# Linear interference and systematic soliton shape modulation by engineering
plane wave background and soliton parameters

Sagardeep Talukdara, Riki Duttaa, Gautam Kumar Sahariaa and
Sudipta Nandy a

aDepartment of Physics, Cotton University, Guwahati 781001, India email:
<EMAIL_ADDRESS>
###### Abstract

We investigate linear interference of a plane wave with different localised
waves using coupled Fokas-Lenells equation(FLE) with four wave mixing (FWM)
term. We obtain localised wave solution of the coupled FLE by linear
superposition of two distinctly independent wave solutions namely plane wave
and one soliton solution & plane wave and two soliton solution. We obtain
several nonlinear profiles depending on the relative phase induced by soliton
parameters. We analyse the linear interference profile under four different
conditions on the spatial and temporal phase coefficients of interfering
waves. We further investigate the interaction of two soliton solution and a
plane wave. In this case we notice that asymptotically, two solitons profile
may be similar or different from each other depending on the choices of
soliton parameters in the two cases. The results obtained by us might be
useful for applications in soliton control, a fiber amplifier, all optical
switching, and optical computing. Further we believe that the present
investigation would be useful to study the linear interference pattern of
other localised waves of FLE.

Keywords: Fokas-Lenells equation, Four Wave Mixing, Soliton, Linear
interference

##  1 Introduction

It is well known that the nonlinear Schrödinger type equations namely,
nonlinear schrödinger equation (NLSE)[1], derivative NLSE (DNLSE)[2], higher
order NLSE (HNLSE)[3], Fokas-Lenells equation(FLE) can describe the
propagation of ultrashort pulses through an optical fiber. In comparison to
other equations, the significance of FLE is due to the presence of the spatio-
temporal dispersion term. FLE was first shown to be integrable by Fokas and
Lenells in their original paper[4]. Bright and dark soliton solutions of FLE
are obtained using a handful of methods namely, Hirota bilinear method [5, 6,
7, 8], neural network method[9] and by Backlund transformation [10]. Breather
solution [11], rogue wave solutions [12, 13], algebro-geometric solution [14]
are among the other reported results.
FWM is an important nonlinear physical phenomena having significance in both
nonlinear optics and Bose Einstein condensation (BEC). In nonlinear optics,
FWM is caused by the dependence of the refractive index on the intensity of
optical fiber but it critically depends on the channel spacing and fiber
chromatic dispersion. In BEC, FWM is defined as a pair transition term. The
study of FWM in BEC is important because it can bring in an abundance of
nonlinear matter wave structures with different backgrounds.
FWM in nonlinear optics and in BEC has been studied earlier with coupled NLSE
[15, 16, 17, 18, 19, 20, 21, 22], and coupled Hirota model [23].
The study on multi-component FLE specially in the presence of FWM however, has
not been done earlier. It is worthwhile to mention that the study of coupled
optical fields is specially important for the Wavelength Division
Multiplexing(WDM) technology where channel spacing plays an important role to
increase bandwidth.
It is interesting to note that in presence of FWM term in the nonlinear
Schrödinger equation the linear superposition of solutions is possible which
is uncharacteristic to nonlinear differential systems. Untill now studies have
been carried out for NLSE and higher order NLSE with FWM term. So it is
natural to ask whether or not the linear superposition of solutions exists for
coupled FLE with FWM term. However, to the best of our knowledge the study on
the linear interference between two component FLE systems in the presence of
the FWM effect is absent so far.

Thus, in this manuscript, our objective is to study the linear interference
between two component FLE systems in the presence of the FWM effect and
explore possible localized structure and the mechanism to obtain them. The
present study is important because through this analysis it may be possible to
obtain pulses with customized shapes.
The structure of the manuscript is the following. In the following section we
introduce the model equation. The review of the methodology for obtaining the
pair transition of the various localised structures is given in the third
section. The results and discussion and concluding remarks will be in the
fourth section.

