# Physical Design and Experimental Verification of a Huygens’ Metasurface Two-
lens System for Phased-array Scan-angle Enhancement

Jaemin Kim, Gleb A. Egorov, and George V. Eleftheriades The authors are with
the Edward S. Rogers Sr. Department of Electrical and Computer Engineering,
University of Toronto, Toronto, Canada (e-mail<EMAIL_ADDRESS>gelefth@waves.utoronto.ca).

###### Abstract

Over the past decades, many radome designs to extend the angular scan range of
phased-array antennas have been devised by utilizing dielectric materials and
metamaterials. More recently, metasurface technology such as planar lenses and
beam deflectors have been applied to phased arrays, enabling scan-angle
enhancers to have a low profile. In this work, a physical Huygens’ metasurface
(HMS) two-lens system for scan-angle doubling of a phased array is presented.
For the HMS unit cells, the wire-loop topology is deployed to achieve high
transmission for the required phase-angle shift. The proposed two-lens system
is analyzed by full-wave simulations and experiments. The simulation results
demonstrate that the scan angle doubles when the incident angle is below
$None$ in accordance to the design specification. Furthermore, the directivity
degradation of the refracted beams by the two-HMS lenses is in good agreement
with theory. Finally, a fabricated two-lens system with two $15\lambda$ long
by $15\lambda$ wide metasurface lenses and a $16\times 16$-element patch
antenna array as a source is experimentally verified at
$10\text{\,}\mathrm{GHz}$. The experimental results are in good agreement with
the simulated results by showing angle-doubling performance with $None$ scan
errors.

###### Index Terms:

Phased arrays, beam steering, Huygens’ metasurface, metasurface lens.

##  I Introduction

Phased array antennas are being used in mobile and satellite applications to
scan a beam electronically [1]. However, the scanning range of a phased array
can be limited due to various issues such as directivity degradation and the
presence of grating lobes for larger element spacings such as the case of sub-
arraying [1]. Much research has been conducted in the last decades to enhance
the scan range of phased arrays. A cylindrical dome antenna with a phase-
incurring layer was one of the early theoretical solutions for scan-angle
enhancement [2]. A dielectric dome with a linear source array was realized
experimentally to extend the range of scan angle by optimizing the shape and
thickness of the dome using geometrical optics [3]. Moreover, radomes based on
recent research topics such as transformation optics and negative-index
metamaterials have been studied [4, 5]. Nevertheless, the thick profile of
these radomes impede the realization of compact angle-enhancement systems.

To reduce the thickness of the antenna architecture, metasurface technology
has been applied to scan-angle enhancement of phased arrays [6, 7, 8]. In [6],
a curve shaped metasurface dome (meta-radome) has been used to deflect the
angle of the incident beam and improve the scan range of a phased array. This
approach reduces the volume of the angle enhancement system, but the phased-
array elements need to be excited with non-uniform magnitudes and non-linear
phase shifts for directive radiation of the beam through the meta-radome.

More recently, Huygens’ metasurface (HMS) theory has been utilized for
realizing scan-angle enhancement systems [7, 8]. The HMSs are subwavelength
thin sheets comprising scatterers or unit cells which contain co-located
orthogonal electric and magnetic dipole moments [9]. HMSs have the ability to
refract a given incident field to a desired transmitted field – the particular
incident and transmitted fields are chosen during the design of a particular
HMS. The discontinuity in the fields is supported by the electric and magnetic
surface current densities which are excited by the incident electromagnetic
field in steady state [9]. Thus, they can provide the desired spatially-
varying phase shift between the incident and transmitted fields. The HMSs have
been used in a wide range of applications such as reflectionless wide-angle
refraction and beamforming with bianisotropic surfaces [10, 11, 12]. More
recently, HMSs have demonstrated the ability of independenet phase and
amplitude control for precise antenna beamforming through the excitation of
auxialiary surface waves [13, 14]. In terms of extending the scan range of
phased arrays, HMS planar lenses with ray optical theory were discussed in [7,
8]. Compared with the meta-domes and radomes in [2, 3, 6], the structures with
HMS lenses are planar. Furthermore, the array elements can be linearly phased
as usually. However, these HMS lenses have been based on ideal impedance
sheets.

