# Fokas-Lenells equation dark soliton and gauge equivalent spin equation

Riki Dutta, Sagardeep Talukdar, Gautam Kumar Saharia and
Sudipta Nandy a email<EMAIL_ADDRESS>
Abstract

We propose the Hirota bilinearization of the Fokas-Lenells derivative
nonlinear Schrödinger equation with a non-vanishing background. The bilinear
method is applied using an auxilary function to obtain the dark one soliton
solution, dark two soliton solution and eventually the scheme for obtaining
dark N soliton solutions. The use of auxilary function in bilinearization
makes the method simpler than the ones reported earlier. Later, we have
introduced a Lax pair for this integrable equation and using a transformation
we have shown that this system is gauge equivalent to a spin system, namely
the Landau-Lifshitz equation.

##  1 Introduction

Fokas-Lenells equation (FLE) [1, 2] is an integrable equation which describes
the propagation of ultra-short pulses in an optical medium. FLE equation is a
special case of a more generalized nonlinear Schrödinger equation (NLSE) which
includes higher order effects. There are only a few known integrable equations
of special cases, namely NLSE [3, 4, 5], higher order NLSE (HNLSE) [6, 7],
derivative NLSE (DNLSE) [8] and the fourth equation is the FLE. A common
property in all the equations is the appearance of solitons. Soliton arise as
a result of a balance between the nonlinear and dispersive terms of the wave
equations. While the bright soliton appear as a hump on the zero background of
light, the dark soliton appear as a dip on a continuous background.

The significance of FLE is due to the presence of a spatio-temporal dispersion
term along with a cubic nonlinear self-steepening term which account the
higher order nonlinear effects in an ultra-short optical pulse. While the
study on NLSE, HNLSE and DNLSE is done extensively, the study on FLE is
relatively sparse. A few of the recent notable contributions on FLE are
solitary wave and elliptical solutions of FLE in presence of perturbation and
modulation instability [9], combined optical solitary waves of FLE [10],
dynamical behaviour of soliton solutions of FLE [11], inverse transform of FLE
with non zero boundary condition [12], derivation of solitons of dimensionless
FLE with perturbation term [13] and more.

The dimensionless expression for FLE [1, 2] is

$\displaystyle iU_{t}+a_{1}U_{xx}-a_{2}U_{xt}+b|U|^{2}U+iba_{2}|U|^{2}U_{x}=0$
(1)

where $U$ is the field function that can describe the waveform of an ultra-
short pulse. The suffix $x$ and $t$ denote the partial differentiations with
respect to $x$ and $t$ respectively. $U_{t}$ is the temporal evolution of the
pulse, $U_{xx}$ is the group velocity dispersion, $U_{xt}$ is the spatio-
temporal dispersion, $|U|^{2}U$ is the Kerr nonlinearity effect and
$|U|^{2}U_{x}$ is the cubic nonlinear self-steepening effect of the medium.

Various methods, namely inverse scattering method, Hirota bilinearization and
others [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 13, 26] are proposed
to obtain the different form of soliton solutions. In this manuscript our
objective is to propose an alternate simplified and systematic scheme, namely
bilinearization by introducing an auxiliary function and obtain a generalized
expression for dark and bright dark soliton solutions.

It is interesting to note that the family of NLSE show gauge equivalence with
spin system called Landau-Lifshitz equation (LLE) [27, 28, 29, 30]. FLE also
belongs to this class of nonlinear Schrödinger type equations so it is worth
to investigate the gauge connection of FLE with the spin system. To the best
of our knowledge in the literature no such work has been reported earlier with
FLE.

The structure of this manuscript is that in the following section we consider
a gauge transformation that will convert eqn. 1 into a simplified form. Then
using Hirota bilinearization on this simplified form, we shall derive a dark
soliton solution and multi dark soliton solutions and discuss their properties
and also mention the condition under which we can get bright soliton
solutions. Thereafter in the third section, we propose a gauge transformation
of the Lax pair for this FLE to obtain a spin system which is known as the
Landau-Lifshitz spin system. Fourth section is the concluding one.

##  2 Bilinearization of FLE with vanishing background

Assuming $n=\frac{1}{a_{2}}$ and $m=\frac{a_{1}}{a_{2}}>0$, consider the gauge
transformation

$\displaystyle U=\sqrt{\frac{m}{|b|}}ne^{i(nx+2mnt)}u$ (2)

followed by the transformation of variables ($\xi,\tau$)

$\displaystyle\xi=2(x+mt),\quad\tau=-\frac{mn^{2}}{2}t$ (3)

we get the following equation

$\displaystyle u_{\xi\tau}=u-2i\sigma|u|^{2}u_{\xi}$ (4)

here $u$ is the field function corresponding to the new transformed system and
$\sigma$ = $\frac{b}{|b|}$. Assuming $b$ is positive, we can write eqn. 4 as

$\displaystyle u_{\xi\tau}=u-2i|u|^{2}u_{\xi}$ (5)

notice that eqn. 5 is the first negative hierarchy of DNLSE. For dark soliton
solution of eqn. 5 we assume a non vanishing background condition
$u\rightarrow\rho\ e^{i\ (\kappa\xi+\omega\tau)}$ as
$\xi\rightarrow\pm\infty$. Under this condition we expect a dark soliton
solution with the bilinearization.