##  2 Linear interference

In dimensionless form coupled FLE with FWM term:

$\displaystyle iU_{t}$
$\displaystyle+a_{1}U_{xx}+a_{2}U_{xt}+b(|U|^{2}+2|V|^{2})U+bV^{2}U^{*}$
$\displaystyle+i\rho(|U|^{2}+|V|^{2})U_{x}+i\rho(UV^{*}+VU^{*})V_{x}=0$ (1)
$\displaystyle iV_{t}$
$\displaystyle+a_{1}V_{xx}+a_{2}V_{xt}+b(|V|^{2}+2|U|^{2})V+bU^{2}V^{*}$
$\displaystyle+i\rho(|U|^{2}+|V|^{2})V_{x}+i\rho(U^{*}V+UV^{*})U_{x}=0$ (2)

where $U$ and $V$ are complex envelopes of two wave fields. The coupled
equation can describe the ultrafast pulse propagation of two mutually
orthogonal optical fields in a medium where the beam is allowed to diffract
along longitudinal and transverse directions.
Under the gauge transformation, namely

$\displaystyle U=\sqrt{\frac{m}{|b|}}ne^{i(nx+2mnt)}u\quad
V=\sqrt{\frac{m}{|b|}}ne^{i(nx+2mnt)}v$ (3)

followed by the transformation of variables, namely

$\displaystyle\xi=x+mt;\quad\tau=-mn^{2}t$ (4)

where $n=\frac{-1}{a_{2}}$ and $m=\frac{a_{1}}{a_{2}}$ Eqs. 2-2 are
transformed into a simplified coupled equation:

$\displaystyle u_{\xi\tau}$
$\displaystyle=u-i2\sigma(|u|^{2}+|v|^{2})u_{\xi}-i2\sigma(v^{*}u+vu^{*})v_{\xi}$
(5) $\displaystyle v_{\xi\tau}$
$\displaystyle=v-i2\sigma(|u|^{2}+|v|^{2})v_{\xi}-i2\sigma(v^{*}u+vu^{*})u_{\xi}$
(6)

where $\sigma=sgn(b)$. It is important to note that eqs. 5 -6 are the linear
transformation of gauged FLE. That is taking $b>0$ the equations can be
decoupled into two identical gauged FLEs, namely

$\displaystyle q_{1\xi\tau}=q_{1}-2i|q_{1}|^{2}q_{1\xi}$ (7) $\displaystyle
q_{2\xi\tau}=q_{2}-2i|q_{2}|^{2}q_{2\xi}$ (8)

such that $u=\frac{1}{2}(q_{1}+q_{2})$ and $v=\frac{1}{2}(q_{1}-q_{2})$ where
$q_{1}$ and $q_{2}$ are two different independent solutions. It is worthwhile
to mention that eqs. 7-8 are the first negative hierarchy of the DNLSE. In
other words, the eqs. 7-8 are integrable even though the coupled eqs. 5-6 are
not.

Let us consider the solution of eq. 7, a plane wave solution,

$\displaystyle q_{1}=Ae^{i[k\ \xi-(\frac{1}{k}\ +2A^{2})\tau]}$ (9)

where $k$ is the wave vector and $\omega=\frac{1}{k}\ +2A^{2}$ is the
frequency and $A$ is the amplitude of the plane wave.

We consider the second solution $q_{2}$, a soliton solution obtained by
applying a direct method, namely the Hirota bilinear method [24, 25] as given
in [7].

$\displaystyle
q_{2}=\frac{\alpha_{1}e^{\theta+i\chi}}{\beta_{1}+\frac{|\alpha_{1}|^{2}}{4a^{2}\beta^{*}_{1}}(a+ib)^{2}(b+ia)e^{2\theta}}$
(10)

where $\theta=a(\xi+c\tau)$ and $\chi=b(\xi-c\tau)$. The parameter
$c=\frac{1}{a^{2}+b^{2}}$ is the soliton velocity.
Earlier, linear superposition of solutions of NLSE[23] and HNLSE have been
reported. Here, we investigate the solutions of the coupled FLE with FWM term
using the linear superposition of component FLEs.

