# Complex short pulse and coupled complex short pulse equations

Bao-Feng Feng  Department of Mathematics
The University of Texas-Pan American
Edinburg, TX, 78541-2999, USA

###### Abstract

In the present paper, we propose a complex short pulse equation and a coupled
complex short equation to describe ultra-short pulse propagation in optical
fibers. They are integrable due to the existence of Lax pairs and infinite
number of conservation laws. Furthermore, we find their multi-soliton
solutions in terms of pfaffians by virtue of Hirota’s bilinear method. One-
and two-soliton solutions are investigated in details, showing favorable
properties in modeling ulta-short pulses with a few optical cycles.
Especially, same as the coupled nonlinear Schrödinger equation, there is an
interesting phenomenon of energy redistribution in soliton interactions. It is
expected that, for the ultra-short pulses, the complex and coupled complex
short pulses equation will play the same roles as the nonlinear Schrödinger
equation and coupled nonlinear Schrödinger equation.

Keywords: Complex short pulse equation; Coupled complex short pulse equation;
Hirota bilinear method; Pfaffian; Envelope soliton; Soliton interaction.

##  1 Introduction

The nonlinear Schrödinger (NLS) equation, as one of the universal equations
that describe the evolution of slowly varying packets of quasi-monochromatic
waves in weakly nonlinear dispersive media, has been very successful in many
applications such as nonlinear optics and water waves [1, 2, 3, 4]. The NLS
equation is integrable, which can be solved by the inverse scattering
transform [5].

However, in the regime of ultra-short pulses where the width of optical pulse
is in the order of femtosecond ($10^{-15}$ s), the NLS equation becomes less
accurate [6]. Description of ultra-short processes requires a modification of
standard slow varying envelope models based on the NLS equation. There are
usually two approaches to meet this requirement in the literature. The first
one is to add several higher-order dispersive terms to get higher-order NLS
equation [2]. The second one is to construct a suitable fit to the frequency-
dependent dielectric constant $\epsilon(\omega)$ in the desired spectral
range. Several models have been proposed by this approach including the short-
pulse (SP) equation [7, 8, 9, 10].

Recently, Schäfer and Wayne derived a so-called short pulse (SP) equation [7]

$u_{xt}=u+\frac{1}{6}\left(u^{3}\right)_{xx}\,,$ (1)

to describe the propagation of ultra-short optical pulses in nonlinear media.
Here, $u=u(x,t)$ is a real-valued function, representing the magnitude of the
electric field, the subscripts $t$ and $x$ denote partial differentiation.
Apart from the context of nonlinear optics, the SP equation has also been
derived as an integrable differential equation associated with pseudospherical
surfaces [11]. The SP equation has been shown to be completely integrable [11,
12, 13, 14, 15]. The periodic and soliton solutions of the SP equation were
found in [16, 17, 18]. The connection between the SP equation and the sine-
Gordon equation through the hodograph transformation was clarified, and then
the $N$-soliton solutions including multi-loop and multi-breather ones were
given in [19, 20] by using the Hirota bilinear method [21]. The integrable
discretization of the SP equation was studied in [22], the geometric
interpretation of the SP equation, as well as its integrable discretization,
was given in [23]. The higher-order corrections to the SP equation was studied
in [24] most recently.

Similar to the case of the NLS equation [25], it is necessary to consider its
two-component or multi-component generalizations of the SP equation for
describing the effects of polarization or anisotropy. As a matter of fact,
several integrable coupled short pulse have been proposed in the literature
[26, 27, 28, 29, 30, 31]. Most recently, the bi-Hamiltonian structures for the
above two-component SP equations were obtained by Brunelli [32].

In the present paper, we propose and study a complex short pulse (CSP)
equation

$q_{xt}+q+\frac{1}{2}\left(|q|^{2}q_{x}\right)_{x}=0\,,$ (2)

and its two-component generalization

$\displaystyle
q_{1,xt}+q_{1}+\frac{1}{2}\left((|q_{1}|^{2}+|q_{2}|^{2})q_{1,x}\right)_{x}=0\,,$
(3) $\displaystyle
q_{2,xt}+q_{2}+\frac{1}{2}\left((|q_{1}|^{2}+|q_{2}|^{2})q_{2,x}\right)_{x}=0\,.$
(4)

As will be revealed in the present paper, both the CSP equation and its two-
component generalization are integrable guaranteed by the existence of Lax
pairs and infinite number of conservation laws. They have $N$-soliton
solutions which can be constructed via Hirota’s bilinear method.

The outline of the present paper is organized as follows. In section 2, we
derive the CSP equation and coupled complex short pulse (CCSP) equation from
the physical context. In section 3, by providing the Lax pairs, the
integrability of the proposed two equations are confirmed, and further, the
conservation laws, both local and nonlocal ones, are investigated. Then
$N$-soliton solutions to both the CSP and CCSP equations are constructed in
terms of pfaffians by Hirota’s bilinear method in section 4. In section 5,
soliton-interaction for coupled complex short pulse equation is investigated
in details, which shows rich phenomena similar to the coupled nonlinear
Schrödinger equatoin. In particular, they may undergo either elastic or
inelastic collision depending on the initial conditions. For inelastic
collisions, there is an energy exchange between solitons, which can allow the
generation or vanishing of soliton. The dynamics is more richer in compared
with the single component case. The paper is concluded by comments and remarks
in section 6.

##  2 The derivation of the complex short pulse and coupled complex short
pulse equations

In this section, following the procedure in [2, 7], we derive the complex
short pulse equation (2) and its two-component generalization that governs the
propagation of ultra short pulse packet along optical fibers.

###  2.1 The complex short pulse equation

We start with a wave equation for electric field

$\nabla^{2}\mathbf{E}-\frac{1}{c^{2}}\mathbf{E}_{tt}=\mu_{0}\mathbf{P}_{tt}\,,$
(5)

originated from the Maxwell equation. Here $\mathbf{E}(\mathbf{r},t)$ and
$\mathbf{P}(\mathbf{r},t)$ represent the electric field and the induced
polarization, respectively, $\mu_{0}$ is the vacuum permeability, $c$ is the
speed of light in vacuum. If we assume the local medium response and only the
third-order nonlinear effects governed by $\chi^{(3)}$, the induced
polarization consists of two parts,
$\mathbf{P}(\mathbf{r},t)=\mathbf{P}_{L}(\mathbf{r},t)+\mathbf{P}_{NL}(\mathbf{r},t)$,
where the linear part

$\mathbf{P}_{L}(\mathbf{r},t)=\epsilon_{0}\int_{-\infty}^{\infty}\chi^{(1)}(t-t^{\prime})\cdot\mathbf{E}(\mathbf{r},t^{\prime})\,dt^{\prime}\,,$
(6)

and the nonlinear part

$\mathbf{P}_{NL}(\mathbf{r},t)=\epsilon_{0}\int_{-\infty}^{\infty}\chi^{(3)}(t-t_{1},t-t_{2},t-t_{3})\times\mathbf{E}(\mathbf{r},t_{1})\mathbf{E}(\mathbf{r},t_{2})\mathbf{E}(\mathbf{r},t_{3})\,dt_{1}dt_{2}dt_{3}\,.$
(7)

Here $\epsilon_{0}$ is the vacuum permittivity and $\chi^{(j)}$ is the $j$th-
order susceptibility. Since the nonlinear effects are relatively small in
silica fibers, $\mathbf{P}_{NL}$ can be treated as a small perturbation.
Therefore, we first consider (5) with $\mathbf{P}_{NL}=0$. Furthermore, we
restrict ourselves to the case that the optical pulse maintains its
polarization along the optical fiber, and the transverse diffraction term
$\Delta_{\perp}\mathbf{E}$ can be neglected. In this case, the electric field
can be considered to be one-dimensional and expressed as

$\mathbf{E}=\frac{1}{2}\mathbf{e_{1}}\left(E(z,t)+c.c.\right)\,,$ (8)

where $\mathbf{e_{1}}$ is a unit vector in the direction of the polarization,
$E(z,t)$ is the complex-valued function, and $c.c.$ stands for the complex
conjugate. Conducting a Fourier transform on (5) leads to the Helmholtz
equation

$\tilde{E}_{zz}(z,\omega)+\epsilon(\omega)\frac{\omega^{2}}{c^{2}}\tilde{E}(z,\omega)=0\,,$
(9)

where $\tilde{E}(z,\omega)$ is the Fourier transform of $E(z,t)$ defined as

$\tilde{E}(z,\omega)=\int_{-\infty}^{\infty}{\ E}(z,t)e^{\mathrm{i}\omega
t}\,dt\,,$ (10)

$\epsilon(\omega)$ is called the frequency-dependent dielectric constant
defined as

$\epsilon(\omega)=1+\tilde{\chi}^{(1)}(\omega)\,,$ (11)

where $\tilde{\chi}^{(1)}(\omega)$ is the Fourier transform of $\chi^{(1)}(t)$

$\tilde{\chi}^{(1)}(\omega)=\int_{-\infty}^{\infty}\chi^{(1)}(t)e^{\mathrm{i}\omega
t}\,dt\,.$ (12)

Now we proceed to the consideration of the nonlinear effect. Assuming the
nonlinear response is instantaneous so that $P_{NL}$ is given by
$P_{NL}(z,t)=\epsilon_{0}\epsilon_{NL}E(z,t)$ [2] where the nonlinear
contribution to the dielectric constant is defined as

$\epsilon_{NL}=\frac{3}{4}\chi^{(3)}_{xxxx}|E(z,t)|^{2}\,.$ (13)

In this case, the Helmholtz equation (9) can be modified as

$\tilde{E}_{zz}(z,\omega)+\tilde{\epsilon}(\omega)\frac{\omega^{2}}{c^{2}}\tilde{E}(z,\omega)=0\,,$
(14)

where

$\tilde{\epsilon}(\omega)=1+\tilde{\chi}^{(1)}(\omega)+\epsilon_{NL}\,.$ (15)

As pointed out in [7, 3, 8], the Fourier transform $\tilde{\chi}^{(1)}$ can be
well approximated by the relation
$\tilde{\chi}^{(1)}=\tilde{\chi}_{0}^{(1)}-\tilde{\chi}_{2}^{(1)}\lambda^{2}$
if we consider the propagation of optical pulse with the wavelength between
1600 nm and 3000 nm. It then follows that the linear equation (9) written in
Fourier transformed form becomes

$\tilde{E}_{zz}+\frac{1+\tilde{\chi}_{0}^{(1)}}{c^{2}}\omega^{2}\tilde{E}-(2\pi)^{2}\tilde{\chi}_{2}^{(1)}\tilde{E}+\epsilon_{NL}\frac{\omega^{2}}{c^{2}}\tilde{E}=0\,.$
(16)