To write eqn. 5 in the bilinear form let us assume

$\displaystyle u=\frac{g}{f}$ (6)

where $g$ and $f$ are two complex functions of ($\xi,\tau$). Consequently eqn.
5 becomes

$\frac{1}{f^{2}}(D_{\xi}D_{\tau}-1)g.f-\frac{g}{f^{3}}D_{\xi}D_{\tau}(f.f)+\frac{2i|g|^{2}}{f^{3}f^{*}}D_{\xi}(g.f)+\frac{g\lambda
f.f}{f^{3}}-\frac{\lambda g.f}{f^{2}}+\\\
\frac{s|g|^{2}}{f^{3}}-\frac{s|g|^{2}f^{*}}{f^{3}f^{*}}=0$ (7)

where $D_{\xi}$, $D_{\tau}$ are Hirota derivatives [6] and are defined as

$\displaystyle
D_{\xi}^{m}D_{\tau}^{n}g(\xi,\tau).f(\xi,\tau)=(\frac{\partial}{\partial\xi}-\frac{\partial}{\partial\xi^{\prime}})^{m}(\frac{\partial}{\partial\tau}-\frac{\partial}{\partial\tau^{\prime}})^{n}g(\xi,\tau).f(\xi^{\prime},\tau^{\prime})\Bigg{|}_{(\xi=\xi^{\prime})(\tau=\tau^{\prime})}$
(8)

Notice that the last two terms in eqn. 7 contains an auxiliary function $s$
which is introduced so that the multilinear eqn. 7 can be cast into two
bilinear equations, namely eqns. 10 and 11. Here $\lambda$ is a constant to be
determined by solving the bilinear eqns. 9 \- 11. Consequently, the proposed
bilinear equations in terms of $g$, $f$ and $s$ are

$\displaystyle(D_{\xi}D_{\tau}-1-\lambda)g.f$ $\displaystyle=0$ (9)
$\displaystyle(D_{\xi}D_{\tau}-\lambda)f.f$ $\displaystyle=sg^{*}$ (10)
$\displaystyle 2iD_{\xi}(g.f)$ $\displaystyle=sf^{*}$ (11)

To obtain the soliton solution, $g$ and $f$ are expanded with respect to an
arbitrary parameter $\epsilon$ as follows

$\displaystyle g$ $\displaystyle=g_{0}\
(1+\epsilon^{2}g_{2}+\epsilon^{4}g_{4}+...),\quad\quad\quad
f=1+\epsilon^{2}f_{2}+\epsilon^{4}f_{4}+...$ (12)

and the auxiliary function $s$ is expanded as

$\displaystyle s=s_{0}\ (1+\epsilon^{2}s_{2}+\epsilon^{4}s_{4}+...)$ (13)

###  2.1 Dark one soliton Solution

The dark one soliton solution (1SS) is obtained by dropping terms of order
greater than or equal to $\epsilon^{3}$ in $g$, $f$ and $s$. Thus from eqn. 6
the dark 1SS of eqn. 5 is

$\displaystyle u$
$\displaystyle=\frac{g_{0}(1+\epsilon^{2}g_{2}^{(1)})}{1+\epsilon^{2}f_{2}^{(1)}}\Big{|}_{\epsilon=1}$
(14)

Let us consider the expressions for $g_{0}$, $s_{0}$, $g_{2}^{(1)}$,
$s_{2}^{(1)}$ and $f_{2}^{(1)}$ are as follows

$\displaystyle g_{0}$ $\displaystyle=\rho\ e^{i\ (\kappa\xi+\omega\tau)}$ (15)
$\displaystyle s_{0}$ $\displaystyle=\rho_{s}\ e^{i\ (\kappa\xi+\omega\tau)}$
(16) $\displaystyle g_{2}^{(1)}$ $\displaystyle=K\ e^{\theta+\theta^{*}}$ (17)
$\displaystyle s_{2}^{(1)}$ $\displaystyle=M\ e^{\theta+\theta^{*}}$ (18)
$\displaystyle f_{2}^{(1)}$ $\displaystyle=T\ e^{\theta+\theta^{*}}$ (19)

where $\theta$ = $p\ \xi+\Omega\ \tau$. $p$, $\Omega$, $K$, $M$, $T$ are
complex parameters, $\xi$, $\omega$ are real constants and $\rho$, $\rho_{s}$
are positive constants. Let us consider $p$ = $p_{r}+i\ p_{i}$ and $T$ =
$T_{r}+i\ T_{i}$ where $p_{r}$, $p_{i}$, $T_{r}$ and $T_{i}$ are real. On
substituting the above equations into eqn. 14, we have

$\displaystyle u$ $\displaystyle=\rho\ e^{i\ (\kappa\xi+\omega\tau)}\
\frac{1+K\ e^{\theta+\theta^{*}}}{1+T\ e^{\theta+\theta^{*}}}$ (20)

further putting eqns. 15 \- 19 into eqns. 9 \- 11 yield the following
expressions