##  3 Soliton and plane wave pair transition

###  3.1  W-shaped, anti-dark, dark-bright soliton in plane wave background

Using the linear superposition of plane wave and soliton, the analytical
solutions of eqs. 5 and 6 are given as,

$\displaystyle
u_{1}=\frac{1}{2}[Ae^{i\chi_{pw}}+\frac{\alpha_{1}(\beta^{*}_{1}\
e^{\theta}+\beta^{*}_{2}\
e^{3\theta})}{|\beta_{1}|^{2}+(\beta_{1}\beta^{*}_{2}+\beta_{2}\beta^{*}_{1})e^{2\theta}+|\beta_{2}|^{2}e^{4\theta}}e^{i\chi_{sol}}]$
(11) $\displaystyle
v_{1}=\frac{1}{2}[Ae^{i\chi_{pw}}-\frac{\alpha_{1}(\beta^{*}_{1}\
e^{\theta}+\beta^{*}_{2}\
e^{3\theta})}{|\beta_{1}|^{2}+(\beta_{1}\beta^{*}_{2}+\beta_{2}\beta^{*}_{1})e^{2\theta}+|\beta_{2}|^{2}e^{4\theta}}e^{i\chi_{sol}}]$
(12)

where the first term in eqs. 11 and 12 is the plane wave solution ($q_{1}$) of
eq. 7 having amplitude $A$ and phase

$\displaystyle\chi_{pw}=k\ \xi-(\frac{1}{k}\ +2A^{2})\tau$ (13)

The second term in eqs. 11 and 12 is the bright one soliton solution ($q_{2}$)
of eq. 8.

The phase of the soliton however, is more complex due to the presence of
complex coefficients, namely $\alpha_{1},\ \beta_{1}$ and $\beta_{2}$.
Consequently the analysis becomes more involved. For convenience however, we
identify the part of the phase function which determines soliton velocity and
wave vector, namely

$\displaystyle\chi_{sol}=b\ \xi-\frac{b}{a^{2}+b^{2}}\tau$ (14)

To investigate the effect of linear interference we present the solution
namely $q_{2}$ in the following convenient form.

$\displaystyle|u|^{2}=\frac{2|A||\alpha_{1}||\cosh\ z_{1}||\cosh\
z_{2}|}{\sqrt{|\beta_{1}||\beta_{2}|}\ \cosh\
(\theta+ln\frac{|\beta_{2}|}{|\beta_{1}|}+\beta_{1}\beta^{*}_{2}+\beta^{*}_{1}\beta_{2})}$
(15)

where,

$\displaystyle z_{1}=$ $\displaystyle X_{1}+i\ Y_{1};\qquad
z_{2}=X_{2}+iY_{2}$ (16) $\displaystyle X_{1}=$ $\displaystyle\
\theta+\frac{1}{2}ln(\frac{|\beta_{1}|}{|\beta_{2}|})$ (17) $\displaystyle
Y_{1}=$ $\displaystyle\ \frac{\beta_{1}-\beta_{2}}{2}$ (18) $\displaystyle
X_{2}=$ $\displaystyle\theta+\frac{1}{2}\
ln(\frac{2|\alpha_{1}|\sqrt{|\beta_{1}||\beta_{2}|}|\cosh\ z_{1}|}{A(\cosh\
(\theta+ln\frac{|\beta_{2}|}{|\beta_{1}|}+\beta_{1}\beta^{*}_{2}+\beta^{*}_{1}\beta_{2})})$
(19) $\displaystyle Y_{2}=$
$\displaystyle\frac{(\alpha_{1}-\frac{\beta_{1}+\beta_{2}}{2})}{2}\qquad\qquad\qquad\qquad\qquad\text{Case-I}$
(20) $\displaystyle Y_{2}=$
$\displaystyle\frac{(\alpha_{1}-\frac{\beta_{1}+\beta_{2}}{2})}{2}+[\chi_{x(sol)}-\chi_{x(pw)}]\qquad\text{Case-
II}$ (21) $\displaystyle Y_{2}=$
$\displaystyle\frac{(\alpha_{1}-\frac{\beta_{1}+\beta_{2}}{2})}{2}+[\chi_{t(sol)}-\chi_{t(pw)}]\qquad\text{Case-
III}$ (22) $\displaystyle Y_{2}=$
$\displaystyle\frac{(\alpha_{1}-\frac{\beta_{1}+\beta_{2}}{2})}{2}+[\chi_{sol}-\chi_{pw}]\qquad\qquad\text{Case-
IV}$ (23)