Applying the inverse Fourier transform to (16) yields a single nonlinear wave
equation

$E_{zz}-\frac{1}{c_{1}^{2}}E_{tt}=\frac{1}{c_{2}^{2}}E+\frac{3}{4}\chi^{(3)}_{xxxx}\left(|E|^{2}E\right)_{tt}=0\,.$
(17)

Similar to [7], we focus on only a right-moving wave packet and make a
multiple scales ansatz

$E(z,t)=\epsilon
E_{0}(\phi,z_{1},z_{2},\cdots)+\epsilon^{2}E_{1}(\phi,z_{1},z_{2},\cdots)+\cdots\,,$
(18)

where $\epsilon$ is a small parameter, $\phi$ and $z_{n}$ are the scaled
variables defined by

$\phi=\frac{t-\frac{x}{c_{1}}}{\epsilon},\quad z_{n}=\epsilon^{n}z\,.$ (19)

Substituting (18) with (19) into (17), we obtain the following partial
differential equation for $E_{0}$ at the order $O(\epsilon)$:

$-\frac{2}{c_{1}}\frac{\partial^{2}E_{0}}{\partial{\phi}\partial{z_{1}}}=\frac{1}{c_{2}^{2}}E_{0}+\frac{3}{4}\chi^{(3)}_{xxxx}\frac{\partial}{\partial{\phi}}\left(|E_{0}|^{2}\frac{\partial
E_{0}}{\partial{\phi}}\right)\,.$ (20)

Finally, by a scale transformation

$x=\frac{c_{1}}{2}\phi,\quad t={c_{2}}z_{1},\quad
q=\frac{c_{1}\sqrt{6c_{2}\chi^{(3)}_{xxxx}}}{4}E_{0}\,,$ (21)

we arrive at the normalized form of the complex short pulse equation (2).

###  2.2 Coupled complex short pulse equation

In the previous subsection, a major simplification made in the derivation of
the complex short pulse equation is to assume that the polarization is
preserved during its propagating inside an optical fiber. However, this is not
really the case in practice. For birefringent fibers, two orthogonally
polarized modes have to be considered. Therefore, similar to the extension of
coupled nonlinear Schrödinger equation from the NLS equation, an extension to
a two-component version of the complex short pulse equation (2) is needed to
describe the propagation of ultra-short pulse in birefringent fibers. In fact,
several generalizations have been proposed for the short pulse equation [26,
27, 28, 29, 30, 31]. Particularly, by taking into account the effects of
anisotropy and polarization, Pietrzyk et. al. have derived a general two-
component short-pulse equation from the physical context [26]. We follow the
approach by Pietrzyk et. al. to derive a two-component complex short pulse
equation. However, as shown in subsequent section, the two-component complex
short pulse equation admits multi-soliton solutions which reveals richer
dynamics in soliton interactions in compared with the real SP equation.

We first consider the linear birefringent fiber such that the electric field
with an arbitrarily polarized optical fiber can be expressed as

$\mathbf{E}=\frac{1}{2}\left(\mathbf{e_{1}}E_{1}(z,t)+\mathbf{e_{2}}E_{2}(z,t)\right)+c.c.\,,$
(22)

where $\mathbf{e_{1}}$, $\mathbf{e_{2}}$ are two unit vectors along positive
$x$\- and $y$-direction in the transverse plane perpendicular to the optical
fiber, respectively, $E_{1}$ and $E_{2}$ are the complex amplitudes of the
polarization components correspondingly. Without the presence of nonlinear
polarization ($P_{NL}=0$) and the transverse diffraction, the Fourier
transform converts (5) into a pair of Helmholtz equations

$\tilde{E}_{1,zz}(z,\omega)+\epsilon(\omega)\frac{\omega^{2}}{c^{2}}\tilde{E_{1}}(z,\omega)=0\,,$
(23)

$\tilde{E}_{2,zz}(z,\omega)+\epsilon(\omega)\frac{\omega^{2}}{c^{2}}\tilde{E_{2}}(z,\omega)=0\,.$
(24)

Same as the scalar case, the frequency-dependent dielectric constant
$\epsilon(\omega)=1+\tilde{\chi}^{(1)}(\omega)$, where $\tilde{\chi}^{(1)}$
can be well approximated by the relation
$\tilde{\chi}^{(1)}=\tilde{\chi}_{0}^{(1)}-\tilde{\chi}_{2}^{(1)}\lambda^{2}$
for the propagation of optical pulse with the wavelength between 1600 nm and
3000 nm.

As indicated in [2], the nonlinear part of the induced polarization
$\mathbf{P}_{NL}$ can be written as

$\mathbf{P}_{NL}=\frac{1}{2}\left(\mathbf{e_{1}}P_{1}(z,t)+\mathbf{e_{2}}P_{2}(z,t)\right)+c.c.\,,$
(25)

where

$P_{1}=\frac{3\epsilon_{0}}{4}\chi_{xxxx}^{(3)}\left[\left(|E_{1}|^{2}+\frac{2}{3}|E_{2}|^{2}\right)E_{1}+\frac{1}{3}(E_{1}^{\ast}E_{2})E_{2}\right]\,,$
(26)
$P_{2}=\frac{3\epsilon_{0}}{4}\chi_{xxxx}^{(3)}\left[\left(|E_{2}|^{2}+\frac{2}{3}|E_{1}|^{2}\right)E_{2}+\frac{1}{3}(E_{2}^{\ast}E_{1})E_{1}\right]\,.$
(27)

The last term in Eqs. (26) and (27) leads to the degenerate four-wave mixing.
In highly birefringent fibers, the four-wave-mixing term can often be
neglected. In this case, we arrive at a coupled nonlinear wave equation

$E_{1,zz}-\frac{1}{c_{1}^{2}}E_{1,tt}=\frac{1}{c_{2}^{2}}E_{1}+\frac{3}{4}\chi_{xxxx}^{(3)}\left[\left(|E_{1}|^{2}+\frac{2}{3}|E_{2}|^{2}\right)E_{1}\right]_{tt}\,,$
(28)

$E_{2,zz}-\frac{1}{c_{1}^{2}}E_{2,tt}=\frac{1}{c_{2}^{2}}E_{2}+\frac{3}{4}\chi_{xxxx}^{(3)}\left[\left(|E_{2}|^{2}+\frac{2}{3}|E_{1}|^{2}\right)E_{2}\right]_{tt}\,.$
(29)

Similar to the scalar case, by a multiple scales expansion and an appropriate
scaling transformation, a couple complex short pulse equation can be obtained
from (28)–(29)

$q_{1,xt}+q_{1}+\frac{1}{2}\left((|q_{1}|^{2}+\frac{2}{3}|q_{2}|^{2})q_{1,x}\right)_{x}=0\,,$
(30)

$q_{2,xt}+q_{2}+\frac{1}{2}\left((|q_{2}|^{2}+\frac{2}{3}|q_{1}|^{2})q_{2,x}\right)_{x}=0\,.$
(31)

More generally, we can consider the coupled short pulse equation for
elliptically birefringent fibers. In this case, the electric field can be
written as

$\mathbf{E}=\frac{1}{2}\left(\mathbf{e_{x}}E_{x}(z,t)+\mathbf{e_{y}}E_{y}(z,t)\right)+c.c.\,,$
(32)

where $\mathbf{e_{x}}$ and $\mathbf{e_{y}}$ are orthonormal polarization
eigenvectors

$\mathbf{e_{x}}=\frac{\mathbf{e_{1}}+ir\mathbf{e_{2}}}{\sqrt{1+r^{2}}},\quad\mathbf{e_{y}}=\frac{r\mathbf{e_{1}}-i\mathbf{e_{2}}}{\sqrt{1+r^{2}}}\,.$
(33)

The parameter $r$ represents the ellipticity. It is common to introduce the
ellipticity angle $\theta$ as $r=\tan(\theta/2)$. The case $\theta=0$ and
$\pi/2$ correspond to linearly and circularly birefringent fibers,
respectively.

Following a procedure similar to the case of linearly birefringent fibers, one
can drive the normalized form for the coupled complex short pulse equation

$q_{1,xt}+q_{1}+\frac{1}{2}\left((|q_{1}|^{2}+B|q_{2}|^{2})q_{1,x}\right)_{x}=0\,,$
(34)
$q_{2,xt}+q_{2}+\frac{1}{2}\left((|q_{2}|^{2}+B|q_{1}|^{2})q_{2,x}\right)_{x}=0\,.$
(35)

where the parameter $B$ is related to the ellipticity angle $\theta$ as

$B=\frac{2+2\sin^{2}\theta}{2+\cos^{2}\theta}\,.$ (36)

For a linearly birefringent fiber ($\theta=0$), $B=\frac{2}{3}$, and Eqs.
(34)– (35) reduces to Eqs. (30)– (31). For a circularly birefringent fiber
($\theta=\pi/2$), $B=2$. In general, the coupling parameter $B$ depends on the
ellipticity angle $\theta$ and can vary from $\frac{2}{3}$ to $2$ for values
of $\theta$ in the range from $0$ to $\pi/2$. Note that $B=1$ when
$\theta\approx 35^{\circ}$. As discussed in the subsequent section, this case
is of particular interest because the coupled system is integrable and admits
$N$-soliton solution.

##  3 Lax pairs and conservation laws for the complex and coupled complex
short pulse equations

###  3.1 Lax pairs and integrability

In [26], a matrix generalization for the SP equation is given based on zero-
curvature representation, from which the Lax pairs for several integrable two-
component SP equations are explicitly provided. In this subsection, we will
show the integrability of the complex short pulse and coupled complex short
pulse equations by finding their Lax pairs constructed from another matrix
generalization of the SP equation.

The Lax pair for the complex short pulse equation (2) can be expressed as

$\Psi_{x}=U\Psi,\quad\Psi_{t}=V\Psi\,,$ (37)

with

$\displaystyle U=\lambda\left(\begin{array}[]{cc}1&q_{x}\\\
q^{*}_{x}&-1\end{array}\right),\quad
V=\left(\begin{array}[]{cc}-\frac{\lambda}{2}|q|^{2}-\frac{1}{4\lambda}&-\frac{\lambda}{2}|q|^{2}q_{x}+\frac{q}{2}\\\
-\frac{\lambda}{2}|q|^{2}q^{*}_{x}-\frac{q^{*}}{2}&\frac{\lambda}{2}|q|^{2}+\frac{1}{4\lambda}\end{array}\right)\,.$
(38)

It can be easily shown that the compatibility condition
$U_{t}-V_{x}+[U,\,V]=0$ gives the complex short pulse equation (2).