In this paper, we demonstrate a physical design, including losses, for a two-
HMS lens system for angle doubling by full-wave simulations and experiments.
Here, the desired scan range of the angle enhancement system is from $None$ to
$None$, whereas the source array steers its beam electronically between $None$
and $None$. The HMS unit cells are designed with a wire-loop topology to
exhibit high transmittance over all required phase angles. Note that the
stacked-layer unit cell topology is also a popular approach for designing HMSs
by utilizing the equivalent transmission-line model [9]. However, the stacked-
layer HMS unit cells can suffer from significant loss when the phase angle of
$\mathrm{S}_{21}$ is near $None$ due to resonance [15]. Hence the power
transmission efficiency of HMSs can be compromised. On the other hand, the
wire-loop unit cells we utilize in this work exhibit high transmittance for
the desired phase angles of $\mathrm{S}_{21}$ including $None$ as shown with
full-wave simulations. Additionally, the scan angle of the lossy two-HMS scan-
angle doubler is enhanced almost twice and with low scan error, when the
incident beam angle is between $None$ and $None$, according to the desired
specification.

Figure 1: A schematic of the proposed two-HMS lens angle-enhancing system with
coordinate system vectors and angles for the incident and refracted beams.
Parameters such as distances and focal lengths are specified. The dotted and
solid ray paths of the system depict beam scanning when the incident beam
angle is $None$ and $\theta_{\mathrm{inc}}$, respectively.

##  II Two-lens system for scan enhancement

Two-HMS lens scan-angle enhancement is achieved with a converging lens and a
diverging lens which are placed in the near-field region of a source array as
depicted in Fig. 1 [8]. Ray tracing through the two-lens system can be
expressed by ray transfer matrix analysis [16]. The ray transfer matrix for a
two-lens system is shown in (1) and (2), which gives the position and angle of
a ray when passing through the lenses.

$\begin{bmatrix}x^{\prime}\\\
\theta_{\mathrm{ref}}\end{bmatrix}=\begin{bmatrix}A&B\\\
C&D\end{bmatrix}\begin{bmatrix}x\\\ \theta_{\mathrm{inc}}\end{bmatrix},$ (1)

where A, B, C, and D are given by

$\begin{split}&A=1-\frac{d_{l}}{f_{1}}\\\
&B=d\left(1-\frac{d_{l}}{f_{1}}\right)+d_{l}\\\
&C=\frac{1}{f_{1}f_{2}}(d_{l}-f_{1}-f_{2})\\\
&D=1-\frac{d_{l}}{f_{2}}+\frac{d}{f_{1}f_{2}}(d_{l}-f_{1}-f_{2}),\\\
\end{split}$ (2)

where $d$ is the distance between a source and the first lens, $d_{l}$ is the
distance between the two lenses, and $f_{1}$ and $f_{2}$ are the focal lengths
of each lens. For the two-lens system to work as a scan enhancer, $C$ and $D$
in the transfer matrix must satisfy the condition in (3).

$\begin{split}&d_{l}-f_{1}-f_{2}=0\\\ \end{split}$ (3)

As a result, the desired angle of a ray passing through the two-lens system
can be obtained by (4), where $\alpha$ is the angular scan enhancement factor
for the two-lens system.

$\begin{split}&\theta_{\mathrm{ref}}=\left(1-\frac{d_{l}}{f_{2}}\right)\theta_{\mathrm{inc}}=\alpha\theta_{\mathrm{inc}}\\\
\end{split}$ (4)

To fulfill the requirement for the scan-angle doubler in the two-lens system,
$\alpha$ has to be 2, and it leads to $f_{1}=2d_{l}=f_{c}$ and
$f_{2}=-d_{l}=f_{d}$, where the $f_{c}$ and $f_{d}$ are the focal lengths of
the converging and diverging lens, respectively. Here, the angle-doubling
system takes $d_{l}=4\lambda$ at $10\text{\,}\mathrm{GHz}$ resulting in
$f_{c}=8\lambda$ and $f_{d}=-4\lambda$ as shown in Fig. 1. The source array is
located at a distance $d=4\lambda$ away from the first converging lens. Note
that the source array can be placed at any $d$ from the lens as the theory
shows, but $d=4\lambda$ was chosen for our experimental convenience. The array
comprises 16-element infinitely long electric current line sources,
$\lambda/2$-spaced, to propagate transverse electric (TE) polarized fields.
The array is phased to create off-broadside beams between $None$ and $None$.

