystyle\psi_{j}^{(n)}(k,K,l,L+1)-\psi_{j}^{(n)}(k,K,l,L)=B\psi_{j}^{(n-1)}(k,K,l,L),$

where $k$, $K$, $l$, $L$ are discrete independent variables and $a$, $A$, $b$,
$B$ are difference intervals. In order to derive the breather solution, we
assume

$\displaystyle m_{ij}^{(n)}(k,K,l,L)$
$\displaystyle\quad=\sum_{\nu=1}^{2}\sum_{\mu=1}^{2}\frac{a_{i\nu}b_{j\mu}}{p_{i\nu}+q_{j\mu}}\left(-\frac{p_{i\nu}}{q_{j\mu}}\right)^{n}\left(\frac{1+aq_{j\mu}}{1-ap_{i\nu}}\right)^{k}\left(\frac{1+A/q_{j\mu}}{1-A/p_{i\nu}}\right)^{K}\left(\frac{1+bq_{j\mu}}{1-bp_{i\nu}}\right)^{l}\left(\frac{1+B/q_{j\mu}}{1-B/p_{i\nu}}\right)^{L},$
$\displaystyle\phi_{i}^{(n)}(k,K,l,L)=\sum_{\nu=1}^{2}a_{i\nu}p_{i\nu}^{n}(1-ap_{i\nu})^{-k}(1-A/p_{i\nu})^{-K}(1-bp_{i\nu})^{-l}(1-B/p_{i\nu})^{-L},$
$\displaystyle\psi_{j}^{(n)}(k,K,l,L)=\sum_{\mu=1}^{2}b_{j\mu}q_{j\mu}^{n}(1+aq_{j\mu})^{k}(1+A/q_{j\mu})^{K}(1+bq_{j\mu})^{l}(1+B/q_{j\mu})^{L},$

where $p_{i\nu}$, $q_{j\mu}$, $a_{i\nu}$, $b_{j\mu}$ are arbitrary constants.
Obviously, the $\phi_{i}^{(n)}$ and $\psi_{j}^{(n)}$ defined satisfy the above
linear dispersion relations, therefore the determinant with above defined
element $m_{ij}^{(n)}(k,K,l,L)$ satisfy the bilinear equations (8).

Next, we proceed to the reductions. By imposing

$p_{i2}=-\frac{A}{a}\frac{1-ap_{i1}}{p_{i1}-A},\quad
q_{j2}=-\frac{A}{a}\frac{1+aq_{j1}}{q_{j1}+A},$

we can show that $\tau_{n}$ satisfies the following reduction condition,

$\tau_{n}(k+1,K-1,l,L)=\tau_{n}(k,K,l,L)\prod_{i=1}^{N}\frac{q_{i1}q_{i2}}{p_{i1}p_{i2}},$

it follows that the bilinear equations (8) are reduced to

$\left\\{\begin{array}[]{l}(1-aA)\sigma_{k+1}^{t}(n)\sigma_{k-1}^{t}(n)-\sigma_{k}^{t}(n)\sigma_{k}^{t}(n)+aA\sigma_{k}^{t}(n+1)\sigma_{k}^{t}(n-1)=0,\\\\[5.0pt]
A(a-b)(1-aB)\sigma_{k}^{t+1}(n+1)\sigma_{k}^{t}(n)-a(A-B)(1-Ab)\sigma_{k}^{t}(n+1)\sigma_{k}^{t+1}(n)\\\\[5.0pt]
\quad+Ab(1-aA)\sigma_{k-1}^{t+1}(n+1)\sigma_{k+1}^{t}(n)-aB(1-aA)\sigma_{k+1}^{t}(n+1)\sigma_{k-1}^{t+1}(n)=0,\end{array}\right.$
(9)

by taking $\sigma_{k}^{t}(n)=\tau_{n}(k,0,t,t)$. Furthermore by taking

$A=-\epsilon\bar{a},\quad B=-\epsilon\bar{b},\quad
q_{j1}=\frac{\bar{a}}{a}\frac{\bar{p}_{j1}+\epsilon
a}{1-\bar{a}\bar{p}_{j1}},\quad
b_{j1}=\frac{\bar{a}_{j1}}{a(1-\bar{a}\bar{p}_{j1})},\quad
b_{j2}=\frac{\bar{a}_{j2}}{a(1-\bar{a}\bar{p}_{j2})},$

where $\epsilon=+1$ or $-1$, $\sigma_{k}^{t}(n)$ satisfies the complex
conjugate condition,

$\sigma_{k}^{t}(-n)G=\overline{\sigma_{k}^{t}(n)G},$

with a gauge factor $G$ which is given below in eq.(11). In order to simplify
the final expression, let us parametrise by