$\displaystyle\rho_{s}$ $\displaystyle=-2\ \kappa\rho$ (21)
$\displaystyle\lambda$ $\displaystyle=2\ \kappa\rho^{2}$ (22)
$\displaystyle\omega$ $\displaystyle=-\frac{1}{\kappa}-2\ \rho^{2}$ (23)
$\displaystyle\Omega$ $\displaystyle=\frac{h}{p}$ (24) $\displaystyle M$
$\displaystyle=\frac{T^{2}}{K^{*}}$ (25) $\displaystyle K$
$\displaystyle=\gamma\ T^{*}$ (26)

where $h$ is real and $\gamma$ is complex and are represented as

$\displaystyle\gamma$ $\displaystyle=\frac{\kappa\ T_{i}+p_{r}\ T}{\kappa\
T_{i}+p_{r}\ T^{*}}$ (27) $\displaystyle h$ $\displaystyle=\frac{|p|^{2}\
\kappa\rho^{2}\ T_{i}^{2}}{|p_{r}\ T+\kappa\ T_{i}|^{2}}$ (28)

and the system obeys the constraint

$\displaystyle\kappa^{2}\ (1+\kappa\rho^{2})T_{i}^{2}+2p_{r}\ \kappa\
(1+\kappa\rho^{2})\ T_{i}\ T_{r}+p_{r}^{2}\ |T|^{2}$ $\displaystyle=0$ (29)

now keeping one of $T_{r}$ or $T_{i}$ fixed the other can be calculated from
eqn. 29. If we fix $T_{r}$ then $T_{i}$ can be expressed as

$\displaystyle T_{i}$ $\displaystyle=-\frac{\kappa\
(1+\kappa\rho^{2})+\sqrt{-p_{r}^{2}+\kappa^{3}\rho^{2}\
(1+\kappa\rho^{2})}}{p_{r}^{2}+\kappa^{2}+\kappa^{3}\rho^{2}}\ p_{r}\ T_{r}$
(30)

from eqn. 30 it follows that the condition $\ |p_{r}|\ \leq\
\sqrt{\kappa^{3}\rho^{2}\ (1+\kappa\rho^{2})}$ must satisfy to obtain 1SS.

####  2.1.1 Properties of dark one soliton Solution

In this part we shall discuss about the properties of 1SS (eqn. 20) namely
velocity, width inverse, the criteria upon which the nature of the one soliton
obtained will be dark or bright and also discuss the amplitude. One thing to
point out that there is always a background present so instead of calling
’bright soliton’, an ’anti-dark soliton’ term is much more accurate but we
shall refer to it as ’bright soliton’ for convenience. At first, let us
parametrise $\theta$ as

$\displaystyle\theta$ $\displaystyle=p\ x+\Omega\ t$ (31)
$\displaystyle\Rightarrow\theta$ $\displaystyle=p_{r}\ (x+v\ t)+i\ p_{i}\
(x-v\ t)$ (32)

here $2p_{r}$ represents the width inverse and $v$ denotes the velocity of 1SS
and is represented as

$\displaystyle v$ $\displaystyle=\frac{h}{p_{r}^{2}+p_{i}^{2}}$ (33)
$\displaystyle\Rightarrow v$ $\displaystyle=\frac{\kappa\rho^{2}\
T_{i}^{2}}{|p_{r}\ T+\kappa\ T_{i}|^{2}}$ (34)

eqn. 34 shows that $\kappa$ can control both the sign and magnitude of $v$,
graphically represented in figure 1(a). In brief the figure 1(a) interprets
that magnitude of $v$ at first increases with the magnitude of $\kappa$ and
then decreases. And figure 1(b) shows that the magnitude of $v$ decreases with
$p_{r}$ and the sign of $p_{r}$ has no effect on $v$, in other words $v$ is
symmetric to $p_{r}$. $T_{r}$ has no effect on $v$ as $T_{i}$ is proportional
to $T_{r}$ (from eqn. 30) so $T_{r}$ is a common term in both numerator and
denominator of eqn. 34. In both the graphs $v$ tends to zero at extreme points
of $\kappa$ and $p_{r}$. These characteristics of $v$ is irrespective to the
nature of soliton being dark or bright.

(a) $v$ vs $\kappa$

(b) $v$ vs $p_{r}$

Figure 1: Variation of velocity (a) with respect to $\kappa$ and fixed $p_{r}$
= $1$ and $2$ and (b) with respect to $p_{r}$ and fixed $\kappa$ = $\pm 5$,
$\pm 10$. In both the graphs we have fixed $\rho$ = $2$ and $T_{r}$ can have
any value say $\pm 1$, $\pm 2$, etc. the graphs will remain the same.

The amplitude ($A$) of 1SS is expressed as

$\displaystyle A$ $\displaystyle=\rho\ \Big{|}\frac{T+\gamma\
|T|}{T+|T|}\Big{|}$ (35)

for positive $p_{r}$, when $\kappa$ and $T_{i}$ have different signs we get
$A$ smaller than $\rho$ and $A$ is bigger than $\rho$ for the same sign of
$\kappa$ and $T_{i}$. For the case of negative $p_{r}$ the criteria become
vice versa. These are the criteria for the soliton to be dark and bright
respectively. Figures 2(a) and 2(b) show the variation of $A$ with respect to
$\kappa$ and $p_{r}$ respectively for dark and bright solitons. Figure 2(b)
shows that the amplitude of the dark soliton at first decreases then increases
with $p_{r}$ and for bright soliton the amplitude decreases monotonically with
$p_{r}$ and in both dark and bright cases the amplitude $A\rightarrow\rho$ as
$p_{r}\to\pm\sqrt{\kappa^{3}\rho^{2}\ (1+\kappa\rho^{2})}$. One important fact
to notice is that in the limit $p_{r}\to 0$, the dark soliton reduces to a
plane wave that means algebraic dark soliton does not exist. However algebraic
bright soliton exists. Figure 2(b) shows the same. In the same graph we see
that at certain values of $p_{r}$ the amplitude of dark soliton reduces to
zero indicating the occurrence of a black soliton. Figures 3(a) and 3(b)
represent the propagation of dark and bright soliton respectively in 2D
density plot.