##  III Huygens’ metasurface lens design

To realize the scan-angle enhancer with the two lenses, we utilize Huygens’
metasurface (HMS) lenses because the HMS unit cells can be designed with high
magnitude of $\mathrm{S}_{21}$ over all required phase angles [15, 17]. The
phase angles of $\mathrm{S}_{21}$ of the HMS lenses should be specified by (5)
as the quadratic phase profile for a lens [7],

Figure 2: Two kinds of wire-loop Huygens’ metasurface unit cells and
parameters used for S-parameter optimization. A loop structure is realized
with printed capacitors on the outer layers in (a) and (b). For the wire
structure in the middle layer, a printed capacitor in (a) and a printed
inductor in (b) are utilized. The unit cell area in the $xy$-plane is
$4\text{\,}\mathrm{mm}$ $\times$ $4\text{\,}\mathrm{mm}$.

$\phi(x)=\text{sgn}(f)\frac{2\pi}{\lambda}\left(\sqrt{x^{2}+f^{2}}-\lvert
f\rvert\right),$ (5)

where $f$ is the focal length of a lens and $x$ is the position from the
center of the lens. For sgn$(f)$, $+$ and $-$ are specified for the converging
and diverging lens. The HMS unit cell contains co-located orthogonal electric
and magnetic dipole moments to satisfy the boundary conditions for a desired
wave transformation. The electric and magnetic dipoles in the unit cell are
represented by a scalar surface electric impedance and magnetic admittance for
TE polarized fields [9].

In this work, we make use of a wire-loop topology to design the HMS unit cells
[18]. The unit cells have certain surface impedances and admittances for them
to synthesize the required $\mathrm{S}_{21}$ phase angles. The surface
impedances of the wire and loop are determined by printed capacitors or
inductors in each structure. Fig. 2 describes the HMS unit cell layout, which
consists of physical wire and loop structures with simulation parameters for a
Rogers RO3003 substrate (thickness = $1.524\text{\,}\mathrm{mm}$,
$\varepsilon_{r}=3$, and $\tan\delta=0.0013$) and 2929 bondply (thickness =
$0.0508\text{\,}\mathrm{mm}$ and $\varepsilon_{r}=2.94$). The thickness of the
copper layer on the substrates to form the wire and loop structures is
$18\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The wire structure is shaped by a
printed capacitor or inductor on the inner layer of one of the substrates. The
loop structure is formed by printed capacitors on the outer layers of the two
substrates with two vias. The two substrates are attached by a bonding
material. The area of the unit cell surface is $4\text{\,}\mathrm{mm}$
$\times$ $4\text{\,}\mathrm{mm}$.

The magnetic field in the $x$ direction only polarizes the loop. However, the
electric field in the $y$ direction excites not only the wire in the middle
layer but also the loop. Therefore, the design procedure is required to set
the desired magnetic admittance of the unit cell by tuning the printed
capacitor in the loop, and then the electric impedance is adjusted by tuning
the printed capacitor or inductor in the wire [18].

Based on the design procedure above, a wire-loop unit-cell library covering
the entire $\mathrm{S}_{21}$ phase range is created. The unit cells were
synthesized by the full-wave electromagnetics solver CST microwave studio. In
calculation, infinite periodic array analysis with these unit cells was
utilized. Fig. 3(a) and (b) show the magnitude and phase of $\mathrm{S}_{21}$
of the HMS unit cells at the corresponding unit cell position for the
converging lens ($f=8\lambda)$ and the diverging lens ($f=-4\lambda$),
respectively. Moreover, the required phase profile, $\phi(x)$, for the two
lenses is shown with the black curve. As shown $\lvert\mathrm{S}_{21}\rvert$
of the HMS unit cells for the converging lens and the diverging lens are 0.97
and 0.98 on average, respectively.

Figure 3: Phase angle and magnitude of $\mathrm{S}_{21}$ for each unit cell
forming the converging lens ($f=8\lambda$) in (a) and the diverging lens
($f=-4\lambda$) in (b). The simulated phase angle of $\mathrm{S}_{21}$ (black
dots) matches the desired phase angle calculated by (5). The simulated
magnitude of $\mathrm{S}_{21}$ is expressed with blue dots. Figure 4: The
simulation structure with the location of the lenses and the source array, and
boundary condition settings. Figure 5: Full-wave simulation results: the far-
field radiation pattern of the incident beam and the refracted beam (left),
and the absolute value of the refracted electric field in the $y$ direction,
$\lvert\mathrm{Re}\\{E_{y}\\}\rvert$ (right). (a) Incident beam angle is
$None$. (b) Incident beam angle is $None$.