$|a|^{2}=\epsilon\frac{r-2+1/r}{4},\quad
b=\frac{2a}{1-r}\frac{1-rc}{1+c},\quad
p_{i1}=\frac{1-1/r}{2a}\frac{1-rp_{i}}{1+p_{i}},$
$a_{i1}=\epsilon\frac{r-1/r}{4}a_{i},\quad a_{i2}=-a(\bar{a}+\epsilon
p_{i2})b_{i},\quad$

where $r$ is a real constant, $c$ is a complex constant, $p_{i}$ represents
the complex wave number of the $i$th breather, $a_{i}$ and $b_{i}$ stand for
the complex phase constants of the $i$th breather. We comment that $\epsilon$
is equal to the sign of $r$ because of the positivity of $|a|^{2}$. Then we
have

$\displaystyle\tilde{m}_{ij}^{n}(k,K,l,L)$
$\displaystyle:=m_{ij}^{n}(k,K,l,L)\frac{-4}{(1-r)(1+p_{i})(1+\bar{p}_{j})}\left(-r\frac{1-\bar{p}_{j}^{2}}{1-r^{2}\bar{p}_{j}^{2}}\right)^{n}r^{K-k}(rc)^{-l}\left(r\bar{c}\frac{1-p_{i}^{2}/\bar{c}^{2}}{1-r^{2}p_{i}^{2}}\frac{1-\bar{p}_{j}^{2}}{1-r^{2}\bar{c}^{2}\bar{p}_{j}^{2}}\right)^{L}$
$\displaystyle=\frac{a_{i}\bar{a}_{j}}{1-rp_{i}\bar{p}_{j}}\left(\frac{1-rp_{i}}{1+p_{i}}\frac{1+\bar{p}_{j}}{1-r\bar{p}_{j}}\right)^{n}\left(\frac{1+p_{i}}{1+rp_{i}}\frac{1+\bar{p}_{j}}{1+r\bar{p}_{j}}\right)^{k}\left(\frac{1-rp_{i}}{1-p_{i}}\frac{1-r\bar{p}_{j}}{1-\bar{p}_{j}}\right)^{K}$
$\displaystyle\hskip
50.0pt\times\left(\frac{1+p_{i}}{1+rcp_{i}}\frac{1+\bar{p}_{j}/c}{1+r\bar{p}_{j}}\right)^{l}\left(\frac{1+p_{i}/\bar{c}}{1+rp_{i}}\frac{1+\bar{p}_{j}}{1+r\bar{c}\bar{p}_{j}}\right)^{L}$
$\displaystyle-\frac{a_{i}\bar{b}_{j}}{1+rp_{i}\bar{p}_{j}}\left(\frac{1-rp_{i}}{1+p_{i}}\frac{1-\bar{p}_{j}}{1+r\bar{p}_{j}}\right)^{n}\left(\frac{1+p_{i}}{1+rp_{i}}\frac{1-\bar{p}_{j}}{1-r\bar{p}_{j}}\right)^{k}\left(\frac{1-rp_{i}}{1-p_{i}}\frac{1+r\bar{p}_{j}}{1+\bar{p}_{j}}\right)^{K}$
$\displaystyle\hskip
50.0pt\times\left(\frac{1+p_{i}}{1+rcp_{i}}\frac{1-\bar{p}_{j}/c}{1-r\bar{p}_{j}}\right)^{l}\left(\frac{1+p_{i}/\bar{c}}{1+rp_{i}}\frac{1-\bar{p}_{j}}{1-r\bar{c}\bar{p}_{j}}\right)^{L}$
$\displaystyle-\frac{b_{i}\bar{a}_{j}}{1+rp_{i}\bar{p}_{j}}\left(\frac{1+rp_{i}}{1-p_{i}}\frac{1+\bar{p}_{j}}{1-r\bar{p}_{j}}\right)^{n}\left(\frac{1-p_{i}}{1-rp_{i}}\frac{1+\bar{p}_{j}}{1+r\bar{p}_{j}}\right)^{k}\left(\frac{1+rp_{i}}{1+p_{i}}\frac{1-r\bar{p}_{j}}{1-\bar{p}_{j}}\right)^{K}$
$\displaystyle\hskip
50.0pt\times\left(\frac{1-p_{i}}{1-rcp_{i}}\frac{1+\bar{p}_{j}/c}{1+r\bar{p}_{j}}\right)^{l}\left(\frac{1-p_{i}/\bar{c}}{1-rp_{i}}\frac{1+\bar{p}_{j}}{1+r\bar{c}\bar{p}_{j}}\right)^{L}$
$\displaystyle+\frac{b_{i}\bar{b}_{j}}{1-rp_{i}\bar{p}_{j}}\left(\frac{1+rp_{i}}{1-p_{i}}\frac{1-\bar{p}_{j}}{1+r\bar{p}_{j}}\right)^{n}\left(\frac{1-p_{i}}{1-rp_{i}}\frac{1-\bar{p}_{j}}{1-r\bar{p}_{j}}\right)^{k}\left(\frac{1+rp_{i}}{1+p_{i}}\frac{1+r\bar{p}_{j}}{1+\bar{p}_{j}}\right)^{K}$
$\displaystyle\hskip
50.0pt\times\left(\frac{1-p_{i}}{1-rcp_{i}}\frac{1-\bar{p}_{j}/c}{1-r\bar{p}_{j}}\right)^{l}\left(\frac{1-p_{i}/\bar{c}}{1-rp_{i}}\frac{1-\bar{p}_{j}}{1-r\bar{c}\bar{p}_{j}}\right)^{L},$