Notice that in eq. 15, the factor $|\cosh\ z_{2}|$ carries the phase
information of soliton and the plane wave. The factor varies with the
differences in the phases of the two superposing wave solutions. It accounts
for the modulation of the soliton into different shapes. Notice that soliton
phase changes with the changes in the soliton parameters $\alpha_{1},\
\beta_{1},\ a\ b$ which in turn changes $|\cosh\ z_{2}|$. Hence we may obtain
a variety of linear interference structures by changing the soliton parameters
depending on the phase relations between soliton phase given by eq. 14 and
plane wave phase given by eq. 13. We narrow down the linear interference
between soliton and plane wave to the following following four cases depending
on their relative phases. Case-I: when the plane wave phase matches with the
soliton phase. That is,

$\displaystyle k$ $\displaystyle=b$ (24) $\displaystyle\frac{1}{k}+2A^{2}$
$\displaystyle=\frac{b}{a^{2}+b^{2}}$ (25)

Case-II: when $t$ component of plane wave phase matches with that of soliton.
That is, eq. 25 is satisfied but eq. 24 is not satisfied. Case-III: when $x$
component of plane wave phase matches with that of soliton. That is, when eq.
25 is not satisfied but eq. 24 is satisfied. Case-IV: when plane wave phase
does not matches with that of soliton. None of the eqs. 24, 25 are satisfied.

We further note that from eq. 25 it follows that $A$ is real when $kb\ >\
a^{2}+b^{2}$, imaginary when $kb\ <\ a^{2}+b^{2}$. In other words for the
second condition the plane wave has an additional phase of $\frac{\pi}{2}$.
Again, the phase of soliton depends on the soliton parameters, namely
$\alpha_{1},\beta_{1},a,b$. Thus due to these additional phases we observe
noticeable alteration of soliton shape profile.

We represent the different cases of soliton and plane wave linear interference
using phase diagrams 1-1. Figure 1 shows $A$ (Plane wave amplitude) vs $k$
(wave vector) plot drawn with a choice of soliton parameters ($a,\ b$) such
that the condition $kb\ >\ a^{2}+b^{2}$ in eq. 25 is satisfied and figure 1
shows $Ae^{\frac{i\pi}{2}}$ vs $k$ plot as an example when $kb\ <\
a^{2}+b^{2}$ is satisfied. In the figures 1 \- 1 the blue dashed curves
represent the plot of the function $A=f(k)$, obtained from the temporal phase
matching condition, namely eq. 25 and the line $k=b(-ve)$ in figure 1 and
$k=b(+ve)$ in figure 1 represent the spatial phase matching condition. The
interaction points of the two lines, represented by the circled red spot in
each of the figures 1 \- 1 belongs to the case-I. All Points on the curve,
$A=f(k)$ except the intersection point, belong to the case-II and represented
by the circled blue spots. All the points on the line $k=b(\pm)$ except the
intersection points, belong to the case-III, and represented by the circled
green spots. All other points in the $A-k$ phase space other than the curve
$A=f(k)$ and the lines $k=b(\pm)$, belong to the case-IV, and represented by
the circled orange spot.

|
---|---
(a) | (b)
Figure 1: Figure 1 shows $A$ v/s $k$ plot drawn with $a=-1$, $b=-6$ satisfying
the condition $kb>a^{2}+b^{2}$; figure 1 shows $Ae^{i\frac{\pi}{2}}$ v/s $k$
plot drawn with $a=-1$, $b=6$ satisfying the condition $kb<a^{2}+b^{2}$. Red
circles corresponds to the Case-I soliton with values $a=-1$, $b=k=-6$ and
$A=0.0474579$. The blue circle corresponds to values $a=-1$, $b=-6$, $k=-1$
and $A=0.647239$ representing Case-II soliton. While the green circle with
parameters $a=\pm 1$, $k=b=-6$ and $A=1$ is case-III breather and orange
circle is a point signifying case-IV breather with parameters $a=-1$, $b=-6$,
$k=4$ and $A=1.2$.

case I: When both the spatial components and temporal components of the phase
are matching, that is, $k=b$ and $\frac{1}{k}+2A^{2}=\frac{b}{a^{2}+b^{2}}$.
Figure 2, 2 depicts the corresponding linear interference profiles. Notice
that even when the spatial and temporal coefficients matching conditions are
met, there arise a phase difference between the plane wave and the soliton.
For instance if $b>0$ the amplitude of the plane wave has an additional phase
of $\frac{\pi}{2}$. Again, the phase of soliton changes with change in the
parameters $a$ and $b$. Hence, depending on the values of the $a$ and $b$ the
plane wave and soliton may have a wide range of phase differences.
Accordingly, we obtain soliton profiles changing from ant-dark to dark to
’W’-shaped soliton. As an example an anti-dark soliton is seen in figure 2.
We present all the wave profiles obtained along with their conditions in the
table below.