##  IV Full-wave simulation results of the two-lens HMS system

In this section, the designed two-lens HMS system as an angle-doubler with the
wire-loop unit cells is full-wave simulated by CST microwave studio. In the
simulation, the realized periodicity of the 1D lens array and the source array
are in the $y$ direction. Moreover, horizontal upper and lower perfect
electric conductor (PEC) walls were set at $y=2$ $\mathrm{m}\mathrm{m}$ and
$-2$ $\mathrm{m}\mathrm{m}$ as boundary conditions. The specific simulation
setup is shown in Fig. 4. Fig. 5(a) depicts the radiation pattern of the
incident beam and the refracted beam by the two-lens system in the H-plane and
the electric field distribution $\lvert\mathrm{Re}\\{E_{y}\\}\rvert$ when the
incident beam is at broadside. The transmission efficiency of the two-HMS lens
system defined as the ratio of the normally transmitted beam power passing
through the two lenses to the incident beam power at broadside is $85\%$.

Figure 6: (a) The scan error ($\mathrm{2\theta_{inc}}-\mathrm{\theta_{ref}}$)
of the two-HMS lens angle doubling system. (b) The directivities of the
incident beam (black square solid line) and the refracted beam by the two-HMS
lenses at each $\theta_{\mathrm{ref}}$ (blue circle solid line). (c)
Directivity degradation ($D_{\mathrm{inc}}-D_{\mathrm{ref}}$) in the
simulation (blue circle solid line) calculated with the values in (b) and in
the theory (red square solid line) calculated by (6). The simulation of the
HMSs with losses utilize the material constants of copper and Rogers RO3003.
The simulation of the HMSs without losses, whose results are shown with dotted
lines in the three plots, is conducted with PEC and lossless dielectric
materials.

In addition, the reflectance of the broadside beam to the system is $3.4\%$.
As shown, at broadside the incident beam from the phased array becomes a
diverging beam with a directivity degradation of $3.8\text{\,}\mathrm{dB}$.
Note that theoretically the directivity reduction when passing through the
two-lens system is given by (6) [8].

$\frac{D_{\mathrm{ref}}(\theta_{\mathrm{ref}})}{D_{\mathrm{inc}}(\theta_{\mathrm{inc}})}=\frac{1}{\alpha}\cdot\frac{\cos{\theta_{\mathrm{ref}}}}{\cos{\theta_{\mathrm{inc}}}},$
(6)

where $D_{\mathrm{ref}}$ is the directivity of the beam passing through the
two lenses, $D_{\mathrm{inc}}$ is the directivity of the beam from the source
array, and $\alpha$ is the angular scan enhancement factor. Accordingly, for
angle doubling, a from $3$ to $3.5\text{\,}\mathrm{dB}$ degradation of the
directivity of this system is unavoidable, when the incident beam angle is
between $None$ and $None$. Fig. 5(b) shows the same plot as Fig. 5(a), but the
incident beam angle is changed to $None$ off broadside. It is shown that the
beam angle is extended from $None$ to $None$ by the two HMS lenses.
Furthermore, the directivity difference between the incident beam and the
refracted beam is $3.63\text{\,}\mathrm{dB}$, which is in good agreement with
the expected degradation at this incident angle, which is
$3.48\text{\,}\mathrm{dB}$ by (6).

Figure 7: Experimental setup, showing the source array illuminating the two
lenses. The E-plane lies vertically as shown in the vector diagram. Figure 8:
Measured source array pattern with the lenses removed (see Fig. 7).