and the bilinear equations (9) are reduced to

$\left\\{\begin{array}[]{l}(1+r)^{2}\tau_{k+1}^{t}(n)\tau_{k-1}^{t}(n)-4r\tau_{k}^{t}(n)\tau_{k}^{t}(n)-(1-r)^{2}\tau_{k}^{t}(n+1)\tau_{k}^{t}(n-1)=0,\\\\[5.0pt]
(1-c)(1+r\bar{c})\tau_{k}^{t+1}(n+1)\tau_{k}^{t}(n)-(1-\bar{c})(1+rc)\tau_{k}^{t}(n+1)\tau_{k}^{t+1}(n)\\\\[5.0pt]
\quad-(1+\bar{c})(1-rc)\tau_{k-1}^{t+1}(n+1)\tau_{k+1}^{t}(n)+(1+c)(1-r\bar{c})\tau_{k+1}^{t}(n+1)\tau_{k-1}^{t+1}(n)=0,\end{array}\right.$
(10)

where $\tau_{k}^{t}(n)$ is given by

$\tau_{k}^{t}(n)=\det_{1\leq i,j\leq
N}\left(\tilde{m}_{ij}^{n}(k,0,t,t)\right)=\sigma_{k}^{t}(n)G,$
$G=\prod_{i=1}^{N}\frac{-4}{(1-r)(1+p_{i})(1+\bar{p}_{i})}\left(-r\frac{1-\bar{p}_{i}^{2}}{1-r^{2}\bar{p}_{i}^{2}}\right)^{n}r^{-k}\left(\frac{\bar{c}}{c}\frac{1-p_{i}^{2}/\bar{c}^{2}}{1-r^{2}p_{i}^{2}}\frac{1-\bar{p}_{i}^{2}}{1-r^{2}\bar{c}^{2}\bar{p}_{i}^{2}}\right)^{t},$
(11)

which is nothing but $\tau_{k}^{t}(n)$ in Theorem 1. For
$f_{k}^{t}=\tau_{k}^{t}(0)$, $g_{k}^{t}=\tau_{k}^{t}(1)$,
$\bar{g}_{k}^{t}=\tau_{k}^{t}(-1)$, the bilinear equations are written as

$\left\\{\begin{array}[]{l}(1+r)^{2}f_{k+1}^{t}f_{k-1}^{t}-4rf_{k}^{t}f_{k}^{t}-(1-r)^{2}g_{k}^{t}\bar{g}_{k}^{t}=0,\\\\[5.0pt]
(1-c)(1+r\bar{c})g_{k}^{t+1}f_{k}^{t}-(1-\bar{c})(1+rc)g_{k}^{t}f_{k}^{t+1}\\\\[5.0pt]
\quad-(1+\bar{c})(1-rc)g_{k-1}^{t+1}f_{k+1}^{t}+(1+c)(1-r\bar{c})g_{k+1}^{t}f_{k-1}^{t+1}=0,\end{array}\right.$

from which the fully discrete NLS eq.(1) is straightforwardly derived through
the variable transformation (2). This completes the proof of Theorem 1. ∎

In the case of $N=1$, we have the 1-breather solution,

$\displaystyle\tau_{k}^{t}(n)=\frac{1}{1-r|p|^{2}}\left(|a|^{2}\varphi_{n}(p)\overline{\varphi_{-n}(p)}+|b|^{2}\varphi_{n}(-p)\overline{\varphi_{-n}(-p)}\right)$
$\displaystyle\qquad-\frac{1}{1+r|p|^{2}}\left(a\bar{b}\varphi_{n}(p)\overline{\varphi_{-n}(-p)}+\bar{a}b\varphi_{n}(-p)\overline{\varphi_{-n}(p)}\right),$

with (2)-(7) where $p$, $a$, $b$ are complex constants (the index $1$ of
$p_{1}$, $a_{1}$, $b_{1}$ are omitted for notational simplicity). The
Akhmediev breather which is the breather solution localized in time $t$ can be
derived by taking the wave number $p$ pure imaginary. If we require the
regularity of the solution only on the lattice points, i.e., $f_{k}^{t}\neq 0$
for integers $k$ and $t$, then $r$ can be either positive or negative and the
discrete NLS equations (1) of both focusing type and defocusing type admit
regular breather solutions by locating the singularities (if exist) off the
lattice points. On the other hand if we require the solution to be regular in
whole real space, i.e., $f_{k}^{t}$ is non-zero for all real numbers $k$ and
$t$, then $r$ has to be positive and only the focusing discrete NLS admits the
regular breathers for generic parameters. Some exceptional regular solutions
for $r<0$ can be derived from the above $\tau_{k}^{t}(n)$ but usually they are
not called breathers and we don’t discuss about such solutions.