Table 1: Table showing conditions for various soliton profiles obtained for
case-$I$.

Condition | $b>|a|$ | $0<b<|a|$ | $b<-|a|$ | $-|a|<b<0$
---|---|---|---|---
|  | $a>0$ | $a<0$ |  |
$\frac{1}{2}(q_{1}-q_{2})$ | Anti-dark | Anti-dark | Dark |  $W$-shaped | Dark
$\frac{1}{2}(q_{1}+q_{2})$ | Anti-dark | Dark | Anti-dark | Anti-dark | Anti-dark

The set of values chosen in the above table are $a=-1,b=-6$,
$\alpha=1,\beta=1$. Interestingly the profile of the soliton turns out to be a
$W$-shaped occur when the ratio $\frac{a}{b}<1$.

|  |
---|---|---
(a) | (b) | (c)

|  |
---|---|---
(d) | (e) | (f)

Figure 2: The density profile of two types of soliton. (a) and (b) corresponds
to case-$I$ soliton in components $u_{1}$ and $v_{1}$ respectively. They
correspond to the red circle in figure 1 with parameters $a=-1$, $b=-6$,
$k=-6$, $A=0.04745$. Also (c) and (d) corresponds to case-$II$ soliton, in
component $u_{1}$ and $v_{1}$ respectively. They correspond to the entire
black solid line in figure 1 except for the red circle. The blue circle
falling on the line is a particular value of case-$II$ soliton with parameters
$a=-1$, $b=-6$, $k=-1$, $A=0.647239$. While (e) and (f) represent the shift in
the position of case-$II$ soliton with the change of value in $a$. The
parameters for (e) are $a=1$, $b=-6$, $k=-1$ and for (f) $a=-1$, $b=-6$,
$k=-1$. However, $\alpha_{1}=1$ and $\beta_{1}=1$ are kept constant
throughout.

case II: When the temporal components are equal, but the spatial components
are not matching, that is, $k\not=b$ and
$\frac{1}{k}+2A^{2}=\frac{b}{a^{2}+b^{2}}$. The corresponding linear
interference profile for components $u_{1}$ and $v_{1}$ are shown in figures 2
and 2 respectively. In the phase diagram 1 this case corresponds to the
case-$II$ soliton which is represented by the black solid line. Notice that
due to the spatial phase mismatch the soliton shape does not deviate much from
that at $\xi=0$, when b is positive. This is due to vanishing background
property of the soliton. Temporally the profile remain constant. On the other
hand for $b=-ve$. The plane wave has an additional phase of $\frac{\pi}{2}$.
This results in a multipeak soliton structure which is shown in figures 2 and
2 for component $u_{1}$. The spatial distance between the humps can be
measured and is given by $D=\frac{2\pi}{|b-k|}$. and the number of peaks is
determined by the soliton’s width $|\frac{1}{a}|$.

case III:When the temporal components are not equal, but the spatial
components are matching, that is, $k=b$ and
$\frac{1}{k}+2A^{2}\not=\frac{b}{a^{2}+b^{2}}$. The linear interference
profile is shown in figures 4 and 4. In the phase diagram 1 this case
corresponds to case-$III$ breather and is represented by the straight line
$k=b=-6$. The green circle represents a point on that line with parameters
$a=-1$, $b=k=-6$. This localised wave having breather like structure is
periodic in time and is similar to Kuznetsov-Ma(KM) breathers [28]. The hump
and valley in each component are complimentary as seen in figures 4 and 4. The
profile in this case has no significant change under the condition of inbuilt
relative phase. Moreover, the breathing period is given by
$T=\frac{2\pi}{|\frac{1}{k}+2A^{2}-b\ c|}$. It is seen that the time period of
breathing decreases monotonically with the increase of plane wave amplitude as
shown in figure 3. The time period is maximum when the amplitude of the plane
wave is zero and thus induces a bright soliton.