The Lax pair for the coupled complex short pulse equation (3)–(4) is found to
be of the form:

$\Psi_{x}=U\Psi,\quad\Psi_{t}=V\Psi\,,$ (39)

with

$U=\lambda\left(\begin{array}[]{cc}I_{2}&Q_{x}\\\
R_{x}&-I_{2}\end{array}\right),\quad
V=\left(\begin{array}[]{cc}-\frac{\lambda}{2}QR-\frac{1}{4\lambda}I_{2}&-\frac{\lambda}{2}QRQ_{x}+\frac{1}{2}Q\\\
-\frac{\lambda}{2}RQR_{x}-\frac{1}{2}R&\frac{\lambda}{2}QR+\frac{1}{4\lambda}I_{2}\end{array}\right)\,,$
(40)

where $I_{2}$ is a $2\times 2$ identity matrix, $Q$, $R$ are $2\times 2$
matrices defined as

$Q=\left(\begin{array}[]{cc}q_{1}&q_{2}\\\
-q_{2}^{\ast}&q_{1}^{\ast}\end{array}\right),\quad
R=\left(\begin{array}[]{cc}q_{1}^{\ast}&-q_{2}\\\
q_{2}^{\ast}&q_{1}\end{array}\right)\,.$ (41)

Note that $R=Q^{{\dagger}}$, thus,

$QR=RQ=(|q_{1}|^{2}+|q_{2}|^{2})I_{2}\,,$ (42)

the compatibility condition $U_{t}-V_{x}+[U,\,V]=0$ for (39) gives the coupled
complex short pulse equation (3)–(4).

As a matter of fact, the coupled complex short pulse equation can be
generalized into a multi-component, or a vector complex short pulse equation

$q_{i,xt}+q_{i}+\frac{1}{2}\left(|\mathbf{q}|^{2}q_{i,x}\right)_{x}=0\,,\quad
i=1,\cdots,n,$ (43)

where $\mathbf{q}=(q_{1},q_{2},\cdots,q_{n})$. The integrability of Eq. (43)
can be guaranteed by the Lax pair constructed in a similar way as in [33].

$\Psi_{x}=U\Psi,\quad\Psi_{t}=V\Psi\,,$ (44)

with

$U=\lambda\left(\begin{array}[]{cc}I_{2^{n-1}}&Q_{x}^{(n)}\\\
R_{x}^{(n)}&-I_{2^{n-1}}\end{array}\right),$
$V=\left(\begin{array}[]{cc}-\frac{1}{2}Q^{(n)}R^{(n)}-\frac{1}{4\lambda}I_{2^{n-1}}&-\frac{\lambda}{2}Q^{(n)}R^{(n)}Q_{x}^{(n)}+\frac{1}{2}Q^{(n)}\\\
-\frac{\lambda}{2}R^{(n)}Q^{(n)}R_{x}^{(n)}-\frac{1}{2}R^{(n)}&\frac{1}{2}Q^{(n)}R^{(n)}+\frac{1}{4\lambda}I_{2^{n-1}}\end{array}\right)\,,$

where $I_{2^{n-1}}$ is a $2^{n-1}\times 2^{n-1}$ identity matrix, $Q^{(n)}$
and $R^{(n)}$ are $2^{n-1}\times 2^{n-1}$ matrices can be constructed
recursively as follows

$Q^{(1)}=q_{1},\quad R^{(1)}=q_{1}^{\ast}\,,$ (45)
$Q^{(n+1)}=\left(\begin{array}[]{cc}Q^{(n)}&q_{n+1}I_{2^{n-1}}\\\
-q_{n+1}^{\ast}I_{2^{n-1}}&R^{(n)}\end{array}\right)\,,$ (46)

$R^{(n+1)}=\left(\begin{array}[]{cc}R^{(n)}&-q_{n+1}I_{2^{n-1}}\\\
q^{*}_{n+1}I_{2^{n-1}}&Q^{(n)}\end{array}\right)\,.$ (47)

By the above construction, we have $R^{(n+1)}=(Q^{(n+1)})^{\dagger}$, and
further

$Q^{(n)}R^{(n)}=R^{(n)}Q^{(n)}=\sum_{i=1}^{n}|q_{i}|^{2}I_{2^{n-1}}\,.$ (48)

Therefore, the zero curvature condition $U_{t}-V_{x}+[U,\,V]=0$ gives the
vector complex coupled short pulse equation (43).

###  3.2 Local and nonlocal conservation laws

Following a systematic method developed by in [33, 34, 35, 36], we construct
conservation laws for the vector complex short pulse equation, the
conservation laws for the complex and coupled short pulse equations can be
treated as special cases for $n=1,2$, respectively. To this end, let us
rewrite the Lax pair for the vector complex short pulse equation as follows:

$\left(\begin{array}[]{c}\Psi_{1}\\\
\Psi_{2}\end{array}\right)_{x}=\left(\begin{array}[]{cc}\lambda I&\lambda
Q_{x}\\\ \lambda R_{x}&-\lambda
I\end{array}\right)\left(\begin{array}[]{c}\Psi_{1}\\\
\Psi_{2}\end{array}\right)\,,$ (49)

$\left(\begin{array}[]{c}\Psi_{1}\\\
\Psi_{2}\end{array}\right)_{t}=\left(\begin{array}[]{cc}-\frac{\lambda}{2}QR-\frac{1}{4\lambda}I_{2}&-\frac{\lambda}{2}QRQ_{x}+\frac{1}{2}Q\\\
-\frac{\lambda}{2}RQR_{x}-\frac{1}{2}R&\frac{\lambda}{2}QR+\frac{1}{4\lambda}I_{2}\end{array}\right)\left(\begin{array}[]{c}\Psi_{1}\\\
\Psi_{2}\end{array}\right)\,.$ (50)

Here the size of matrices in the entries of Eqs. (49)–(50) is of
$2^{n-1}\times 2^{n-1}$ and is omitted for brevity. If we define

$\Gamma\equiv\Psi_{2}\Psi_{1}^{-1}\,$ (51)

then we have

$2\lambda Q_{x}\Gamma=\lambda Q_{x}R_{x}-Q_{x}((Q_{x})^{-1}\cdot
Q_{x}\Gamma)_{x}-\lambda(Q_{x}\Gamma)^{2}\,$ (52)

Expanding $Q_{x}\Gamma$ in terms of the spectral parameter $\lambda$ as
follows

$Q_{x}\Gamma=\sum_{n=0}^{\infty}F_{n}\lambda^{-n}\,,$ (53)

and substituting into Eq. (52), we obtain the following relation

$2\lambda
F_{n}=Q_{x}R_{x}\delta_{n,0}-Q_{x}((Q_{x})^{-1}F_{n-1})_{x}-\sum_{l=0}^{n}F_{l}F_{n-l}.$
(54)

The first local conserved density turns out to be

$F_{0}=\left(-1+\sqrt{1+\sum|q_{i,x}|^{2}}\right)I\,,$ (55)

which is associated with a Hamiltonian of

$H_{0}=\int\sqrt{1+|q_{x}|^{2}}\,dx\,,$ (56)

for the complex short pulse equation (2) and

$H_{0}=\int\sqrt{1+|q_{1,x}|^{2}+|q_{2,x}|^{2}}\,dx\,,$ (57)

for the coupled complex short pulse equation (3)–(4).

Following the procedure in [36], we can find the nonlocal conservation laws
for vector complex short pulse equation. To this end, we expand $Q_{x}\Gamma$
as follows

$Q_{x}\Gamma=\sum_{n=1}^{\infty}F_{-n}(2\lambda)^{n}\,.$ (58)

The first two orders in $\lambda$ yield the following equations

$0=Q_{x}R_{x}-Q_{x}\left((Q_{x})^{-1}F_{-1}\right)_{x}\,,$ (59)
$2F_{-1}=-Q_{x}\left((Q_{x})^{-1}F_{-2}\right)_{x}\,,$ (60)

from which, the first two nonlocal conserved densities can be calculated as

$F_{-1}=\frac{1}{2}Q_{x}R\,,$ (61) $F_{-2}=\frac{1}{2}QR\
-\frac{1}{2}\partial_{x}\left(Q\partial_{x}R\right).$ (62)

The first one turns out to be a trivial one, the second one accounts for a
Hamiltonian

$H_{-1}=\frac{1}{2}\int|q|^{2}\,dx\,,$ (63)

for the complex short pulse equation (2) and

$H_{-1}=\frac{1}{2}\int(|q_{1}|^{2}+|q_{2}|^{2})\,dx\,,$ (64)

for the coupled complex short pulse equation (3)–(4).

##  4 Multi-soliton solutions by Hirota’s bilinear method

###  4.1 Bilinear equations and $N$-soliton solution to the complex short
pulse equation

Proposition 4.1. The complex short pulse equation is derived from the
following bilinear equations.

$D_{s}D_{y}f\cdot g=fg\,,$ (65) $D^{2}_{s}f\cdot f=\frac{1}{2}|g|^{2}\,,$ (66)

by dependent variable transformation

$q=\frac{g}{f}\,,$ (67)

and hodograph transformation

$x=y-2(\ln f)_{s}\,,\quad t=-s\,,$ (68)

where $D$ is called Hirota $D$-operator defined by

$D_{s}^{n}D_{y}^{m}f\cdot g=\left(\frac{\partial}{\partial
s}-\frac{\partial}{\partial
s^{\prime}}\right)^{n}\left(\frac{\partial}{\partial
y}-\frac{\partial}{\partial
y^{\prime}}\right)^{m}f(y,s)g(y^{\prime},s^{\prime})|_{y=y^{\prime},s=s^{\prime}}\,.$

.

Proof. Dividing both sides by $f^{2}$, the bilinear equations (65)– (66) can
be cast into

$\left\\{\begin{array}[]{l}\displaystyle\left(\frac{g}{f}\right)_{sy}+2\frac{g}{f}\left(\ln
f\right)_{sy}=\frac{g}{f}\,,\\\\[5.0pt] \displaystyle\left(\ln
f\right)_{ss}=\frac{1}{4}\frac{|g|^{2}}{f^{2}}\,.\end{array}\right.$ (69)

From the hodograph transformation and dependent variable transformation, we
then have

$\frac{\partial x}{\partial s}=-2(\ln
f)_{ss}=-\frac{1}{2}|q|^{2}\,,\qquad\frac{\partial x}{\partial y}=1-2(\ln
f)_{sy}\,,$

which implies

${\partial_{y}}=\rho^{-1}{\partial_{x}}\,,\qquad{\partial_{s}}=-{\partial_{t}}-\frac{1}{2}|q|^{2}{\partial_{x}}\,$
(70)

by letting $1-2(\ln f)_{sy}=\rho^{-1}$.