Figure 1: One-breather solution with parameters $r=2.0$, $c=3+2\rm{i}$,
$p=0.5\rm{i}$

##  3 Rogue wave solution for fully discrete NLS equation

The rogue wave solution of rational function type can be derived as a limit of
the breather solution. In Theorem 1, we take

$a_{i}=\frac{1}{2}\Big{(}1+\sum_{\nu=1}^{i}c_{\nu}p_{i}^{2\nu-1}\Big{)},\quad
b_{i}=\frac{1}{2}\Big{(}1-\sum_{\nu=1}^{i}c_{\nu}p_{i}^{2\nu-1}\Big{)},$

scale the $\tau$ function by
$\displaystyle\tau_{k}^{t}(n)/\prod_{i=1}^{N}(p_{i}\bar{p}_{i})^{2i-1}$ and
finally take the limit $p_{i}\to 0$ successively for $i=1,2,\ldots,N$. Then
the leading order of $\tau_{k}^{t}(n)$ in $p_{i}$’s gives a polynomial of $k$
and $t$ which turns to be the rogue wave solution. This result is summarized
in the following theorem.

######  Theorem 2.

The $N$th order rogue wave solution for the fully discrete NLS equation (1) is
given by

$\tau_{k}^{t}(n)=\det_{1\leq i,j\leq N}\left(B_{ij}^{(n)}(k,t)\right),\qquad
B_{ij}^{(n)}=\sum_{\nu=1}^{2\min(i,j)}r^{\nu-1}\Phi_{2i-\nu}^{(n)}\overline{\Phi_{2j-\nu}^{(-n)}},$
$\Phi_{i}^{(n)}=S_{i}(x(n))+\sum_{\mu=1}^{\left[\frac{i+1}{2}\right]}c_{\mu}S_{i+1-2\mu}(x(n)),\qquad
x(n)=(x_{1}(n),x_{2}(n),\ldots,x_{h}(n),\ldots),$
$x_{h}(n)=\frac{(-1)^{h}}{h}\left((1-(-r)^{h})n-(1-r^{h})k-(1-r^{h}+(1/\bar{c})^{h}-(rc)^{h})t\right),\quad\hbox{for
}h=1,2,\ldots,$

with the variable transformations (2)-(4), where $[\ ]$ means the Gauss symbol
and $S_{\mu}(x)$ is the so-called elementary Schur function defined by
$\displaystyle\sum_{\mu=0}^{\infty}S_{\mu}(x)\lambda^{\mu}=\exp\sum_{h=1}^{\infty}x_{h}\lambda^{h}$.
This solution has $N$ complex parameters $c_{\mu}$, $\mu=1,2,\ldots,N$.

###### Proof.

Firstly $\varphi_{n}(p)$ in (7) is written as
$\displaystyle\varphi_{n}(p)=\sum_{\mu=0}^{\infty}S_{\mu}(x(n))p^{\mu}$. We
rewrite $A_{ij}^{(n)}$ in Theorem 1 as

$\displaystyle
A_{ij}^{(n)}=\frac{1}{1-(rp_{i}\bar{p}_{j})^{2}}\Big{(}a_{i}\varphi_{n}(p_{i})-b_{i}\varphi_{n}(-p_{i})\Big{)}\Big{(}\overline{a_{j}\varphi_{-n}(p_{j})-b_{j}\varphi_{-n}(-p_{j})}\Big{)}$
$\displaystyle\qquad+\frac{rp_{i}\bar{p}_{j}}{1-(rp_{i}\bar{p}_{j})^{2}}\Big{(}a_{i}\varphi_{n}(p_{i})+b_{i}\varphi_{n}(-p_{i})\Big{)}\Big{(}\overline{a_{j}\varphi_{-n}(p_{j})+b_{j}\varphi_{-n}(-p_{j})}\Big{)}.$

Denoting $a_{i}-b_{i}=p_{i}d_{i}$ and $a_{i}+b_{i}=s_{i}$, the four factors in
the above expression are written as