Figure 3: The figure shows variation in the time period of breathing for a
case-$III$ breather with the amplitude of the plane wave. The time period
decreases monotonically with the plane wave amplitude. The time period is seen
to be maximum when the plane wave amplitude is nearly zero.

case IV:When the temporal components, as well as spatial components, are
mismatched, that is, $k\neq b$ and
$\frac{1}{k}+2A^{2}\neq\frac{b}{a^{2}+b^{2}}$. This case corresponds to the
case-$IV$ breather which lies in the entire domain of the figure 1 except the
line and the curve of eqs. 24 and 25. A typical type-IV breather is shown in
the figure 4 with parameters $a=-1$, $b=-6$, $k=4$ and $A=1.2$. Notice that
the condition $b_{1}\neq k$ admits a multi-peak profile while the temporal
condition is responsible for the periodic nature of the wave structure. Hence
we name it as multi-peak breather. The linear interference profile is shown in
figures 4 and 4. It is worthwhile to note that bright, dark, anti-dark,
$W$-shaped profiles has been seen in [26] for coupled NLSE with FWM term.
However, in that case one has to introduce additional phase difference between
plane wave and other localised waves. On the other hand, in the present case,
that is in FLE with FWM the phase difference between the two linearly
superposed waves is inbuilt and depends on $a$ and $b$, $\alpha_{1}$ and
$\beta_{1}$ as well as on the sign of $b$.

|
---|---
(a) | (b)

|
---|---
(c) | (d)

Figure 4: The density distribution of two types of profiles where (a) and (b)
corresponds to case-$III$ breather showing the exchange in the hump and valley
in components $u_{1}$ and $v_{1}$ respectively. The parameters are $a=-1$,
$b=-6$, $k=-6$, $A=1$. While (c) and (d) are case-$IV$ breathers. The
parameters are $a=-1$, $b=-6$, $k=4$, $A=1.2$. However, $\alpha_{1}=1$ and
$\beta_{1}=1$ are kept constant throughout.

We now extend the analysis to the linear interaction of plane wave and multi-
soliton. A multi soliton is the bound state of more than one solitons which
are asymptotically separated.

###  3.2 Plane wave and two soliton interaction

The solution of a two soliton with a plane wave would be as follows.

$\displaystyle u_{2}=$ $\displaystyle\frac{1}{2}[Ae^{i\chi_{pw}}+$ (26)
$\displaystyle\frac{\alpha_{1}e^{\theta_{1}+i\chi{1}}+\alpha_{2}e^{\theta_{2}+i\chi{2}}+\alpha_{3}e^{2\theta_{1}+\theta_{2}+i\chi{2}}+\alpha_{4}e^{2\theta_{2}+\theta_{1}+i\chi{1}}}{\beta_{1}+\beta_{2}e^{2\theta_{1}}+\beta_{3}e^{2\theta_{2}}+\beta_{4}e^{\theta_{1}+\theta_{2}+i(\chi_{1}-\chi_{2})}+\beta_{5}e^{\theta_{1}+\theta_{2}+i(\chi_{2}-\chi_{1})}+\beta_{6}e^{2(\theta_{1}+\theta_{2})}}]$
$\displaystyle v_{2}=$ $\displaystyle\frac{1}{2}[Ae^{i\chi_{pw}}-$ (27)
$\displaystyle\frac{\alpha_{1}e^{\theta_{1}+i\chi{1}}+\alpha_{2}e^{\theta_{2}+i\chi{2}}+\alpha_{3}e^{2\theta_{1}+\theta_{2}+i\chi{2}}+\alpha_{4}e^{2\theta_{2}+\theta_{1}+i\chi{1}}}{\beta_{1}+\beta_{2}e^{2\theta_{1}}+\beta_{3}e^{2\theta_{2}}+\beta_{4}e^{\theta_{1}+\theta_{2}+i(\chi_{1}-\chi_{2})}+\beta_{5}e^{\theta_{1}+\theta_{2}+i(\chi_{2}-\chi_{1})}+\beta_{6}e^{2(\theta_{1}+\theta_{2})}}]$

where $\chi_{pw}$ is the plane wave phase as discussed earlier and the soliton
parameters are $\theta_{1}=a_{1}\xi+\frac{a_{1}\tau}{a^{2}_{1}+b^{2}_{1}}$,
$\theta_{2}=a_{2}\xi+\frac{a_{2}\tau}{a^{2}_{2}+b^{2}_{2}}$,
$\chi_{1}=b_{1}\xi-\frac{b_{1}\tau}{a^{2}_{1}+b^{2}_{1}}$ and
$\chi_{2}=b_{2}\xi-\frac{b_{1}\tau}{a^{2}_{1}+b^{2}_{1}}$