Notice that the first equation in (69) can be rewritten as

$\left(\frac{g}{f}\right)_{sy}=\left(1-2(\ln f)_{sy}\right)\frac{g}{f}\,,$
(71)

or

$\rho\left(\frac{g}{f}\right)_{sy}=\frac{g}{f}\,,$ (72)

which is converted into

$\partial_{x}\left(-\partial_{t}-\frac{1}{2}|q|^{2}\partial_{x}\right)q=q\,,$
(73)

by using (70). Eq. (73) is nothing but the complex short pulse equation (2).

$N$-soliton solution to the bilinear equations (65)–(66) can be expressed by
pfaffians similar to the ones for coupled modified KdV equation [37]. To this
end, we need to define two sets: $B_{\mu}$ ($\mu=1,2$):
$B_{1}=\\{b_{1},b_{2},\cdots,b_{N}\\}$,
$B_{2}=\\{b_{N+1},b_{2},\cdots,b_{2N}\\}$, and an index function of $b_{j}$ by
$index(b_{j})=\mu$ if $b_{j}\in B_{\mu}$.

Theorem 4.2. The pfaffians

$\displaystyle f$ $\displaystyle=$
$\displaystyle\mathrm{Pf}(a_{1},\cdots,a_{2N},b_{1},\cdots,b_{2N})\,,$ (74)
$\displaystyle g$ $\displaystyle=$
$\displaystyle\mathrm{Pf}(d_{0},\beta_{1},a_{1},\cdots,a_{2N},b_{1},\cdots,b_{2N})\,.$
(75)

satisfy the bilinear equations (65)–(66) provided that the elements of the
pfaffians are defined by

$\mathrm{Pf}(a_{j},a_{k})=\frac{p_{j}-p_{k}}{p_{j}+p_{k}}e^{\eta_{j}+\eta_{k}}\,,\quad\mathrm{Pf}(a_{j},b_{k})=\delta_{j,k}\,,$
(76)
$\mathrm{Pf}(b_{j},b_{k})=\frac{1}{4}\frac{\alpha_{j}\alpha_{k}}{p^{-2}_{j}-p^{-2}_{k}}\delta_{\mu+1,\nu}\,,\quad\mathrm{Pf}(d_{l},a_{k})=p_{k}^{l}e^{\eta_{k}}\,,$
(77)

$\mathrm{Pf}(b_{j},\beta_{1})=\alpha_{j}\delta_{\mu,1}\,,\quad\mathrm{Pf}(d_{0},b_{j})=\mathrm{Pf}(d_{0},\beta_{1})=\mathrm{Pf}(a_{j},\beta_{1})=0\,.$
(78)

Here $\mu=index(b_{j})$, $\nu=index(b_{k})$, $\eta_{j}=p_{j}y+p_{j}^{-1}s$
which satisfying $p_{j+N}=\bar{p}_{j}$, $\alpha_{j+N}=\bar{\alpha}_{j}$,
$\bar{p_{j}}$ and $\bar{\alpha}_{j}$ represent the complex conjugates of
$p_{j}$ and ${\alpha}_{j}$, respectively. The same notation will be used
hereafter.

The proof of the Theorem is given in Appendix. Combined with dependent and
hodograph transformations (67)–(68), the above pfaffians (74)–(75) give
$N$-soliton solution to the complex short pulse equation (2) in parametric
form.

###  4.2 One- and two-soliton solutions for the complex short pulse equation

In this subsection, we provide one- and two-soliton to the complex short pulse
equation (2) and give a detailed analysis for their properties.

####  4.2.1 One-soliton solution

Based on (74)–(75), the tau-functions for one-soliton solution ($N=1$) are

$\displaystyle
f=-1-\frac{1}{4}\frac{|\alpha_{1}|^{2}(p_{1}\bar{p}_{1})^{2}}{(p_{1}+\bar{p}_{1})^{2}}e^{\eta_{1}+\bar{\eta}_{1}}\,,$
(79) $g=-\alpha_{1}e^{\eta_{1}}\,.$ (80)

Let $p_{1}=p_{1R}+\mathrm{i}p_{1I}$, and we assume $p_{1R}>0$ without loss of
generality, then the one-soliton solution can be expressed in the following
parametric form

$q=\frac{\alpha_{1}}{|\alpha_{1}|}\frac{2p_{1R}}{|p_{1}|^{2}}e^{\mathrm{i}\eta_{1I}}\mbox{sech}\left(\eta_{1R}+\eta_{10}\right)\,,$
(81)
$x=y-\frac{2p_{1R}}{|p_{1}|^{2}}\left(\tanh\left(\eta_{1R}+\eta_{10}\right)+1\right)\,,\quad
t=-s\,,$ (82)

where

$\eta_{1R}=p_{1R}y+\frac{p_{1R}}{|p_{1}|^{2}}s,\quad\eta_{1I}=p_{1I}y-\frac{p_{1I}}{|p_{1}|^{2}}s\,,\quad\eta_{10}=\ln\frac{|\alpha_{1}||p_{1}|^{2}}{4p_{1R}}\,.$
(83)

Eq. (81) represents an envelope soliton of amplitude $2p_{1R}/|p_{1}|^{2}$ and
phase $\eta_{1I}$. To analyze the property for the one-soliton solution, we
calculate out

$\frac{\partial x}{\partial
y}=1-\frac{2p^{2}_{1R}}{|p_{1}|^{2}}{\mbox{sech}}^{2}(\eta_{1R}+\eta_{10})\,.$
(84)

Therefore, $\partial x/\partial y\to 1$ as $y\to\pm\infty$. Moreover, it
attains a minimum value of
$({p^{2}_{1I}-p^{2}_{1R}})/({p^{2}_{1I}+p^{2}_{1R}})$ at the peak point of
envelope soliton where $\eta_{1R}+\eta_{10}=0$. Since ${\partial|q|}/{\partial
x}=\frac{\partial|q|/\partial y}{\partial x/\partial y}$, we can classify this
one-soliton solution as follows:

  * 1.

smooth soliton: when $|p_{1R}|<|p_{1I}|$, ${\partial x}/{\partial y}$ is
always positive, which leads to a smooth envelope soliton similar to the
envelope soliton for the nonlinear Schrödinger equation. An example with
$p_{1}=1+1.5\mathrm{i}$ is illustrated in Fig. 1 (a).

  * 2.

loop soliton: when $|p_{1R}|>|p_{1I}|$, the minimum value of ${\partial
x}/{\partial y}$ at the peak point of the soliton becomes negative. In view of
the fact that $\partial x/\partial y\to 1$ as $y\to\pm\infty$, ${\partial
x}/{\partial y}$ has two zeros at both sides of the peak of the envelope
soliton. Moreover, ${\partial x}/{\partial y}<0$ between these two zeros. This
leads to a loop soliton for the envelope of $q$. An example is shown in Fig.
(b) with $p_{1}=1+0.5\mathrm{i}$.

  * 3.

cuspon soliton: when $|p_{1R}|=|p_{1I}|$, ${\partial x}/{\partial y}$ has a
minimum value of zero at $\eta_{1R}+\eta_{10}=0$, which makes the derivative
of the envelope $|q|$ with respect to $x$ going to infinity at the peak point.
Thus, we have a cusponed envelope soliton, which is illustrated in Fig. 1 (c)
with $p_{1}=1+\mathrm{i}$.

(a)(b)

(c)

Figure 1: Envelope soliton for the complex short pulse equation (2), solid
line: $Re(q)$, dashed line: $|q|$; (a) smooth soliton with
$p_{1}=1+1.5\mathrm{i}$, (b) loop soliotn with $p_{1}=1+0.5\mathrm{i}$, (c)
cuspon soliton with $p_{1}=1+\mathrm{i}$.

Remark 4.3. The one-soliton solution to the short pulse equation (1) is of
loop-type, which lacks physical meaning in the context of nonlinear optics.
However, the one-soliton solution to the complex short pulse equation (2) is
of breather-type, which allows physical meaning for optical pulse.

Remark 4.4. When $|p_{1R}|<|p_{1I}|$, there is no singularity for one-soliton
solution. Moreover, in view of $\eta_{1R}$ associated with the width of
envelope soliton and $\eta_{1I}$ associated with the phase, it is obvious that
this nonsingular envelope soliton can only contain a few optical cycle. This
property coincides with the fact that the complex short pulse equation is
derived for the purpose of describing ultra-short pulse propagation. When
$|p_{1R}|=|p_{1I}|$, the soliton becomes cuspon-like one, which agrees with
the results in [10] derived from a bidirectional model.

####  4.2.2 Two-soliton solution

Based on the $N$-soliton solution of the complex short pulse equation from
(74)–(75), the tau-functions for two-soliton solution can be expanded for
$N=2$

$\displaystyle f=\mathrm{Pf}(a_{1},a_{2},a_{3},a_{4},b_{1},b_{2},b_{3},b_{4})$
$\displaystyle\quad=1+a_{1\bar{1}}e^{\eta_{1}+\bar{\eta}_{1}}+a_{1\bar{2}}e^{\eta_{1}+\bar{\eta}_{2}}+a_{2\bar{1}}e^{\eta_{2}+\bar{\eta}_{1}}+a_{2\bar{2}}e^{\eta_{2}+\bar{\eta_{2}}}$
$\displaystyle\qquad+|P_{12}|^{2}\left(a_{1\bar{1}}a_{2\bar{2}}P_{1\bar{2}}P_{2\bar{1}}-a_{1\bar{2}}a_{2\bar{1}}P_{1\bar{1}}P_{2\bar{2}}\right)e^{\eta_{1}+\eta_{2}+\bar{\eta}_{1}+\bar{\eta}_{2}}\,,$
(85)

$\displaystyle
g=\mathrm{Pf}(d_{0},\beta_{1},a_{1},a_{2},a_{3},a_{4},b_{1},b_{2},b_{3},b_{4})$
$\displaystyle\quad=\alpha_{1}e^{\eta_{1}}+\alpha_{2}e^{\eta_{2}}+P_{12}\left(\alpha_{1}P_{1\bar{1}}a_{2\bar{1}}-\alpha_{2}P_{2\bar{1}}a_{1\bar{1}}\right)e^{\eta_{1}+\eta_{2}+\bar{\eta}_{1}}$
$\displaystyle\qquad+P_{12}\left(\alpha_{1}P_{1\bar{2}}a_{2\bar{2}}-\alpha_{2}P_{2\bar{2}}a_{1\bar{2}}\right)e^{\eta_{1}+\eta_{2}+\bar{\eta}_{2}}\,,$
(86)

where

$P_{ij}=\frac{p_{i}-p_{j}}{p_{i}+p_{j}}\,,\quad
P_{i\bar{j}}=\frac{p_{i}-\bar{p}_{j}}{p_{i}+\bar{p}_{j}}\,,\quad
a_{i\bar{j}}=\frac{\alpha_{i}\bar{\alpha}_{j}(p_{i}\bar{p}_{j})^{2}}{4(p_{i}+\bar{p}_{j})^{2}}\,,$
(87)

and $\eta_{j}=p_{j}y+p_{j}^{-1}s$,
$\bar{\eta}_{j}=\bar{p}_{j}y+\bar{p}_{j}^{-1}s$.