$\displaystyle
a_{i}\varphi_{n}(p_{i})-b_{i}\varphi_{n}(-p_{i})=\sum_{\mu=0}^{\infty}S_{\mu}(x(n))p_{i}^{\mu}(a_{i}-(-1)^{\mu}b_{i})$
$\displaystyle\qquad=(p_{i}d_{i},p_{i}s_{i},p_{i}^{3}d_{i},p_{i}^{3}s_{i},\cdots)\,{}^{t}(S_{0}(x(n)),S_{1}(x(n)),S_{2}(x(n)),S_{3}(x(n)),\cdots),$
$\displaystyle\overline{a_{j}\varphi_{-n}(p_{j})-b_{j}\varphi_{-n}(-p_{j})}=\sum_{\mu=0}^{\infty}\overline{S_{\mu}(x(-n))p_{j}^{\mu}(a_{j}-(-1)^{\mu}b_{j})}$
$\displaystyle\qquad=(S_{0}(\overline{x(-n)}),S_{1}(\overline{x(-n)}),S_{2}(\overline{x(-n)}),S_{3}(\overline{x(-n)}),\cdots)\,{}^{t}(\bar{p}_{j}\bar{d}_{j},\bar{p}_{j}\bar{s}_{j},\bar{p}_{j}^{3}\bar{d}_{j},\bar{p}_{j}^{3}\bar{s}_{j},\cdots),$
$\displaystyle
a_{i}\varphi_{n}(p_{i})+b_{i}\varphi_{n}(-p_{i})=\sum_{\mu=0}^{\infty}S_{\mu}(x(n))p_{i}^{\mu}(a_{i}+(-1)^{\mu}b_{i})$
$\displaystyle\qquad=(s_{i},p_{i}^{2}d_{i},p_{i}^{2}s_{i},p_{i}^{4}d_{i},\cdots)\,{}^{t}(S_{0}(x(n)),S_{1}(x(n)),S_{2}(x(n)),S_{3}(x(n)),\cdots),$
$\displaystyle\overline{a_{j}\varphi_{-n}(p_{j})+b_{j}\varphi_{-n}(-p_{j})}=\sum_{\mu=0}^{\infty}\overline{S_{\mu}(x(-n))p_{j}^{\mu}(a_{j}+(-1)^{\mu}b_{j})}$
$\displaystyle\qquad=(S_{0}(\overline{x(-n)}),S_{1}(\overline{x(-n)}),S_{2}(\overline{x(-n)}),S_{3}(\overline{x(-n)}),\cdots)\,{}^{t}(\bar{s}_{j},\bar{p}_{j}^{2}\bar{d}_{j},\bar{p}_{j}^{2}\bar{s}_{j},\bar{p}_{j}^{4}\bar{d}_{j},\cdots),$

where ${}^{t}v$ means transpose of $v$. Thus we obtain

$\displaystyle
A_{ij}^{(n)}=\sum_{\lambda=0}^{\infty}(rp_{i}\bar{p}_{j})^{2\lambda}(p_{i}d_{i},p_{i}s_{i},p_{i}^{3}d_{i},p_{i}^{3}s_{i},\cdots)\,{}^{t}(S_{0}(x(n)),S_{1}(x(n)),S_{2}(x(n)),S_{3}(x(n)),\cdots)$
$\displaystyle\hskip
40.0pt\times(S_{0}(\overline{x(-n)}),S_{1}(\overline{x(-n)}),S_{2}(\overline{x(-n)}),S_{3}(\overline{x(-n)}),\cdots)\,{}^{t}(\bar{p}_{j}\bar{d}_{j},\bar{p}_{j}\bar{s}_{j},\bar{p}_{j}^{3}\bar{d}_{j},\bar{p}_{j}^{3}\bar{s}_{j},\cdots)$
$\displaystyle\qquad+\sum_{\lambda=0}^{\infty}(rp_{i}\bar{p}_{j})^{2\lambda+1}(s_{i},p_{i}^{2}d_{i},p_{i}^{2}s_{i},p_{i}^{4}d_{i},\cdots)\,{}^{t}(S_{0}(x(n)),S_{1}(x(n)),S_{2}(x(n)),S_{3}(x(n)),\cdots)$
$\displaystyle\hskip
40.0pt\times(S_{0}(\overline{x(-n)}),S_{1}(\overline{x(-n)}),S_{2}(\overline{x(-n)}),S_{3}(\overline{x(-n)}),\cdots)\,{}^{t}(\bar{s}_{j},\bar{p}_{j}^{2}\bar{d}_{j},\bar{p}_{j}^{2}\bar{s}_{j},\bar{p}_{j}^{4}\bar{d}_{j},\cdots)$
$\displaystyle\qquad=\begin{pmatrix}p_{i}d_{i}&p_{i}s_{i}&p_{i}^{3}d_{i}&p_{i}^{3}s_{i}&\cdots\end{pmatrix}\raisebox{-30.0pt}{$\begin{pmatrix}S_{0}(x(n))&&&&\raisebox{-5.0pt}{\smash{\Large
0}}\\\ S_{1}(x(n))&S_{0}(x(n))\\\ S_{2}(x(n))&S_{1}(x(n))&S_{0}(x(n))\\\
S_{3}(x(n))&S_{2}(x(n))&S_{1}(x(n))&S_{0}(x(n))\\\
\vdots&\vdots&\vdots&&\ddots\end{pmatrix}$}$
$\displaystyle\qquad\times\begin{pmatrix}1&&&&\raisebox{-5.0pt}{\smash{\Large
0}}\\\ &r\\\ &&r^{2}\\\ &&&r^{3}\\\ \mbox{\smash{\Large
0}}&&&&\ddots\end{pmatrix}\begin{pmatrix}S_{0}(\overline{x(-n)})&S_{1}(\overline{x(-n)})&S_{2}(\overline{x(-n)})&S_{3}(\overline{x(-n)})&\cdots\\\
&S_{0}(\overline{x(-n)})&S_{1}(\overline{x(-n)})&S_{2}(\overline{x(-n)})&\cdots\\\
&&S_{0}(\overline{x(-n)})&S_{1}(\overline{x(-n)})&\cdots\\\
&&&S_{0}(\overline{x(-n)})\\\ \mbox{\smash{\Large
0}}&&&&\ddots\end{pmatrix}\begin{pmatrix}\bar{p}_{j}\bar{d}_{j}\\\
\bar{p}_{j}\bar{s}_{j}\\\ \bar{p}_{j}^{3}\bar{d}_{j}\\\
\bar{p}_{j}^{3}\bar{s}_{j}\\\ \vdots\end{pmatrix}.$