Here, one of the two components represent the plane wave while the other
represents a two soliton wave. However, it is well known that a two soliton is
the nonlinear superposition of two solitons. For convenience, we can view this
as the interaction of the plane wave with two distinct bright solitons, that
is each of the solitons interacts with the plane wave following the relations
namely case-$I$, case-$II$, case-$III$ and case-$IV$ as described in the
previous section. It is important to notice that, the interaction between
solitons induces a phase shift in the envelope as well as the oscillatory part
[7].

|  |
---|---|---
(a) | (b) | (c)

|
---|---
(d) | (e)

Figure 5: The density distribution of several types of profiles where (a) and
(b) corresponds to an anti-dark and $W$-shaped interaction, two dark-bright
interaction in the component $u_{2}$ and $v_{2}$ respectively. While (c) shows
the interaction between two $W$-shaped solitons, (d)and (e) corresponds to the
collision of multi-peak and anti-dark soliton and breather and anti-dark
soliton respectively. The parameters for (a) are
$a_{1}=-1,b_{1}=-6,a_{2}=-1,b_{2}=-6.01$ and for (b) are $a_{1}=3$,
$b_{1}=-1$, $a_{2}=-3$, $b_{1}=-1.01$. The parameters for (c) are $a_{1}=1$,
$b_{1}=-6$, $a_{2}=-1$, $b_{1}=-6.01$ while for (d) the parameters are set as
$a_{1}=1$, $b_{1}=3$, $a_{2}=0.5$, $b_{1}=-1.5$ and for (e) $a_{1}=1$,
$b_{1}=3$, $a_{2}=1.5$, $b_{1}=-0.5$. Meanwhile
$\alpha_{1}=1,\alpha_{2}=1,\beta_{1}=1$ along with
$k=b_{1},A=\sqrt{\frac{b_{1}}{2(a^{2}_{1}+b^{2}_{1})}-\frac{1}{2b_{1}}}$ are
kept constant throughout all the cases.

Consequently, even if both the solitons have nearly identical parameters they
may not have nearly identical profiles after interaction with the plane wave
that is because the second soliton has an additional phase in comparison to
the phase of the first soliton. This additional phase occurs due to the
nonlinear interaction of two solitons. Consequently after linear interaction
with the plane wave profile of each of the solitons may considerably differ
from each other.

Conditions | Components
---|---
$\frac{1}{2}(q_{1}-q_{2})$ | $\frac{1}{2}(q_{1}+q_{2})$
$0<b_{1}<|a_{1}|$ $0<b_{2}<|a_{2}|$ |  $a_{1}>0$, $a_{2}>0$ |  Anti-dark+ Dark bright |  Dark bright+ Anti-dark
$a_{1}>0$, $a_{2}<0$ |  Dark bright+ Dark bright |  Dark bright+ Dark bright
$a_{1}<0$, $a_{2}>0$ |  Dark bright+ Dark bright |  Dark bright+ Dark bright
$a_{1}<0$, $a_{2}<0$ |  Anti-dark+ Dark bright |  Dark bright+ Anti-dark
$b_{1}<-|a_{1}|$ $b_{2}<-|a_{2}|$ |  $a_{1}>0$, $a_{2}>0$ |  W-shaped+ Anti-dark |  Anti-dark+ W-shape
$a_{1}>0$, $a_{2}<0$ |  Anti-dark+ Anti-dark |  W-shaped+ W-shaped
$a_{1}<0$, $a_{2}>0$ |  Anti-dark+ Anti-dark |  W-shaped+ W-shaped
$a_{1}<0$, $a_{2}<0$ |  Anti-dark+ W-shaped |  W-shaped+ Anti-dark
$-|a_{1}|<b_{1}<0$ $-|a_{2}|<b_{2}<0$ |  $a_{1}>0$, $a_{2}>0$ |  Dark+ Anti-dark |  Anti-dark+ Dark
$a_{1}>0$, $a_{2}<0$ |  Dark bright+ Dark bright |  Dark bright+ Dark bright
$a_{1}<0$, $a_{2}>0$ |  Dark bright+ Dark bright |  Dark bright+ Dark bright
$a_{1}<0$, $a_{2}<0$ |  Anti-dark+ Dark |  Dark+ Anti-dark
$b_{1}>|a_{1}|$ $b_{2}>|a_{2}|$ |  $a_{1}>0$, $a_{2}>0$ |  Anti-dark+ Dark bright |  Dark bright+ Anti-dark
$a_{1}>0$, $a_{2}<0$ |  Dark bright+ Dark bright |  Dark bright+ Dark bright
$a_{1}<0$, $a_{2}>0$ |  Dark bright+ Dark bright |  Dark bright+ Dark bright
$a_{1}<0$, $a_{2}<0$ |  Dark bright+ Anti-dark |  Anti-dark+ Dark bright
Table 2: Conditions for various profiles obtained during the interaction of
two soliton and plane wave.