(a)(b)

Figure 2: Two-soliton solution to the complex short pulse equation (a) contour
plot; (b) profiles at $t=-80$, $80$.

To avoid the singularity of the envelope solitons, the conditions
$|p_{1R}|<|p_{1I}|$ and $|p_{2R}|<|p_{2I}|$ need to be satisfied. When two
solitons stay apart, the amplitude of each soliton is of
$2|p_{iR}|/|p_{i}|^{2}$, and the velocity is of $-1/|p_{i}|^{2}$ in the
$ys$-coordinate system. Therefore, the soliton of larger velocity will catch
up with and collide with the soliton of smaller velocity if it is initially
located on the left. Furthermore, the collision is elastic, and there is no
change in shape and amplitude of solitons except a phase shift. In Fig. 2, we
illustrate the contour plot for the collision of two solitons (a), as well as
the profiles (b) before and after the collision. The parameters are taken as
$\alpha_{1}=\alpha_{2}=1.0$, $p_{1}=1+1.2\mathrm{i}$ and
$p_{2}=1+2\mathrm{i}$.

Since the velocity of single envelope soliton is $-1/|p_{i}|^{2}$ in the
$ys$-coordinate system, a bound state can be formed under the condition of
$|p_{1}|^{2}=|p_{2}|^{2}$ if two solitons stay close enough and move with the
same velocity. Such a bound state is shown in Fig. 3 for parameters chosen as
$\alpha_{1}=\alpha_{2}=1.0$, $p_{1}=1.3+1.8193\mathrm{i}$,
$p_{2}=1+2\mathrm{i}$. It is interesting that the envelope of the bound state
oscillates periodically as it moves along $x$-axis.

(a)(b)

Figure 3: Bound state to the complex short pulse equation: (a) 3D plot (b)
profiles at $t=-100$, $40$.

###  4.3 Bilinear equations and $N$-soliton solutions to the coupled complex
short pulse equation

Proposition 4.5. The coupled complex short pulse equation is derived from
bilinear equations

$D_{s}D_{y}f\cdot g_{i}=fg_{i},\quad i=1,2\,,$ (88) $D^{2}_{s}f\cdot
f=\frac{1}{2}\left(|g_{1}|^{2}+|g_{2}|^{2}\right)\,,$ (89)

by dependent variable transformation

$q_{1}=\frac{g_{1}}{f},\quad q_{2}=\frac{g_{2}}{f}\,,$ (90)

and hodograph transformation

$x=y-2(\ln f)_{s}\,,\quad t=-s\,,$ (91)

Proof. Dividing both sides of Eqs. (88)–(89) by $f^{2}$, we have

$\left(\frac{g_{i}}{f}\right)_{sy}+2\frac{g_{i}}{f}\left(\ln
f\right)_{sy}=\frac{g_{i}}{f}\,,$ (92) $\left(\ln
f\right)_{ss}=\frac{1}{4}\left(\frac{|g_{1}|^{2}}{f^{2}}+\frac{|g_{2}|^{2}}{f^{2}}\right)\,.$
(93)

From dependent variable and hodograph transformations (90)–(91), we obtain

$\frac{\partial x}{\partial s}=-2(\ln
f)_{ss}=-\frac{1}{2}\left(|q_{1}|^{2}+|q_{2}|^{2}\right)\,,\qquad\frac{\partial
x}{\partial y}=1-2(\ln f)_{sy}\,,$

which implies

${\partial_{y}}=\rho^{-1}{\partial_{x}}\,,\qquad{\partial_{s}}=-{\partial_{t}}-\frac{1}{2}\left(|q_{1}|^{2}+|q_{2}|^{2}\right){\partial_{x}}\,$
(94)

by letting $1-2(\ln f)_{sy}=\rho^{-1}$.

With the use of (94), Eq. (92) can be recast into

$\rho\left(\frac{g_{i}}{f}\right)_{sy}=\frac{g_{i}}{f}\,,\quad i=1,2\,,$ (95)

which can be further converted into

$\partial_{x}\left(-\partial_{t}-\frac{1}{2}(|q_{1}|^{2}+|q_{2}|^{2})\partial_{x}\right)q_{i}=q_{i}\,,\quad
i=1,2\,.$ (96)

Eq. (96) is, obviously, equivalent to the coupled complex short pulse equation
(3)–(4).

$N$-soliton solution for the coupled complex short pulse equation is given in
a similar way as the complex short pulse equation by the following theorem.

Theorem 4.6. The coupled complex short pulse equation admits the following
$N$-soliton solution

$q_{i}=\frac{g_{i}}{f},\quad x=y-2(\ln f)_{s}\,,\quad t=-s\,,$

where $f$, $g_{i}$ are pfaffians defined as

$\displaystyle f$ $\displaystyle=$
$\displaystyle\mathrm{Pf}(a_{1},\cdots,a_{2N},b_{1},\cdots,b_{2N})\,,$ (97)
$\displaystyle g_{i}$ $\displaystyle=$
$\displaystyle\mathrm{Pf}(d_{0},\beta_{i},a_{1},\cdots,a_{2N},b_{1},\cdots,b_{2N})\,,$
(98)

and the elements of the pfaffians are determined as

$\mathrm{Pf}(a_{j},a_{k})=\frac{p_{j}-p_{k}}{p_{j}+p_{k}}e^{\eta_{j}+\eta_{k}}\,,\quad\mathrm{Pf}(a_{j},b_{k})=\delta_{j,k}\,,$
(99)
$\mathrm{Pf}(b_{j},b_{k})=\frac{1}{4}\frac{\sum^{2}_{i=1}\alpha^{(i)}_{j}\alpha^{(i)}_{k}}{p^{-2}_{j}-p^{-2}_{k}}\delta_{\mu+1,\nu}\,,\quad\mathrm{Pf}(d_{l},a_{k})=p_{k}^{l}e^{\eta_{k}}\,,$
(100)

$\mathrm{Pf}(b_{j},\beta_{i})=\alpha^{(i)}_{j}\delta_{\mu,i}\,,\quad\mathrm{Pf}(d_{0},b_{j})=\mathrm{Pf}(d_{0},\beta_{i})=\mathrm{Pf}(a_{j},\beta_{i})=0\,.$
(101)

Here $\mu=index(b_{j})$, $\nu=index(b_{k})$,
$\eta_{j}=p_{j}y+p_{j}^{-1}s+\eta_{j,0}$ which satisfying
$p_{j+N}=\bar{p}_{j}$, $\alpha_{j+N}=\bar{\alpha}_{j}$.

The proof of the Theorem is given in the Appendix. In the subsequent section,
based on the $N$-soliton solution of coupled complex short pulse equation, we
will investigate the dynamics of one- and two-solitons in details.

Remark 4.7. Through the transformations

$x=y-2(\ln f)_{s}\,,\quad t=-s\,,\quad q_{i}=\frac{g_{i}}{f}\,,$ (102)

the vector complex short pulse equation (43) can be decomposed into the
following bilinear equations

$D_{s}D_{y}f\cdot g_{i}=fg_{i},\quad i=1,\cdots,n\,,$ (103) $D^{2}_{s}f\cdot
f=\frac{1}{2}\left(\sum^{n}_{i=1}|g_{i}|^{2}\right)\,.$ (104)

The parametric form of $N$-soliton solution in terms of pfaffians to the
vector complex short pulse equation (43) can be given in a very similar from
as to to the coupled complex short pulse equation. Here, we omit the details
and will report the results later on.

##  5 Dynamics of solitons to the coupled complex short pulse equation

###  5.1 One-soliton solution

The tau-functions for one-soliton solution to the coupled complex short pulse
equation are obtained from (97)–(98) for $N=1$

$f=-1-\frac{1}{4}\frac{\sum_{i=1}^{2}|\alpha^{(i)}_{1}|^{2}(p_{1}\bar{p}_{1})^{2}}{(p_{1}+\bar{p}_{1})^{2}}e^{\eta_{1}+\bar{\eta}_{1}}\,,$
(105) $g_{1}=-\alpha^{(1)}_{1}e^{\eta_{1}}\,,\quad
g_{2}=-\alpha^{(2)}_{1}e^{\eta_{1}}\,.$ (106)

Let $p_{1}=p_{1R}+\mathrm{i}p_{1I}$, the one-soliton solution can be expressed
in the following parametric form

$\left(\begin{array}[]{c}q_{1}\\\
q_{2}\end{array}\right)=\left(\begin{array}[]{c}A_{1}\\\
A_{2}\end{array}\right)\frac{2p_{1R}}{|p_{1}|^{2}}e^{\mathrm{i}\eta_{1I}}{\mbox{sech}}\left(\eta_{1R}+\eta_{10}\right)\,,$
(107)
$x=y-\frac{2p_{1R}}{|p_{1}|^{2}}\left(\tanh(\eta_{1R}+\eta_{10})+1\right)\,,\quad
t=-s\,,$ (108)

where

$\eta_{1R}=p_{1R}\left(y+\frac{1}{|p_{1}|^{2}}s\right),\quad\eta_{1I}=p_{1I}\left(y-\frac{1}{|p_{1}|^{2}}s\right)\,,$
(109)
$A_{i}=\frac{\alpha_{1}^{(i)}}{\sqrt{\sum_{i=1}^{2}|\alpha_{1}^{(i)}|^{2}}}\,,\quad\eta_{10}=\ln\frac{\sqrt{\sum_{i=1}^{2}|\alpha_{1}^{(i)}|^{2}}|p_{1}|^{2}}{4|p_{1R}|}\,.$
(110)

The amplitudes of the single soliton in each component are
${2|A_{1}|p_{1R}}/{|p_{1}|^{2}}$ and ${2|A_{2}|p_{1R}}/{|p_{1}|^{2}}$,
respectively. Note that $|A_{1}|^{2}+|A_{2}|^{2}=1$. Same as the analysis for
one-soliton solution of complex short pulse equation, if $|p_{1R}|<|p_{1I}|$,
the envelope for one-soliton in each of the component is smooth, whereas, if
$|p_{1R}|>|p_{1I}|$, it becomes a loop (multi-valued) soliton, if
$|p_{1R}|=|p_{1I}|$, it is a cuspon.