(a) $A$ vs $\kappa$

(b) $A$ vs $p_{r}$

Figure 2: Variation of amplitude (a) with respect to $\kappa$ and fixed
$p_{r}$ = $1$ and (b) with respect to $p_{r}$ and fixed $\kappa$ = $\pm 5$,
$\pm 10$. In both the graphs we fixed $\rho$ = $2$ and $T_{r}$ = $6$ and $-6$
for dark and bright soliton respectively.

(a) Dark 1SS

(b) Bright 1SS

Figure 3: (a) represents dark soliton with $\kappa$ = $-5$ and $T_{r}$ = $6$
and (b) represents bright soliton with $\kappa$ = $5$ and $T_{r}$ = $-6$ in
density plot. In both the figures we have fixed $\rho$ = $2$.

###  2.2 Dark two soliton Solution

The dark two soliton solution (2SS) is obtained by dropping the terms of order
greater than or equal to $\epsilon^{5}$ in $g$, $f$ and $s$. Thus from eqn. 6
the dark 2SS of eqn. 5 is

$\displaystyle u$
$\displaystyle=\frac{g_{0}(1+\epsilon^{2}g_{2}^{(2)}+\epsilon^{4}g_{4}^{(2)})}{1+\epsilon^{2}f_{2}^{(2)}+\epsilon^{4}f_{4}^{(2)}}\Big{|}_{\epsilon=1}$
(36)

Let us consider the expressions for $g_{2}^{(2)}$,
$g_{4}^{(2)}$,$s_{2}^{(2)}$, $s_{4}^{(2)}$, $f_{2}^{(2)}$ and $f_{4}^{(2)}$ in
2SS are as follows

$\displaystyle g_{2}^{(2)}$ $\displaystyle=K_{1}\
e^{\theta_{1}+\theta_{1}^{*}}+K_{2}\ e^{\theta_{2}+\theta_{2}^{*}}$ (37)
$\displaystyle g_{4}^{(2)}$ $\displaystyle=K_{12}\
e^{\theta_{1}+\theta_{1}^{*}+\theta_{2}+\theta_{2}^{*}}$ (38) $\displaystyle
s_{2}^{(2)}$ $\displaystyle=M_{1}\ e^{\theta_{1}+\theta_{1}^{*}}+M_{2}\
e^{\theta_{2}+\theta_{2}^{*}}$ (39) $\displaystyle s_{4}^{(2)}$
$\displaystyle=M_{12}\
e^{\theta_{1}+\theta_{1}^{*}+\theta_{2}+\theta_{2}^{*}}$ (40) $\displaystyle
f_{2}^{(2)}$ $\displaystyle=T_{1}\ e^{\theta_{1}+\theta_{1}^{*}}+T_{2}\
e^{\theta_{2}+\theta_{2}^{*}}$ (41) $\displaystyle f_{4}^{(2)}$
$\displaystyle=T_{12}\
e^{\theta_{1}+\theta_{1}^{*}+\theta_{2}+\theta_{2}^{*}}$ (42)

where $\theta_{j}$ = $p_{j}\ \xi+\Omega_{j}\ \tau$. $p_{j}$, $\Omega_{j}$,
$K_{j}$, $M_{j}$, $T_{j}$, $K_{12}$, $M_{12}$, $T_{12}$ are complex parameters
($j=1,2$). Let us consider $p_{j}$ = $p_{r_{j}}+i\ p_{i_{j}}$ and $T_{j}$ =
$T_{r_{j}}+i\ T_{i_{j}}$ where $p_{r_{j}}$, $p_{i_{j}}$, $T_{r_{j}}$ and
$T_{i_{j}}$ are real. On substituting the above equations into eqn. 36, we
have

$\displaystyle u$ $\displaystyle=\rho\ e^{i\
(\kappa\xi+\omega\tau)}\frac{1+K_{1}\ e^{\theta_{1}+\theta_{1}^{*}}+K_{2}\
e^{\theta_{2}+\theta_{2}^{*}}+K_{12}\
e^{\theta_{1}+\theta_{1}^{*}+\theta_{2}+\theta_{2}^{*}}}{1+T_{1}\
e^{\theta_{1}+\theta_{1}^{*}}+T_{2}\ e^{\theta_{2}+\theta_{2}^{*}}+T_{12}\
e^{\theta_{1}+\theta_{1}^{*}+\theta_{2}+\theta_{2}^{*}}}$ (43)

now putting eqns. 37 \- 42 into eqns 9 \- 11 yield the following expressions:

$\displaystyle\Omega_{j}$ $\displaystyle=\frac{h_{j}}{p_{j}}$ (44)
$\displaystyle M_{j}$ $\displaystyle=\frac{T_{j}^{2}}{K_{j}^{*}}$ (45)
$\displaystyle M_{12}$ $\displaystyle=\frac{T_{12}^{2}}{K_{12}^{*}}$ (46)
$\displaystyle K_{j}$ $\displaystyle=\gamma_{j}\ T_{j}^{*}$ (47)
$\displaystyle K_{12}$ $\displaystyle=\gamma_{1}\ \gamma_{2}\ T_{12}^{*}$ (48)
$\displaystyle T_{12}$ $\displaystyle=c\ T_{1}\ T_{2}$ (49)

where $h_{j}$’s are real and $\gamma_{j}$’s are complex and are represented as

$\displaystyle\gamma_{j}$ $\displaystyle=\frac{\kappa\ T_{i_{j}}+p_{r_{j}}\
T_{j}}{\kappa\ T_{i_{j}}+p_{r_{j}}\ T_{j}^{*}}$ (50) $\displaystyle h_{j}$
$\displaystyle=\frac{|p_{j}|^{2}\ \kappa\rho^{2}\ T_{i_{j}}^{2}}{|p_{r_{j}}\
T_{j}+\kappa\ T_{i_{j}}|^{2}}$ (51)

and $c$ is real expressed as

$\displaystyle c$ $\displaystyle=\frac{(p_{1_{r}}\ T_{1_{r}}\
T_{2_{i}}-p_{2_{r}}\ T_{1_{i}}\ T_{2_{r}})^{2}+(p_{1_{r}}-p_{2_{r}})^{2}\
T_{1_{i}}^{2}\ T_{2_{i}}^{2}}{(p_{1_{r}}\ T_{1_{r}}\ T_{2_{i}}-p_{2_{r}}\
T_{1_{i}}\ T_{2_{r}})^{2}+(p_{1_{r}}+p_{2_{r}})^{2}\ T_{1_{i}}^{2}\
T_{2_{i}}^{2}}$ (52)

the system obeys the constraints

$\displaystyle\kappa^{2}\ (1+\kappa\rho^{2})T_{i_{j}}^{2}+2p_{r_{j}}\ \kappa\
(1+\kappa\rho^{2})\ T_{i_{j}}\ T_{r_{j}}+p_{r_{j}}^{2}\ |T_{j}|^{2}$
$\displaystyle=0$ (53)

if we fix $T_{r_{j}}$ then $T_{i_{j}}$ can be calculated from eqn. 53 as

$\displaystyle T_{i_{j}}$ $\displaystyle=-\frac{\kappa\
(1+\kappa\rho^{2})+\sqrt{-p_{r_{j}}^{2}+\kappa^{3}\rho^{2}\
(1+\kappa\rho^{2})}}{p_{r_{j}}^{2}+\kappa^{2}+\kappa^{3}\rho^{2}}\ p_{r_{j}}\
T_{r_{j}}$ (54)

from eqn. 54, the conditions $\ |p_{r_{j}}|\ \leq\ \sqrt{\kappa^{3}\rho^{2}\
(1+\kappa\rho^{2})}$ must satisfy to obtain 2SS.

####  2.2.1 Properties of dark two soliton solution

In a similar way as we have done in section 2.1.1, to investigate the
properties of 2SS we first parametrise $\theta_{j}$ ($j=1,2$) as

$\displaystyle\theta_{j}$ $\displaystyle=p_{j}\ x+\Omega_{j}\ t$ (55)
$\displaystyle\Rightarrow\theta_{j}$ $\displaystyle=p_{r_{j}}\ (x+v_{j}\ t)+i\
p_{i_{j}}\ (x-v_{j}\ t)$ (56)

where $v_{j}$ denotes the velocity of the jth soliton and represented as

$\displaystyle v_{j}$
$\displaystyle=\frac{h_{j}}{p_{r_{j}}^{2}+p_{i_{j}}^{2}}$ (57)
$\displaystyle\Rightarrow v_{j}$ $\displaystyle=\frac{\kappa\rho^{2}\
T_{i_{j}}^{2}}{|p_{r_{j}}\ T_{j}+\kappa\ T_{i_{j}}|^{2}}$ (58)

and $2p_{r_{j}}$ represents the width inverse of the jth soliton of 2SS. The
amplitude, $A_{j}$ of the jth soliton is given as

$\displaystyle A_{j}$ $\displaystyle=\rho\ \Big{|}\frac{T_{j}+\gamma_{j}\
|T_{j}|}{T_{j}+|T_{j}|}\Big{|}$ (59)

The velocity and amplitude of the jth soliton obey the same characteristics as
that of 1SS obey as discussed in the section 2.1.1. Interaction of two
individual solitons of 2SS is demonstrated in figures 4(a) and 4(b) for a
dark-dark case and bright-bright case respectively in 2D density plot.

(a) Dark-dark 2SS

(b) Bright-bright 2SS

Figure 4: (a) represent the dark-dark 2SS, here $p_{r_{1}}$ = $3$, $p_{r_{2}}$
= $4$, $T_{r_{1}}$ = $6$, $T_{r_{2}}$ = $9$ and $\kappa$ = $-5$ (b) represent
the bright-bright 2SS, here $p_{r_{1}}$ = $3$, $p_{r_{2}}$ = $4$, $T_{r_{1}}$
= $-6$, $T_{r_{2}}$ = $-9$ and $\kappa$ = $5$. In both the graphs we fixed
$\rho=2$.