Therefore $\tau_{k}^{t}(n)$ in Theorem 1 is given in the form of the following
determinant,

$\displaystyle\tau_{k}^{t}(n)=\left|\raisebox{8.0pt}{$\begin{pmatrix}p_{1}d_{1}&p_{1}s_{1}&p_{1}^{3}d_{1}&p_{1}^{3}s_{1}&\cdots\\\
p_{2}d_{2}&p_{2}s_{2}&p_{2}^{3}d_{2}&p_{2}^{3}s_{2}&\cdots\\\
\vdots&\vdots&\vdots&\vdots\\\
p_{N}d_{N}&p_{N}s_{N}&p_{N}^{3}d_{N}&p_{N}^{3}s_{N}&\cdots\end{pmatrix}$}\begin{pmatrix}S_{0}(x(n))&&&&\raisebox{-5.0pt}{\smash{\Large
0}}\\\ S_{1}(x(n))&S_{0}(x(n))\\\ S_{2}(x(n))&S_{1}(x(n))&S_{0}(x(n))\\\
S_{3}(x(n))&S_{2}(x(n))&S_{1}(x(n))&S_{0}(x(n))\\\
\vdots&\vdots&\vdots&&\ddots\end{pmatrix}\right.$ $\displaystyle\hskip
40.0pt\times\begin{pmatrix}1&&&&\raisebox{-5.0pt}{\smash{\Large 0}}\\\ &r\\\
&&r^{2}\\\ &&&r^{3}\\\ \mbox{\smash{\Large
0}}&&&&\ddots\end{pmatrix}\begin{pmatrix}S_{0}(\overline{x(-n)})&S_{1}(\overline{x(-n)})&S_{2}(\overline{x(-n)})&S_{3}(\overline{x(-n)})&\cdots\\\
&S_{0}(\overline{x(-n)})&S_{1}(\overline{x(-n)})&S_{2}(\overline{x(-n)})&\cdots\\\
&&S_{0}(\overline{x(-n)})&S_{1}(\overline{x(-n)})&\cdots\\\
&&&S_{0}(\overline{x(-n)})\\\ \mbox{\smash{\Large 0}}&&&&\ddots\end{pmatrix}$
$\displaystyle\hskip
40.0pt\times\left.\begin{pmatrix}\bar{p}_{1}\bar{d}_{1}&\bar{p}_{2}\bar{d}_{2}&\cdots&\bar{p}_{N}\bar{d}_{N}\\\
\bar{p}_{1}\bar{s}_{1}&\bar{p}_{2}\bar{s}_{2}&\cdots&\bar{p}_{N}\bar{s}_{N}\\\
\bar{p}_{1}^{3}\bar{d}_{1}&\bar{p}_{2}^{3}\bar{d}_{2}&\cdots&\bar{p}_{N}^{3}\bar{d}_{N}\\\
\bar{p}_{1}^{3}\bar{s}_{1}&\bar{p}_{2}^{3}\bar{s}_{2}&\cdots&\bar{p}_{N}^{3}\bar{s}_{N}\\\
\vdots&\vdots&&\vdots\\\ \end{pmatrix}\right|.$

Now let us take

$a_{i}=\frac{1}{2}\Big{(}1+\sum_{\nu=1}^{i}c_{\nu}p_{i}^{2\nu-1}\Big{)},\quad
b_{i}=\frac{1}{2}\Big{(}1-\sum_{\nu=1}^{i}c_{\nu}p_{i}^{2\nu-1}\Big{)}.$