###  5.2 Soliton interactions

Two-soliton solution for coupled complex short pulse equation is obtained from
(97)–(98) for $N=2$. By expanding the pfaffians, the tau-functions for two-
soliton solution are expressed by

$\displaystyle
f=1+e^{\eta_{1}+\bar{\eta}_{1}+r_{1\bar{1}}}+e^{\eta_{1}+\bar{\eta}_{2}+r_{1\bar{2}}}+e^{\eta_{2}+\bar{\eta}_{1}+r_{2\bar{1}}}+e^{\eta_{2}+\bar{\eta}_{2}+r_{2\bar{2}}}$
$\displaystyle\qquad+|P_{12}|^{2}|P_{1\bar{2}}|^{2}P_{1\bar{1}}P_{2\bar{2}}\left(B_{1\bar{1}}B_{2\bar{2}}-B_{2\bar{1}}B_{1\bar{2}}\right)e^{\eta_{1}+\eta_{2}+\bar{\eta}_{1}+\bar{\eta}_{2}}\,,$
(111)

$\displaystyle
g_{1}=\alpha^{(1)}_{1}e^{\eta_{1}}+\alpha^{(1)}_{2}e^{\eta_{2}}+P_{12}P_{1\bar{1}}P_{2\bar{1}}\left(\alpha^{(1)}_{2}B_{1\bar{1}}-\alpha^{(1)}_{1}B_{2\bar{1}}\right)e^{\eta_{1}+\eta_{2}+\bar{\eta}_{1}}$
$\displaystyle\qquad+P_{12}P_{1\bar{2}}P_{2\bar{2}}\left(\alpha^{(1)}_{2}B_{1\bar{2}}-\alpha^{(1)}_{1}B_{2\bar{2}}\right)e^{\eta_{1}+\eta_{2}+\bar{\eta}_{2}}\,,$
(112)

$\displaystyle
g_{2}=\alpha_{1}^{(2)}e^{\eta_{1}}+\alpha_{2}^{(2)}e^{\eta_{2}}+P_{12}P_{1\bar{1}}P_{2\bar{1}}\left(\alpha_{2}^{(2)}B_{1\bar{1}}-\alpha_{1}^{(2)}B_{2\bar{1}}\right)e^{\eta_{1}+\eta_{2}+\bar{\eta}_{1}}$
$\displaystyle\qquad+P_{12}P_{1\bar{2}}P_{2\bar{2}}\left(\alpha_{2}^{(2)}B_{1\bar{2}}-\alpha_{1}^{(2)}B_{2\bar{2}}\right)e^{\eta_{1}+\eta_{2}+\bar{\eta}_{2}}\,,$
(113)

where

$P_{ij}=\frac{p_{i}-p_{j}}{p_{i}+p_{j}}\,,\quad
P_{i\bar{j}}=\frac{p_{i}-\bar{p}_{j}}{p_{i}+\bar{p}_{j}}\,,$
$B_{i\bar{j}}=\frac{\alpha_{i}^{(1)}\bar{\alpha}_{j}^{(1)}+\alpha_{i}^{(2)}\bar{\alpha}_{j}^{(2)}}{4(p_{i}^{-2}-\bar{p}_{j}^{-2})}\,,\quad
e^{r_{i\bar{j}}}=\frac{\alpha_{i}^{(1)}\bar{\alpha}_{j}^{(1)}+\alpha_{i}^{(2)}\bar{\alpha}_{j}^{(2)}}{4(p_{i}^{-1}+\bar{p}_{j}^{-1})^{2}}\,.$

and $\eta_{j}=p_{j}y+p_{j}^{-1}s$, $p_{3}=\bar{p}_{1}$, $p_{4}=\bar{p}_{2}$,
thus, $\eta_{3}=\bar{\eta}_{1}$, $\eta_{4}=\bar{\eta}_{2}$.

Next, we investigate the asymptotic behavior of two-soliton solution. To this
end, we assume $p_{1R}>p_{2R}>0$, $p_{1R}/|p_{1}|^{2}>p_{2R}/|p_{2}|^{2}$
without loss of generality. For the above choice of parameters, we have (i)
$\eta_{1R}\approx 0$, $\eta_{2R}\rightarrow\mp\infty$ as
$t\rightarrow\mp\infty$ for soliton 1 and (ii) $\eta_{2R}\approx 0$,
$\eta_{2R}\rightarrow\pm\infty$ as $t\rightarrow\mp\infty$ for soliton 2\.
This leads to the following asymptotic forms for two-soliton solution.
(i) Before collision ($t\rightarrow-\infty$)
Soliton 1 ($\eta_{1R}\approx 0$, $\eta_{2R}\rightarrow-\infty$):

$\displaystyle\left(\begin{array}[]{c}q_{1}\\\ q_{2}\end{array}\right)$
$\displaystyle\rightarrow$
$\displaystyle\left(\begin{array}[]{c}\alpha_{1}^{(1)}\\\
\alpha_{1}^{(2)}\end{array}\right)\frac{e^{\eta_{1}}}{1+e^{\eta_{1}+\bar{\eta}_{1}+r_{1\bar{1}}}}\,,$
(118) $\displaystyle\rightarrow$
$\displaystyle\left(\begin{array}[]{c}A_{1}^{1-}\\\
A_{2}^{1-}\end{array}\right)\frac{2p_{1R}}{|p_{1}|^{2}}e^{i\eta_{1I}}{\mbox{sech}}\left(\eta_{1R}+\frac{r_{1\bar{1}}}{2}\right)\,,$
(121)

where

$\left(\begin{array}[]{c}A_{1}^{1-}\\\
A_{2}^{1-}\end{array}\right)=\left(\begin{array}[]{c}\alpha_{1}^{(1)}\\\
\alpha_{1}^{(2)}\end{array}\right)\frac{1}{\sqrt{|\alpha_{1}^{(1)}|^{2}+|\alpha_{1}^{(2)}|^{2}}}\,.$
(122)

Soliton 2 ($\eta_{2R}\approx 0$, $\eta_{1R}\to\infty$):

$\left(\begin{array}[]{c}q_{1}\\\
q_{2}\end{array}\right)\to\left(\begin{array}[]{c}A^{2-}_{1}\\\
A^{2-}_{2}\end{array}\right)\frac{2p_{2R}}{|p_{2}|^{2}}e^{i\eta_{2I}}{\mbox{sech}}\left(\eta_{2R}+\frac{r_{1\bar{1}2\bar{2}}-r_{1\bar{1}}}{2}\right)\,,$
(123)

where

$\left(\begin{array}[]{c}A^{2-}_{1}\\\
A^{2-}_{2}\end{array}\right)=\left(\begin{array}[]{c}e^{r^{(1)}_{1\bar{1}2}}\\\
e^{r^{(2)}_{1\bar{1}2}}\end{array}\right)\frac{e^{-(r_{1\bar{1}2\bar{2}}+r_{1\bar{1}}-r_{2\bar{2}})/{2}}}{\sqrt{|\alpha^{(1)}_{2}|^{2}+|\alpha^{(2)}_{2}|^{2}}}\,,$
(124)

with

$e^{r^{(i)}_{1\bar{1}2}}=P_{12}P_{1\bar{1}}P_{2\bar{1}}\left(\alpha^{(i)}_{2}B_{1\bar{1}}-\alpha^{(i)}_{1}B_{2\bar{1}}\right)\,,\quad(i=1,2)$
(125)
$e^{r_{1\bar{1}2\bar{2}}}=|P_{12}|^{2}|P_{1\bar{2}}|^{2}P_{1\bar{1}}P_{2\bar{2}}\left(B_{1\bar{1}}B_{2\bar{2}}-B_{2\bar{1}}B_{1\bar{2}}\right)\,.$
(126)

After collision ($t\to\infty$)
Soliton 1 ($\eta_{1R}\approx 0$, $\eta_{2R}\to\infty$):

$\left(\begin{array}[]{c}q_{1}\\\
q_{2}\end{array}\right)\rightarrow\left(\begin{array}[]{c}A_{1}^{1+}\\\
A_{2}^{1+}\end{array}\right)\frac{2p_{1R}}{|p_{1}|^{2}}e^{i\eta_{1I}}{\mbox{sech}}\left(\eta_{2R}+\frac{r_{1\bar{1}2\bar{2}}-r_{2\bar{2}}}{2}\right)\,,$
(127)

where

$\left(\begin{array}[]{c}A_{1}^{1+}\\\
A_{2}^{1+}\end{array}\right)=\left(\begin{array}[]{c}e^{r_{12\bar{1}}^{(1)}}\\\
e^{r_{12\bar{1}}^{(2)}}\end{array}\right)\frac{e^{-(r_{1\bar{1}2\bar{2}}-r_{1\bar{1}}+r_{2\bar{2}})/{2}}}{\sqrt{|\alpha_{1}^{(1)}|^{2}+|\alpha_{1}^{(2)}|^{2}}}\,,$
(128)

with

$e^{r_{12\bar{1}}^{(i)}}=P_{12}P_{1\bar{2}}P_{2\bar{2}}\left(\alpha_{2}^{(i)}B_{1\bar{2}}-\alpha_{1}^{(i)}B_{2\bar{2}}\right)\,,\quad(i=1,2)\,.$
(129)

Soliton 2 ($\eta_{2R}\approx 0$, $\eta_{1R}\rightarrow-\infty$):

$\left(\begin{array}[]{c}q_{1}\\\
q_{2}\end{array}\right)\rightarrow\left(\begin{array}[]{c}A_{1}^{2+}\\\
A_{2}^{2+}\end{array}\right)\frac{2p_{2R}}{|p_{2}|^{2}}e^{i\eta_{2I}}{\mbox{sech}}\left(\eta_{2R}+\frac{r_{2\bar{2}}}{2}\right)\,,$
(130)

where

$\left(\begin{array}[]{c}A_{1}^{2+}\\\
A_{2}^{2+}\end{array}\right)=\left(\begin{array}[]{c}\alpha_{2}^{(1)}\\\
\alpha_{2}^{(2)}\end{array}\right)\frac{1}{\sqrt{|\alpha_{2}^{(1)}|^{2}+|\alpha_{2}^{(2)}|^{2}}}\,.$
(131)

Similar to the analysis for the CNLS equations [38, 39, 40], the change in the
amplitude of each of the solitons in each component can be obtained by
introducing the transition matrix $T^{k}_{j}$ by
$A_{j}^{k+}=T_{j}^{k}A_{j}^{k-}$, $j,k=1,2$. The elements of transition matrix
is obtained from the above asymptotic analysis as

$T_{j}^{1}=\left(\frac{P_{12}P_{1\bar{2}}}{\bar{P}_{12}\bar{P}_{1\bar{2}}}\right)^{1/2}\frac{1}{\sqrt{1-\lambda_{1}\lambda_{2}}}\left(1-\lambda_{2}\frac{\alpha_{2}^{(j)}}{\alpha_{1}^{(j)}}\right)\,,\quad
j=1,2\,,$ (132)
$T_{j}^{2}=\left(\frac{\bar{P}_{12}P_{1\bar{2}}}{P_{12}\bar{P}_{1\bar{2}}}\right)^{1/2}\sqrt{1-\lambda_{1}\lambda_{2}}\left(1-\lambda_{1}\frac{\alpha_{1}^{(j)}}{\alpha_{2}^{(j)}}\right)^{-1}\,,\quad
j=1,2\,,$ (133)

where $\lambda_{1}=B_{2\bar{1}}/B_{1\bar{1}}$,
$\lambda_{2}=B_{1\bar{2}}/B_{2\bar{2}}$.