####  2.2.2 Asymptotic analysis of 2SS

To study the amplitudes and the phase shift of 2SS we shall perform asymptotic
analysis. When asymptotically apart from each other, 2SS is essentially two
separated 1SS. In the limit of before interaction $\tau\to-\infty$, if we fix
$\theta_{1}$, this implies $|e^{\theta_{1}}|$ is finite and
$|e^{\theta_{2}}|\to\infty$. This corresponds to an asymptotic form of eqn. 43
which is expressed as

$\displaystyle u_{1-}$ $\displaystyle=\rho\ e^{i\ (\kappa\xi+\omega\tau)}\
\frac{K_{2}\ (1+c\ K_{1}\ e^{\theta_{1}+\theta_{1}^{*}})}{T_{2}\ (1+c\ T_{1}\
e^{\theta_{1}+\theta_{1}^{*}})}$ (60) $\displaystyle\Rightarrow u_{1-}$
$\displaystyle=\rho\ e^{i\ (\kappa\xi+\omega\tau)}\ e^{i\ \phi}\
\frac{(1+K_{1}^{\prime}\ e^{\theta_{1}+\theta_{1}^{*}})}{(1+T_{1}^{\prime}\
e^{\theta_{1}+\theta_{1}^{*}})}$ (61)

here $K_{1}^{\prime}=c\ K_{1}$ and $T_{1}^{\prime}=c\ T_{1}$. Eqn. 61 is
nothing but 1SS (from eqn. 20) with an additional phase $\phi$. The arise of
$\phi$ is due to term $\frac{K_{2}}{T_{2}}$ in eqn. 60 and the expression of
$\phi$ is given in a few steps below (in eqn. 64).
In the limit of after interaction $\tau\to\infty$, again we have fixed
$\theta_{1}$ then this will imply $|e^{\theta_{1}}|$ is finite and
$|e^{\theta_{2}}|\to 0$. The corresponding asymptotic form of eqn. 43
expressed as

$\displaystyle u_{1+}$ $\displaystyle=\rho\ e^{i\ (\kappa\xi+\omega\tau)}\
\frac{(1+K_{1}\ e^{\theta_{1}+\theta_{1}^{*}})}{(1+T_{1}\
e^{\theta_{1}+\theta_{1}^{*}})}$ (62)

we see eqn. 62 is same as eqn. 20 with no additional phase. The amplitude of
both the eqns. 61 and 62 is same and can be expressed as

$\displaystyle A_{asymp}$ $\displaystyle=\rho\
\sqrt{\frac{(T_{1}+T_{1}^{*})(K_{1}+K_{1}^{*})-2(|T_{1}|^{2}+|K_{1}|^{2})+2\
\sigma_{1}|K_{1}-T_{1}||K_{1}-T_{1}^{*}|}{(T_{1}-T_{1}^{*})^{2}}}$ (63)

where $\sigma_{1}$ = sign($T_{r_{1}}$). This is one of the important
characteristics of a soliton that upon interaction the amplitude of the
soliton remains same only a phase shift occurs. In this case the phase shift
is $\phi$ and its value is given by

$\displaystyle\phi$ $\displaystyle=tan^{-1}\Big{[}-\frac{2\kappa\
T_{i_{2}}^{2}\ (\kappa\ T_{r_{2}}\ T_{i_{2}}+p_{r_{2}}|T_{2}|^{2})}{(\kappa\
(T_{r_{2}}-T_{i_{2}})\ T_{i_{2}}+p_{r_{2}}|T_{2}|^{2})\ (\kappa\
(T_{r_{2}}+T_{i_{2}})\ T_{i_{2}}+p_{r_{2}}|T_{2}|^{2})}\Big{]}$ (64)

We can perform the same analysis by keeping $\theta_{2}$ fixed. In the limit
$\tau\to-\infty$, the asymptotic form of eqn. 43 becomes

$\displaystyle u_{2-}$ $\displaystyle=\rho\ e^{i\ (\kappa\xi+\omega\tau)}\
\frac{(1+K_{2}\ e^{\theta_{2}+\theta_{2}^{*}})}{(1+T_{2}\
e^{\theta_{2}+\theta_{2}^{*}})}$ (65)

and in the limit $\tau\to\infty$, the asymptotic form of eqn. 43 becomes

$\displaystyle u_{2+}$ $\displaystyle=\rho\ e^{i\ (\kappa\xi+\omega\tau)}\
\frac{K_{1}\ (1+c\ K_{2}\ e^{\theta_{2}+\theta_{2}^{*}})}{T_{1}\ (1+c\ T_{2}\
e^{\theta_{2}+\theta_{2}^{*}})}$ (66)

The amplitude of both $u_{2\pm}$ can be expressed as

$\displaystyle A_{asymp}^{\prime}$ $\displaystyle=\rho\
\sqrt{\frac{(T_{2}+T_{2}^{*})(K_{2}+K_{2}^{*})-2(|T_{2}|^{2}+|K_{2}|^{2})+2\
\sigma_{2}|K_{2}-T_{2}||K_{2}-T_{2}^{*}|}{(T_{2}-T_{2}^{*})^{2}}}$ (67)

where $\sigma_{2}$ = sign($T_{r_{2}}$). The phase shift in this case is
expressed as