Lastly, the angle doubling performance of the two-HMS lens system at various
incident beam angles is analyzed. Fig. 6(a) depicts that the two-HMS lens
system performs well as an angle doubler by showing that the magnitude of the
scan error defined as
$\lvert\mathrm{2\theta_{inc}}-\mathrm{\theta_{ref}}\rvert$ is less than $None$
when the incident angles are below $None$. Fig. 6(b) describes the peak
directivities of the incident beam from the phased array and the refracted
beam passing through the angle-doubler. Fig. 6(c) shows that the directivity
degradation at incident beam angles below $None$ is $3.7\pm 0.7$
$\mathrm{d}\mathrm{B}$ by the simulation. It should be noted that there is a
non-negligible difference between simulation and theory at smaller incident
angles below $None$ in Fig. 6(c). This discrepancy might be due to the fact
that the physical unit cells of the two lenses have a non-uniform
transmittance spatially, in which the difference is $8\%$ at most, as inferred
in Fig. 3. In fact, the simulation results with the analogous system using two
dielectric lenses show similar directivity degradation to the theory below
$None$ of the incident angle. It is also worth noting that material losses of
the HMSs do not affect the directivity degradation, as described in Fig. 6.
Namely, the simulation results conducted with ideal materials such as PEC and
lossless dielectric substrate ($\tan\delta=0$) show very little difference
from those with copper and a lossy dielectric substrate (Rogers RO3003). Here,
the simulated power loss through the two lenses is $11.6\%$.

Figure 9: Comparison of measured and simulated angle-doubled patterns at
$10\text{\,}\mathrm{GHz}$. Angle doubling is observed, and experiment closely
resembles the simulations. Figure 10: Complete measured scan error and power
degradation data for the two-lens doubler.

##  V Experimental results

Two PCB metasurface lenses of size $452\text{\,}\mathrm{mm}$ $\times$
$452\text{\,}\mathrm{mm}$ were fabricated following the validation by full-
wave simulation. The designed metasurface lenses were repeated $113$ times,
which is $15\lambda$ in length at $10\text{\,}\mathrm{GHz}$, along E-plane. To
demonstrate the enhancement of the scan angle range of a phased array with the
two metasurface lens system, an anechoic chamber antenna test was performed.
Fig. 6 shows the overall two-lens system with a source array for incident
beams in our anechoic chamber. For the incident beam source, we have designed
and fabricated a $16\times 16$ patch antenna array with element spacing of
$\lambda/2$, in which the patches were aperture coupled to the corporate feed
network. This array produced a beam at broadside, and the patch array was
rotated around its center in order to experimentally produce off-broadside
beams. Fig. 7 shows measured patterns of the array at
$10\text{\,}\mathrm{GHz}$.

Fig. 9 compares measured and simulated patterns at the particular frequency of
$10\text{\,}\mathrm{GHz}$. To compare the experimental and simulation data
(for which some subtle differences exist) and because we are interested in the
directivity degradation, the plotted patterns are normalized to the peak of
the source of the particular scenario being plotted – for example, in part (b)
the $None$ incidence scenario is plotted, for which the array is tilted to
$None$ but still has the same peak while in simulation the array is steered
and thus has a different peak. Clearly, the experimental setup indeed achieves
angle-doubling with little error. For completeness, Fig. 10 shows obtained (a)
scan error and (b) power degradation for all measured values of $f$ and
$\theta_{\mathrm{inc}}$. It appears that the angle doubler performs the best
for frequencies within $10\pm 0.2$ $\mathrm{GHz}$. The scan error is small,
and the power degradation is $4.7\text{\,}\mathrm{dB}$ on average. These
values suggest that the losses, non-ideal reflections and scattering all
amount to around $1.7\text{\,}\mathrm{dB}$ degradation in directivity
(received power, if speaking in terms of the experiment), which is expected.

##  VI Conclusion

A physical design for a two-Huygens’ metasurface lens system to double the
scan angle of a phased array has been numerically and experimentally
demonstrated. Design parameters for the two-HMS angle doubler are obtained
based on ray optics analysis. The HMS lenses are designed for a quadratic-like
$\mathrm{S}_{21}$ phase profile, and the wire-loop topology is deployed to
implement their unit cells. The HMS-lens doubler is placed in the near-field
of the phased array leading to a compact architecture. The performance of the
system has been verified by full-wave simulations and experiments. First of
all, the magnitude of $\mathrm{S}_{21}$ of the unit cells we utilize for the
converging lens and the diverging lens are 0.97 and 0.98 on the average,
respectively, covering the entire $\mathrm{S}_{21}$ phase angles. Furthermore,
the proposed two-lens HMS angle doubler functions properly showing that the
scan angle of a uniform phased array is enhanced by a factor of two when the
angle of incidence is between $None$ and $None$ which was our design
specification. The simulation results show that the directivities of the
refracted beams by the two-HMS lens system are degraded by nearly
$3.7\text{\,}\mathrm{dB}$, in good agreement with theory. Experimental results
also match well with the simulation by showing that angle-doubling is well
achieved with low scan error, $None$, and around $4.7\text{\,}\mathrm{dB}$
power degradation at $10\text{\,}\mathrm{GHz}$. Finally, it can be pointed out
that if the proposed two-lens HMS system is reversed, it can function as a
beam expander to enhance the directivity of a feeding antenna [19].