Therefore, in general, there is an exchange of energies between two components
of two solitons after the collision. An example is shown in Fig. 4 for the
parameters taken as follows $p_{1}=1+1.2\mathrm{i}$, $p_{2}=1+2\mathrm{i}$,
$\alpha^{(1)}_{1}=\alpha^{(2)}_{1}=1.0$, $\alpha^{(1)}_{2}=2.0$,
$\alpha^{(2)}_{2}=1.0$.

(a)(b)

(c)(d)

Figure 4: Inelastic collision in coupled complex short pulse equation.
(a)-(b): contour plot; (c)-(d): profiles before and after the collision.

However, only for the special case

$\frac{\alpha_{1}^{(1)}}{\alpha_{2}^{(1)}}=\frac{\alpha_{1}^{(2)}}{\alpha_{2}^{(2)}}\,,$
(134)

there is no energy exchange between two compoents of solitons after the
collision. An example is shown in Fig. 5 for the parameters
$p_{1}=1+1.2\mathrm{i}$, $p_{2}=1+2\mathrm{i}$,
$\alpha^{(1)}_{1}=\alpha^{(2)}_{1}=1.0$,
$\alpha^{(1)}_{2}=\alpha^{(2)}_{2}=1.0$.

(a)(b)

(c)(d)

Figure 5: Elastic collision in coupled complex short pulse equation.

It is interesting to note that if we just change the parameters in previous
two examples as $\alpha^{(1)}_{2}=0$, $\alpha^{(2)}_{2}=1.0$, the energy of
one soliton is concentrated in component $q_{2}$ before the collision.
However, component $q_{1}$ gains some energy after the collision. Such an
example is shown in Fig. 6.

(a)(b)

(c)(d)

Figure 6: Inelastic collision in coupled complex short pulse equation for
$p_{1}=1+1.2{\rm i}$, $p_{2}=1+2{\rm i}$,
$\alpha^{(1)}_{1}=\alpha^{(2)}_{1}=1.0$, $\alpha^{(1)}_{2}=0$,
$\alpha^{(2)}_{2}=1.0$. (a)-(b): contour plot; (c)-(d): profiles before and
after the collision.

On the other hand, if we change the parameters as $\alpha^{(1)}_{2}=1.0$,
$\alpha^{(2)}_{2}=0$, then the energy of one soliton, which are distributed
between two components before the collision is concentrated into one component
$q_{2}$ after the collision. The example is shown in Fig. 7.

(a)(b)

(c)(d)

Figure 7: Inelastic collision in coupled complex short pulse equation for
$p_{1}=1+1.2\mathrm{i}$, $p_{2}=1+2\mathrm{i}$,
$\alpha^{(1)}_{1}=\alpha^{(2)}_{1}=1.0$, $\alpha^{(1)}_{2}=1.0$,
$\alpha^{(2)}_{2}=0$. (a)-(b): contour plot; (c)-(d): profiles before and
after the collision.

##  6 Concluding Remarks

In this paper, we proposed a complex short pulse equation and its two-
component generalization. Both of the equations can be used to model the
propagation of ultra-short pulses in optical fibers. We have shown their
integrability by finding the Lax pairs and infinite numbers of conservation
laws. Furthermore, multi-soliton solutions are constructed via Hirota’s
bilinear method. In particular, one-soliton solution for the CSP equation is
an envelope soliton with a few optical cycles under certain condition, which
perfectly match the requirement for the ultra-short pulses. The $N$-solution
for complex short pulse equation and its two-component generalization is a
benchmark for the study of soliton interactions in ultra-short pulses
propagation in optical fibers. It is expected that these analytical solutions
can be confirmed from experiments.

Similar to our previous results for the integrable discretizations of the
short pulse equation [22], how to construct integrable discretizations of the
CSP and coupled CSP equations and how to apply them for the numerical
simulations is also an interesting topic to be studied. It is obviously beyond
the scope of the present paper, we are to report the results on this aspect in
a forthcoming paper.

## Appendix

Proof of Theorem 4.2

Proof. First we define

$(b_{j},\bar{\beta}_{1})=\bar{\alpha}_{j}\delta_{\mu,1}\,,\quad(b_{j},\bar{\beta}_{2})=\bar{\alpha}_{j}\delta_{\mu,2}\,,$

where $index(b_{j})=\mu$ , then from the fact

$\mathrm{Pf}(\bar{a}_{j},a_{k})=\mathrm{Pf}(a_{N+j},a_{N+k})\,,\mathrm{Pf}(\bar{b}_{j},b_{k})=\mathrm{Pf}(b_{N+j},b_{N+k})\,,$

we obtain

$\bar{f}=f\,,\quad\bar{g}=\mathrm{Pf}(d_{0},\bar{\beta}_{1},a_{1},\cdots,a_{2N},b_{1},\cdots,b_{2N})\,.$

Since

$\frac{\partial}{\partial
y}\mathrm{Pf}(a_{j},a_{k})=(p_{j}-p_{k})e^{\eta_{j}+\eta_{k}}=\mathrm{Pf}(d_{0},d_{1},a_{j},a_{k})\,,$

$\frac{\partial}{\partial
s}\mathrm{Pf}(a_{j},a_{k})=(p^{-1}_{k}-p^{-1}_{j})e^{\eta_{j}+\eta_{k}}=\mathrm{Pf}(d_{-1},d_{0},a_{j},a_{k})\,,$

$\frac{\partial^{2}}{\partial
s^{2}}\mathrm{Pf}(a_{j},a_{k})=(p^{-2}_{k}-p^{-2}_{j})e^{\eta_{j}+\eta_{k}}=\mathrm{Pf}(d_{-2},d_{0},a_{j},a_{k})\,,$
$\frac{\partial^{2}}{\partial y\partial
s}\mathrm{Pf}(a_{j},a_{k})=(p_{j}p^{-1}_{k}-p_{k}p^{-1}_{j})e^{\eta_{j}+\eta_{k}}=\mathrm{Pf}(d_{-1},d_{1},a_{i},a_{j})\,,$

we then have

$\frac{\partial f}{\partial y}=\mathrm{Pf}(d_{0},d_{1},\cdots)\,,$

$\frac{\partial f}{\partial s}=\mathrm{Pf}(d_{-1},d_{0},\cdots)\,,$

$\frac{\partial^{2}f}{\partial s^{2}}=\mathrm{Pf}(d_{-2},d_{0},\cdots)\,,$

$\frac{\partial^{2}f}{\partial y\partial
s}=\mathrm{Pf}(d_{-1},d_{1},\cdots)\,.$

Here $\mathrm{Pf}(d_{0},d_{1},a_{1},\cdots,a_{2N},b_{1},\cdots,b_{2N})$ is
abbreviated by $\mathrm{Pf}(d_{0},d_{1},\cdots)$, so as other similar
pfaffians.

Furthermore, it can be shown

$\displaystyle\frac{\partial g}{\partial y}=\frac{\partial}{\partial
y}\left[\sum_{j=1}^{2N}(-1)^{j}\mathrm{Pf}(d_{0},a_{j})\mathrm{Pf}(\beta_{1},\cdots,\hat{a}_{j},\cdots)\right]$
$\displaystyle=\sum_{j=1}^{2N}(-1)^{j}\left[\left({\partial_{y}}\mathrm{Pf}(d_{0},a_{j})\right)\mathrm{Pf}(\beta_{1},\cdots,\hat{a}_{j},\cdots)+\mathrm{Pf}(d_{0},a_{j}){\partial_{y}}\mathrm{Pf}(\beta_{1},\cdots,\hat{a}_{j},\cdots)\right]$
$\displaystyle=\sum_{j=1}^{2N}(-1)^{j}\left[\mathrm{Pf}(d_{1},a_{j})\mathrm{Pf}(\beta_{1},\cdots,\hat{a}_{j},\cdots)+\mathrm{Pf}(d_{0},a_{j})\mathrm{Pf}(\beta_{1},d_{0},d_{1},\cdots,\hat{a}_{j},\cdots)\right]$
$\displaystyle=\mathrm{Pf}(d_{1},\beta_{1},\cdots)+\mathrm{Pf}(d_{0},\beta_{1},d_{0},d_{1},\cdots)$
$\displaystyle=\mathrm{Pf}(d_{1},\beta_{1},\cdots)\,.$

Here $\hat{a}_{j}$ means that the index $j$ is omitted. Similarly, we can show

$\frac{\partial g}{\partial s}=\mathrm{Pf}(d_{-1},\beta_{1},\cdots)\,,$

$\displaystyle\frac{\partial^{2}g}{\partial y\partial
s}=\frac{\partial}{\partial
y}\left[\sum_{j=1}^{2N}(-1)^{j}\mathrm{Pf}(d_{-1},a_{j})\mathrm{Pf}(\beta_{1},\cdots,\hat{a}_{j},\cdots)\right]$
$\displaystyle=\sum_{j=1}^{2N}(-1)^{j}\left[\left({\partial_{y}}\mathrm{Pf}(d_{-1},a_{j})\right)\mathrm{Pf}(\beta_{1},\cdots,\hat{a}_{j},\cdots)+\mathrm{Pf}(d_{-1},a_{j}){\partial_{y}}\mathrm{Pf}(\beta_{1},\cdots,\hat{a}_{j},\cdots)\right]$
$\displaystyle=\sum_{j=1}^{2N}(-1)^{j}\left[\mathrm{Pf}(d_{0},a_{j})\mathrm{Pf}(\beta_{1},\cdots,\hat{a}_{j},\cdots)+\mathrm{Pf}(d_{-1},a_{j})\mathrm{Pf}(\beta_{1},d_{0},d_{1},\cdots,\hat{a}_{j},\cdots)\right]$
$\displaystyle=\mathrm{Pf}(d_{0},\beta_{1},\cdots)+\mathrm{Pf}(d_{-1},\beta_{1},d_{0},d_{1},\cdots)\,.$

An algebraic identity of pfaffian [21]

$\displaystyle\mathrm{Pf}(d_{-1},\beta_{1},d_{0},d_{1},\cdots)\mathrm{Pf}(\cdots)=\mathrm{Pf}(d_{-1},d_{0},\cdots)\mathrm{Pf}(d_{1},\beta_{1},\cdots)$
$\displaystyle\quad-\mathrm{Pf}(d_{-1},d_{1},\cdots)\mathrm{Pf}(d_{0},\beta_{1},\cdots)+\mathrm{Pf}(d_{-1},\beta_{1},\cdots)\mathrm{Pf}(d_{0},d_{1},\cdots)\,,$

implies

$({\partial_{s}}{\partial_{y}}g-g)\times
f={\partial_{s}}f\times{\partial_{y}}g-{\partial_{s}}{\partial_{y}}f\times
g+{\partial_{s}}g\times{\partial_{y}}f\,.$

Therefore, the first bilinear equation is approved.