If the phase matching condition between the first soliton and plane wave is
satisfied but not with the second soliton then the first soliton exhibits a
profile as per case-$I$ and the second soliton profile is given as per
case-$IV$ as discussed in one soliton. However, the time period for the second
soliton may vary from a value greater than zero to a large value when
$|a_{1}|$=$|a_{2}|$ and $|b_{1}|$ nearly equal to $|b_{2}|$. We find various
interaction profiles such as soliton-soliton, soliton-breather, breather-
breather depending on which of the four cases are satisfied with respect to
first soliton. Table 2 shows possible combinations of soliton profiles. Figure
5, shows an anti-dark and $W$-shaped soliton in $u_{2}$ and $v_{2}$ for
$a_{1}=-1$, $b_{1}=-6$, $a_{2}=-1$, $b_{1}=-6.01$, in figure 5 two dark bright
solitons are shown with parameters $a_{1}=3$, $b_{1}=-1$, $a_{2}=-3$,
$b_{1}=-1.01$ in component $u_{2}$ and two $W$-shaped solitons in component
$v_{2}$ 5 with parameters $a_{1}=1$, $b_{1}=-6$, $a_{2}=-1$, $b_{1}=-6.01$.

There are however, several other profiles are also seen for arbitrary soliton
parameters such as multi-peak $\&$ anti-dark soliton, breather $\&$ anti-dark
soliton interaction etc. For instance, multi-peak $\&$ anti-dark interaction
is seen in figure 5 for parameters $a_{1}=1$, $b_{1}=3$, $a_{2}=0.5$,
$b_{1}=-1.5$ and breather $\&$ anti-dark interaction is seen for parameters
$a_{1}=1$, $b_{1}=3$, $a_{2}=1.5$, $b_{1}=-0.5$ as shown in figure 5. The
conditions along with the corresponding soliton profiles are given in the
table 2. Thus from the above analysis we notice that the soliton profile may
be changed by changing the plane wave background. For instance a bright
soliton may be transformed into dark, dark-bright or to other desirable shape
through FWM. In case of two solitons two interacting solitons may have
different shape or same shape again depending upon the soliton parameters.
Thus FWM allows controlling and engineering soliton profile.

##  4 Conclusion and Discussion

In this paper, we have studied the linear interference between a plane wave
and bright soliton in the presence of FWM effect. We show that various
localized structures, namely W-shaped soliton, dark, anti-dark, dark-bright,
breathers and multi-peak solitons are possible through linear interaction
between a plane wave and soliton. These localised structures arise due to the
inbuilt relative phase between the plane wave and bright soliton. The linear
interference has been divided into four cases based on the spatial and
temporal relation of the coefficients of the two waves. The linear interaction
of plane wave and two soliton interaction gives rise to several combinations
of different localized structures since nonlinear interaction between the
soliton alters the soliton’s phase after the interaction. In BEC linear
interaction of waves is normal. In such interaction waves of different
structures is theoretically possible; which may be worthy of investigation
experimentally in future. In nonlinear optics through linear interactions
several pulse structures may be generated and engineered by changing the plane
wave background and soliton parameters. We hope that the present investigation
will be useful in studying the application of FLE in nonlinear optics and
various other fields.

## Acknowledgement

S Talukdar and R Dutta Acknowledge DST, Govt. of India for Inspire fellowship,
grant nos. DST Inspire Fellowship 2020/IF200278; 2020/IF200303.

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