Then we have $\displaystyle d_{i}=\sum_{\nu=1}^{i}c_{\nu}p_{i}^{2\nu-2}$ and
$s_{i}=1$, and the above $\tau_{k}^{t}(n)$ is
$O(p_{1}\bar{p}_{1}p_{2}\bar{p}_{2}\cdots p_{N}\bar{p}_{N})$ as $p_{i}\to 0$
for $1\leq i\leq N$. In order to take the lowest order in $p_{1}$, we consider
the limit, $\displaystyle\tilde{\tau}_{k}^{t}(n):=\lim_{p_{1}\to
0}\tau_{k}^{t}(n)/(p_{1}\bar{p}_{1})$. In this limit, the leading order
becomes $O((p_{2}\bar{p}_{2}\cdots p_{N}\bar{p}_{N})^{3})$, thus for picking
up the lowest order in $p_{2}$, we take the limit,
$\displaystyle\lim_{p_{2}\to
0}\tilde{\tau}_{k}^{t}(n)/(p_{2}\bar{p}_{2})^{3}$. So the leading order
becomes $O((p_{3}\bar{p}_{3}\cdots p_{N}\bar{p}_{N})^{5})$. Repeating this
procedure, finally we obtain the $\tau$ function of rogue wave solution from
that of breather, $\tau_{k}^{t}(n)$,

$\displaystyle\lim_{p_{N}\to 0}\cdots\lim_{p_{2}\to 0}\lim_{p_{1}\to
0}\frac{\tau_{k}^{t}(n)}{p_{1}\bar{p}_{1}p_{2}^{3}\bar{p}_{2}^{3}\cdots
p_{N}^{2N-1}\bar{p}_{N}^{2N-1}}$
$\displaystyle=\left|\begin{pmatrix}c_{1}&1&&&&&&&&\raisebox{-5.0pt}{\smash{\Large
0}}\\\ c_{2}&0&c_{1}&1\\\ c_{3}&0&c_{2}&0&c_{1}&1\\\
\vdots&\vdots&\vdots&\vdots&&&\ddots\\\
c_{N}&0&c_{N-1}&0&\cdots&\cdots&\cdots&c_{1}&1&\end{pmatrix}\begin{pmatrix}S_{0}(x(n))&&&&\raisebox{-5.0pt}{\smash{\Large
0}}\\\ S_{1}(x(n))&S_{0}(x(n))\\\ S_{2}(x(n))&S_{1}(x(n))&S_{0}(x(n))\\\
S_{3}(x(n))&S_{2}(x(n))&S_{1}(x(n))&S_{0}(x(n))\\\
\vdots&\vdots&\vdots&&\ddots\end{pmatrix}\right.$ $\displaystyle\hskip
30.0pt\times\begin{pmatrix}1&&&&\raisebox{-5.0pt}{\smash{\Large 0}}\\\ &r\\\
&&r^{2}\\\ &&&r^{3}\\\ \mbox{\smash{\Large
0}}&&&&\ddots\end{pmatrix}\begin{pmatrix}S_{0}(\overline{x(-n)})&S_{1}(\overline{x(-n)})&S_{2}(\overline{x(-n)})&S_{3}(\overline{x(-n)})&\cdots\\\
&S_{0}(\overline{x(-n)})&S_{1}(\overline{x(-n)})&S_{2}(\overline{x(-n)})&\cdots\\\
&&S_{0}(\overline{x(-n)})&S_{1}(\overline{x(-n)})&\cdots\\\
&&&S_{0}(\overline{x(-n)})\\\ \mbox{\smash{\Large 0}}&&&&\ddots\end{pmatrix}$
$\displaystyle\hskip
30.0pt\times\left.\begin{pmatrix}\bar{c}_{1}&\bar{c}_{2}&\bar{c}_{3}&\cdots&\bar{c}_{N}\\\
1&0&0&\cdots&0\\\ &\bar{c}_{1}&\bar{c}_{2}&\cdots&\bar{c}_{N-1}\\\
&1&0&\cdots&0\\\ &&\bar{c}_{1}&&\vdots\\\ &&1&&\vdots\\\ &&&\ddots&\vdots\\\
&&&&\bar{c}_{1}\\\ &&&&1\\\ \mbox{\smash{\Large 0}}\end{pmatrix}\right|$
$\displaystyle=\left|\begin{pmatrix}\Phi_{1}^{(n)}&\Phi_{0}^{(n)}&&&&&&&&\raisebox{-5.0pt}{\smash{\Large
0}}\\\ \Phi_{3}^{(n)}&\Phi_{2}^{(n)}&\Phi_{1}^{(n)}&\Phi_{0}^{(n)}\\\
\Phi_{5}^{(n)}&\Phi_{4}^{(n)}&\Phi_{3}^{(n)}&\Phi_{2}^{(n)}&\Phi_{1}^{(n)}&\Phi_{0}^{(n)}\\\
\vdots&\vdots&\vdots&\vdots&&&\ddots\\\
\Phi_{2N-1}^{(n)}&\Phi_{2N-2}^{(n)}&\Phi_{2N-3}^{(n)}&\Phi_{2N-4}^{(n)}&\cdots&\cdots&\cdots&\Phi_{1}^{(n)}&\Phi_{0}^{(n)}\end{pmatrix}\right.\begin{pmatrix}1&&&&\raisebox{-5.0pt}{\smash{\Large
0}}\\\ &r\\\ &&r^{2}\\\ &&&r^{3}\\\ \mbox{\smash{\Large
0}}&&&&\ddots\end{pmatrix}$ $\displaystyle\hskip
30.0pt\times\left.\begin{pmatrix}\overline{\Phi_{1}^{(-n)}}&\overline{\Phi_{3}^{(-n)}}&\overline{\Phi_{5}^{(-n)}}&\cdots&\overline{\Phi_{2N-1}^{(-n)}}\\\
\overline{\Phi_{0}^{(-n)}}&\overline{\Phi_{2}^{(-n)}}&\overline{\Phi_{4}^{(-n)}}&\cdots&\overline{\Phi_{2N-2}^{(-n)}}\\\
&\overline{\Phi_{1}^{(-n)}}&\overline{\Phi_{3}^{(-n)}}&\cdots&\overline{\Phi_{2N-3}^{(-n)}}\\\
&\overline{\Phi_{0}^{(-n)}}&\overline{\Phi_{2}^{(-n)}}&\cdots&\overline{\Phi_{2N-4}^{(-n)}}\\\
&&\overline{\Phi_{1}^{(-n)}}&&\vdots\\\
&&\overline{\Phi_{0}^{(-n)}}&&\vdots\\\ &&&\ddots&\vdots\\\
&&&&\overline{\Phi_{1}^{(-n)}}\\\ &&&&\overline{\Phi_{0}^{(-n)}}\\\
\mbox{\smash{\Large 0}}\end{pmatrix}\right|.$