$\displaystyle\phi^{\prime}$ $\displaystyle=tan^{-1}\Big{[}-\frac{2\kappa\
T_{i_{1}}^{2}\ (\kappa\ T_{r_{1}}\ T_{i_{1}}+p_{r_{1}}|T_{1}|^{2})}{(\kappa\
(T_{r_{1}}-T_{i_{1}})\ T_{i_{1}}+p_{r_{1}}|T_{1}|^{2})\ (\kappa\
(T_{r_{1}}+T_{i_{1}})\ T_{i_{1}}+p_{r_{1}}|T_{1}|^{2})}\Big{]}$ (68)

###  2.3 Scheme for dark N soliton solution

To obtain dark N soliton solution (NSS), we have to drop the terms greater
than or equal to $\epsilon^{2N+1}$ in $g$, $f$ and $s$. Eqn. 6 becomes

$\displaystyle u$
$\displaystyle=\frac{g_{0}(1+\epsilon^{2}g_{2}^{(N)}+...+\epsilon^{2N}g_{2N}^{(N)})}{1+\epsilon^{2}f_{2}^{(2)}+...+\epsilon^{2N}f_{2N}^{(N)}}\Big{|}_{\epsilon=1}$
(69)

here the terms in $g^{(N)}$ and $f^{(N)}$ can be expressed as

$\displaystyle g^{(N)}$ $\displaystyle=g_{0}\ \Big{(}1+\sum_{j=1}^{N}\ K_{j}\
e^{\theta_{j}+\theta_{j}^{*}}+\frac{1}{2}\sum_{j,l=1;\ j\neq l}^{N}K_{jl}\
e^{\theta_{j}+\theta_{j}^{*}+\theta_{l}+\theta_{l}^{*}}+...+K_{12...N}\ e^{\
\sum\limits_{j=1}^{N}\theta_{j}+\theta_{j}^{*}}\Big{)}$ (70) $\displaystyle
f^{(N)}$ $\displaystyle=1+\sum_{j=1}^{N}\ T_{j}\
e^{\theta_{j}+\theta_{j}^{*}}+\frac{1}{2}\sum_{j,l=1;\ j\neq l}^{N}T_{jl}\
e^{\theta_{j}+\theta_{j}^{*}+\theta_{l}+\theta_{l}^{*}}+...+T_{12...N}\ e^{\
\sum\limits_{j=1}^{N}\theta_{j}+\theta_{j}^{*}}$ (71)

where $\theta_{j}=p_{j}\ x+\Omega_{j}\ t$. $p_{j}$, $\Omega_{j}$, $K_{j}$,
$K_{jl}$, …, $K_{12...N}$, $T_{j}$, $T_{jl}$, …, $T_{12...N}$, are complex
parameters (subscripts go from $1$ to $N$). These parameters can be calculated
using the bilinear eqns. 9 \- 11. The N soliton system consists of N number of
constraints just like 2SS have two constraints expressed in eqn. 53. The
individual jth soliton of NSS have velocity expression $v_{j}$ from eqn. 58
and amplitude expression $A_{j}$ from eqn. 59 and obey the same
characteristics with respect to the parameters $\rho$, $\kappa$, $p_{r_{j}}$,
$T_{j}$ as discussed in the section 2.1.1.

##  3 Gauge Connected Landau-Lifshitz equation

In this section we derive the gauge connected LLE of eqn. 5 by exploiting the
equivalence of the Lax pairs of the two gauge connected systems. Let us
consider the Lax pair ($L$, $M$) for eqn. 5

$\displaystyle\Psi_{\xi}=L\Psi$ (72) $\displaystyle\Psi_{\tau}=M\Psi$ (73)

where $\Psi$ is the Jost function corresponding to the field function $u$ and
($L$, $M$) are given by

$\displaystyle L=\frac{-i\zeta^{2}}{2}\Sigma+\zeta u_{\xi}$ (74)
$\displaystyle M=\frac{i}{2\zeta^{2}}\Sigma-\frac{i}{\zeta}\Sigma u+i\Sigma
u^{2}$ (75)

$\zeta$ is the spectral parameter and $\Sigma$ and $u$ are $2\times 2$
matrices given as follow

$\quad\Sigma=\left(\begin{tabular}[]{c c}1&0\\\ 0&-1\\\
\end{tabular}\right),\quad\quad u=\left(\begin{tabular}[]{c c}0&q\\\
$-q^{*}$&0\\\ \end{tabular}\right)$

from eqns. 72 and 73, we can write $(\Psi_{\xi})_{\tau}=(\Psi_{\tau})_{\xi}$,
this gives:

$\displaystyle L_{\tau}-M_{\xi}+[L,M]=0$ (76)

eqn. 76 is the compatibility condition also called zero curvature equation.
Under a local gauge transformation we write a matrix $g$ such that:

$\quad g({\xi},{\tau},\zeta_{0})=\Psi({\xi},{\tau},\zeta)|_{\zeta=\zeta_{0}}$

the Jost function $\Psi$ changes to $\Phi$ as:

$\quad\Psi\rightarrow\Phi({\xi},{\tau},\zeta,\zeta_{0})=g({\xi},{\tau},\zeta_{0})^{-1}\Psi({\xi},{\tau},\zeta)$

$\Phi$ is the Jost function corresponding to the spin field function $S$ of
Landau-Lifshitz (LL) system. The new Lax pair ($L^{\prime}$, $M^{\prime}$)
associated with $\Phi$ expressed as:

$\displaystyle\Phi_{\xi}=L^{\prime}\Phi$ (77)
$\displaystyle\Phi_{\tau}=M^{\prime}\Phi$ (78)

and are related to ($L$, $M$) as:

$\displaystyle L^{\prime}=g^{-1}Lg-g^{-1}g_{\xi}$ (79) $\displaystyle
M^{\prime}=g^{-1}Mg-g^{-1}g_{\tau}$ (80)

but

$\displaystyle g_{\xi}g^{-1}=L_{0},\quad\quad\quad g_{\tau}g^{-1}=M_{0}$ (81)

therefore,

$\displaystyle
L^{\prime}=g^{-1}(\frac{-i}{2}(\zeta^{2}-\zeta_{0}^{2})\Sigma+(\zeta-\zeta_{0})u_{\xi})g$
(82) $\displaystyle
M^{\prime}=g^{-1}(\frac{i}{2}(\frac{1}{\zeta^{2}}-\frac{1}{\zeta_{0}^{2}})\Sigma-i(\frac{1}{\zeta}-\frac{1}{\zeta_{0}})\Sigma
u)g$ (83)

where $L_{0}$ and $M_{0}$ are:

$\displaystyle
L_{0}=L|_{\zeta=\zeta_{0}}=\frac{-i\zeta_{0}^{2}}{2}\Sigma+\zeta_{0}u_{\xi}$
(84) $\displaystyle
M_{0}=M|_{\zeta=\zeta_{0}}=\frac{i}{2\zeta_{0}^{2}}\Sigma-\frac{i}{\zeta_{0}}\Sigma
u+i\Sigma u^{2}$ (85)

the compatibility condition eqn. 76 in terms of the new Lax pair
($L^{\prime}$, $M^{\prime}$) takes the form:

$\displaystyle L^{\prime}_{\tau}-M^{\prime}_{\xi}+[L^{\prime},M^{\prime}]=0$
(86)

this is the compatibility condition corresponding to $S$. $S$ being spin field
function of LL system can be represented in terms of $g$ as:

$\displaystyle S=g^{-1}\Sigma g$ (87)

and satisfies

$\displaystyle S^{2}=I$ (88)

using eqn. 81, the derivatives of $S$ namely $S_{\xi}$, $S_{\tau}$ and also
the terms $SS_{\xi}$, $SS_{\tau}$ can be written as:

$\displaystyle S_{\xi}=g^{-1}[\Sigma,L_{0}]g\Rightarrow
S_{\xi}=2\zeta_{0}g^{-1}\Sigma u_{\xi}g$ (89) $\displaystyle
S_{\tau}=g^{-1}[\Sigma,M_{0}]g\Rightarrow
S_{\tau}=-\frac{2i}{\zeta_{0}}g^{-1}ug$ (90) $\displaystyle
SS_{\xi}=2\zeta_{0}g^{-1}u_{\xi}g$ (91) $\displaystyle
SS_{\tau}=-\frac{2i}{\zeta_{0}}g^{-1}\Sigma ug$ (92)

again using the above four expressions in eqns. 82 and 83, we get:

$\displaystyle
L^{\prime}=\frac{-i}{2}(\zeta^{2}-\zeta_{0}^{2})S+(\zeta-\zeta_{0})\frac{1}{2\zeta_{0}}SS_{\xi}$
(93) $\displaystyle
M^{\prime}=\frac{i}{2}(\frac{1}{\zeta^{2}}-\frac{1}{\zeta_{0}^{2}})S+(\frac{1}{\zeta}-\frac{1}{\zeta_{0}})\frac{\zeta_{0}}{2}SS_{\tau}$
(94)

hence, the compatibility condition eqn. 86 yields:

$\displaystyle S_{\xi}+\frac{\zeta_{0}^{2}}{2i}[S,S_{{\tau}{\xi}}]=0$ (95)

this is the gauge connected LLE for the eqn. 5. One important thing to note is
that eqn. 95 is consistent for the system of eqns. 77, 78, 93, 94 in any
matrix $S$ of arbitrary dimension as long as eqn. 88 holds.

##  4 Conclusion

We have bilinearized the Fokas-Lenells derivative nonlinear Schrödinger
equation (FLDE) with a non-vanishing boundary condition. In the proposed
bilinearization we have used an auxiliary function to derive explicitly the
one dark soliton solution and two dark soliton solution and represent the
scheme for obtaining N soliton solution. We have also discussed the criteria
for the soliton to become dark or bright. Our result accepts a wide range of
values for the parameters ($K$’s, $T$’s) unlike the results published earlier.
We have obtained a gauge equivalence between FLDE and Landau-Lifshitz (LL)
spin system with the explicit construction of the new equivalent Lax pair. We
believe that the derived dark (and bright) soliton solutions will be useful in
optical communication and other nonlinear fields of physics where higher order
effects like spatio-temporal evolution and cubic nonlinear self-steepening
effects are taken into account. The gauge equivalent LL equation will be
useful to study the integrability properties of the FLDE.

## Acknowledgement

R Dutta and S Talukdar acknowledge DST, Govt. of India for INSPIRE fellowship
award. Corresponding award number DST/INSPIRE Fellowship/2020/IF200303 and
DST/INSPIRE Fellowship/2020/IF200278.

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