The second bilinear equation can be proved in the same way by Iwao and Hirota
[37].

$\displaystyle\frac{\partial^{2}f}{\partial s^{2}}\times 0-\frac{\partial
f}{\partial s}\frac{\partial f}{\partial s}$
$\displaystyle=\mathrm{Pf}(d_{-2},d_{0},\cdots)\mathrm{Pf}(d_{0},d_{0},\cdots)-\mathrm{Pf}(d_{-1},d_{0},\cdots)\mathrm{Pf}(d_{-1},d_{0},\cdots)$
$\displaystyle=\sum_{i=1}^{2N}(-1)^{i}\mathrm{Pf}(d_{-2},a_{i})\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\sum_{j=1}^{2N}(-1)^{j}\mathrm{Pf}(d_{0},a_{j})\mathrm{Pf}(d_{0},\cdots,\hat{a}_{j},\cdots)$
$\displaystyle-\sum_{i=1}^{2N}(-1)^{i}\mathrm{Pf}(d_{-1},a_{i})\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\sum_{j=1}^{2N}(-1)^{j}\mathrm{Pf}(d_{-1},a_{j})\mathrm{Pf}(d_{0},\cdots,\hat{a}_{j},\cdots)$
$\displaystyle=\sum_{i,j=1}^{2N}(-1)^{i+j}\left[\mathrm{Pf}(d_{-2},a_{i})\mathrm{Pf}(d_{0},a_{j})-\mathrm{Pf}(d_{-1},a_{i})\mathrm{Pf}(d_{-1},a_{j})\right]$
$\displaystyle\quad\times\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{a}_{j},\cdots)$
$\displaystyle=\sum_{i,j=1}^{2N}(-1)^{i+j+1}\left[p_{i}^{-2}+p_{i}^{-1}p_{j}^{-1}\right]\mathrm{Pf}(a_{i},a_{j})\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{a}_{j},\cdots)$

The summation over the second term within the bracket vanishes due to the fact
that

$\displaystyle\sum_{i,j=1}^{2N}(-1)^{i+j+1}p_{i}^{-1}p_{j}^{-1}\mathrm{Pf}(a_{i},a_{j})\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{a}_{j},\cdots)$
$\displaystyle=\sum_{j,i=1}^{2N}(-1)^{j+i+1}p_{j}^{-1}p_{i}^{-1}\mathrm{Pf}(a_{j},a_{i})\mathrm{Pf}(d_{0},\cdots,\hat{a}_{j},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)$
$\displaystyle=-\sum_{i,j=1}^{2N}(-1)^{i+j+1}p_{i}^{-1}p_{j}^{-1}\mathrm{Pf}(a_{i},a_{j})\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{a}_{j},\cdots)\,.$

Therefore,

$\displaystyle-\frac{\partial f}{\partial s}\frac{\partial f}{\partial
s}=\sum_{i,j=1}^{2N}(-1)^{i+j+1}p_{i}^{-2}\mathrm{Pf}(a_{i},a_{j})\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{a}_{j},\cdots)$
$\displaystyle=\sum_{i=1}^{2N}(-1)^{i+1}p_{i}^{-2}\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\left[\sum_{j=1}^{2N}(-1)^{j}\mathrm{Pf}(a_{i},a_{j})\mathrm{Pf}(d_{0},\cdots,\hat{a}_{j},\cdots)\right]$

Further, we note that the following identity can be substituted into the term
within bracket

$\displaystyle\sum_{j=1}^{2N}(-1)^{j}\mathrm{Pf}(a_{i},a_{j})\mathrm{Pf}(d_{0},\cdots,\hat{a}_{j},\cdots)$
$\displaystyle=\mathrm{Pf}(d_{0},a_{i})\mathrm{Pf}(\cdots)+(-1)^{i+1}\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\,$

which is obtained from the expansion of the following vanishing pfaffian
$\mathrm{Pf}(a_{i},d_{0},\cdots)$ on $a_{i}$. Consequently, we have

$\displaystyle-\frac{\partial f}{\partial s}\frac{\partial f}{\partial s}=$
$\displaystyle\sum_{i=1}^{2N}(-1)^{i+1}p_{i}^{-2}\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\left[\mathrm{Pf}(d_{0},a_{i})\mathrm{Pf}(\cdots)+(-1)^{i+1}\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\right]\,,$
$\displaystyle=-\mathrm{Pf}(\cdots)\mathrm{Pf}(d_{-2},d_{0},\cdots)+\sum_{i=1}^{2N}p_{i}^{-2}\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\,,$
(A.1)

which can be rewritten as

$\frac{\partial^{2}f}{\partial s^{2}}f-\frac{\partial f}{\partial
s}\frac{\partial f}{\partial
s}=\sum_{i=1}^{2N}p_{i}^{-2}\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\,.$
(A.2)

Now, we work on the r.h.s of the second bilinear equation.

$\displaystyle\frac{1}{2}|g|^{2}=\frac{1}{2}\mathrm{Pf}(d_{0},\beta_{1},\cdots)\mathrm{Pf}(d_{0},\bar{\beta}_{1},\cdots)$
$\displaystyle=\frac{1}{2}\sum_{i,j}^{2N}(-1)^{i+j}\mathrm{Pf}(b_{i},\beta_{1})\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\mathrm{Pf}(b_{j},\bar{\beta}_{1})\mathrm{Pf}(d_{0},\cdots,\hat{b}_{j},\cdots)$
$\displaystyle=\frac{1}{4}\sum_{i,j}^{2N}(-1)^{i+j}(\alpha_{i}\bar{\alpha}_{j})\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{b}_{j},\cdots)$
$\displaystyle=\sum_{i,j}^{2N}(-1)^{i+j}\left(p_{i}^{-2}-p_{j}^{-2}\right)\mathrm{Pf}(b_{i},b_{j})\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{b}_{j},\cdots)$

Next, the expansion of the vanishing pfaffian
$\mathrm{Pf}(b_{i},d_{0},\cdots)$ on $b_{i}$ yields

$\sum_{j=1}^{2N}(-1)^{i+j}\mathrm{Pf}(b_{i},b_{j})\mathrm{Pf}(d_{0},\cdots,\hat{b}_{j},\cdots)=\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\,,$
(A.4)

which subsequently leads to

$\displaystyle\sum_{i,j}^{2N}(-1)^{i+j}p_{i}^{-2}\mathrm{Pf}(b_{i},b_{j})\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{b}_{j},\cdots)$
$\displaystyle=\sum_{i}^{2N}p_{i}^{-2}\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\,.$
(A.5)

Similarly, we can show that

$\displaystyle-\sum_{i,j}^{2N}(-1)^{i+j}p_{j}^{-2}\mathrm{Pf}(b_{i},b_{j})\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{b}_{j},\cdots)$
$\displaystyle=\sum_{j}^{2N}p_{j}^{-2}\mathrm{Pf}(d_{0},\cdots,\hat{a}_{j},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{b}_{j},\cdots)\,.$
(A.6)

Substituting Eqs. (A.5)–(A.5) into Eq. (Appendix), we arrive at

$\frac{1}{2}|g|^{2}=2\sum_{i}^{2N}p_{i}^{-2}\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\,.$
(A.7)

Consequently we have

$2\frac{\partial^{2}f}{\partial s^{2}}f-2\frac{\partial f}{\partial
s}\frac{\partial f}{\partial s}=\frac{1}{2}|g|^{2}\,,$ (A.8)

which is nothing but the second bilinear equation. Therefore, the proof is
complete.

The proof of Theorem 4.6

Proof. The proof of the first bilinear equation can be done exactly in the
same way as for the complex short pulse equation. In what follows, we prove
the second equation by starting from the r.h.s of this equation. Because

$\bar{g}_{1}=\mathrm{Pf}(d_{0},\bar{\beta}_{1},a_{1},\cdots,a_{2N},b_{1},\cdots,b_{2N})\,,$
$\bar{g}_{2}=\mathrm{Pf}(d_{0},\bar{\beta}_{2},a_{1},\cdots,a_{2N},b_{1},\cdots,b_{2N})\,,$

the r.h.s of the bilinear equation turns out to be

$\displaystyle\frac{1}{2}\left(g_{1}\bar{g}_{1}+g_{2}\bar{g}_{2}\right)$
$\displaystyle=\frac{1}{2}\sum^{2}_{k=1}\sum_{i,j}^{2N}(-1)^{i+j}\mathrm{Pf}(b_{i},\beta_{k})\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\mathrm{Pf}(b_{j},\bar{\beta}_{k})\mathrm{Pf}(d_{0},\cdots,\hat{b}_{j},\cdots)$
$\displaystyle=\frac{1}{4}\sum_{i,j}^{2N}(-1)^{i+j}\sum^{2}_{k=1}(\alpha^{(k)}_{i}\bar{\alpha}^{(k)}_{j})\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{b}_{j},\cdots)$
$\displaystyle=\sum_{i,j}^{2N}(-1)^{i+j}\left(p_{i}^{-2}-p_{j}^{-2}\right)\mathrm{Pf}(b_{i},b_{j})\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{b}_{j},\cdots)$

Similar to the complex short pulse equation, we can show

$\frac{1}{2}\left(|g_{1}|^{2}+|g_{2}|^{2}\right)=2\sum_{i}^{2N}p_{i}^{-2}\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\,.$
(A.10)

Regarding the r.h.s of the bilinear equation, exactly the same as the proof of
the Theorem 4.2, we have

$\frac{\partial^{2}f}{\partial s^{2}}f-\frac{\partial f}{\partial
s}\frac{\partial f}{\partial
s}=\sum_{i}^{2N}p_{i}^{-2}\mathrm{Pf}(d_{0},\cdots,\hat{a}_{i},\cdots)\mathrm{Pf}(d_{0},\cdots,\hat{b}_{i},\cdots)\,.$
(A.11)

Therefore the second bilinear equation is proved.

## Acknowledgements

The author is grateful for the useful discussions with Dr. Yasuhiro Ohta (Kobe
University) and Dr. Kenichi Maruno at Waseda University. This work is
partially supported by the National Natural Science Foundation of China (No.
11428102).

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