By calculating the matrix element, it is easy to see that the above
determinant is equal to $\displaystyle\det_{1\leq i,j\leq
N}\left(B_{ij}^{(n)}(k,t)\right)$. We completed the proof of Theorem 2. ∎

By taking $N=1$ we obtain the fully discrete Peregrine rogue wave solution,

$\displaystyle\tau_{k}^{t}(n)=\left(-(1+r)n+(1-r)k+(1-r+1/\bar{c}-rc)t+c_{1}\right)$
$\displaystyle\hskip
30.0pt\times\left((1+r)n+(1-r)k+(1-r+1/c-r\bar{c})t+\bar{c}_{1}\right)+r,$
(12)

where $c_{1}$ is a complex constant. Similarly to the breather solution, there
are rogue wave solutions regular on the lattice for both focusing case ($r>0$)
and defocusing case ($r<0$), since if there are zeros of $f_{k}^{t}$ we can
avoid explosion of solution by displacing the zeros off the lattice points.
However for regularity of the solution on the real two dimensional space of
$(k,t)$, we have to take $r$ positive. First order and second-order rogue wave
solutions are shown in Figs. 2 and 3, respectively.

Figure 2: First order rogue wave solution (a) $r=2.0$, $c=1+2\rm{i}$,
$c_{1}=2$ (b) $r=0.5$, $c=1+2\rm{i}$, $c_{1}=2$.

Figure 3: Second order rogue wave solution (a) $r=2.0$, $c=1+2\rm{i}$,
$c_{1}=2.0$, $c_{2}=2+2\rm{i}$; (b) $r=0.5$, $c=2+\rm{i}$, $c_{1}=2$,
$c_{2}=2+2\rm{i}$.

There is an exceptional regular solution for $r<0$ which is obtained by taking
$c$ real and $({\rm Im}\,c_{1})^{2}>-r$ in (12), but this is not a rogue wave
solution but a traveling wave solution. An example is shown in Fig. 4.

Figure 4: Traveling wave solution with $r=-0.5$, $c=1.0$, $c_{1}=1+\rm{i}$:
(a) profile, (b) contour plot.

##  4 Concluding Remarks

Even though the study of rogue waves has attracted much attention in more than
one decade, the rogue wave solution in fully discrete integrable systems has
not been reported yet. In this paper, by taking the fully discrete NLS
equation, we firstly constructed its general breather solution via the KP-Toda
reduction method. Then by taking a list to the parameters succeeded in
constructing its general rogue wave solution by taking the limit of $p_{i}\to
0$ successively for $i=1,\cdots,N$.

We expect to construct rogue wave solutions in other discrete systems such as
the discrete complex sine-Gordon equation in the future.

## Acknowledment

BF’s work is partially supported by National Science Foundation (NSF) under
Grant No. DMS-1715991 and U.S. Department of Defense (DoD), Air Force for
Scientific Research (AFOSR) under grant No. W911NF2010276.

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