## Acknowledgment

Computations were performed on the Niagara supercomputer at the SciNet HPC
Consortium. SciNet is funded by: the Canada Foundation for Innovation; the
Government of Ontario; Ontario Research Fund - Research Excellence; and the
University of Toronto.

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# Two-component breather solution of the nonlinear wave equation

G. T. Adamashvili  Technical University of Georgia, Kostava str.77, Tbilisi,
0179, Georgia.
email<EMAIL_ADDRESS>
###### Abstract

A nonlinear wave equation which describes different nonlinear effects in
various fields of research, was considered. In two particular cases, this
equation was reduced to the Sine-Gordon equation and the Born-Infeld equation.
Using the slowly varying envelope approximation and the generalized
perturbative reduction method, the nonlinear wave equation was transformed to
coupled nonlinear Schrödinger equations for auxiliary functions. An explicit
analytical solution of a nonlinear wave equation in the form of a two-breather
molecule was obtained. One breather oscillated with the sum, and the other
with the difference of frequencies and wave numbers. The obtained solution
coincides with the solutions of the two-breather molecule found in a number of
well-known equations from different areas of physics. It is shown that in a
particular case of the small amplitude waves, a solution in the form of a two-
breather molecule for the nonlinear Klein-Gordon equation coincides with the
vector $0\pi$ pulse of the self-induced transparency which is presented under
less stringent conditions compared to the same solution of this equation
obtained earlier. _Keywords:_ Generalized perturbation reduction method, Two-
breather molecule, Born-Infeld equation, Sine-Gordon equation, Nonlinear
Klein-Gordon equation.

###### pacs:

05.45.Yv, 02.30.Jr, 52.35.Mw

##  I Introduction

At the propagation of nonlinear waves, one of the most important and
interesting phenomena is the formation of nonlinear solitary waves of a
stationary form. Although these waves have been studied and observed for a
very long time, interest in them does not fade away. This is because they
express the most fundamental property of strongly nonequilibrium states of
nonlinear systems when the parameters characterizing such states remain
unchanged. Although nonlinear solitary waves occur in various fields of
physics, for waves of a completely different nature and physical functions
characterizing them, for example, the strength of the electric field of
optical waves, the tensor of deformation for sound waves, the displacement of
the surface from the undisturbed state for the water waves and others,
nevertheless their properties are identical. The nonlinear solitary waves
which arise in different nonlinear physical phenomena in various systems, are
described by means of the various nonlinear partial differential equations.
These equations include the Korteweg de Vries equation, the Boussinesq
equation, the Hirota equation, the nonlinear Schrödinger equation, the
Benjamin-Bona-Mahony equation, the Bloch-Maxwell system of equations and many
others [1-9].

Among the nonlinear partial differential equations which have solutions in the
form of nonlinear solitary waves, one can single out the well-known Sine-
Gordon equation and the Born-Infeld equation. Each of these equations
describes nonlinear solitary waves in different areas of research. For
example, the Sine-Gordon equation describes the geometry of surfaces with
Gaussian curvature, the Josephson transition, dislocations in crystals, waves
in ferromagnets associated with the rotation of magnetization, the properties
of elementary particles and the nonlinear phenomena in the various fields of
physics: optics, acoustics, plasma physics, semiconductor quantum dots,
graphene, optical and acoustical metamaterials, and others [1-4]. The Born-
Infeld equation is used to describe the interaction of electromagnetic waves
in nonlinear electrodynamics, various phenomena in the field theory, the
theory of strings, and some of the atomic experiments [1, 10-14]. These
equations have been studied in sufficient detail by various mathematical
methods and their solutions are well known.

Although each of these two equations describes a fairly wide range of
absolutely different phenomena in various fields of physics, nevertheless,
both of these equations are nonlinear Maxwell wave equations but with
different nonlinear terms. Consequently arises the natural question: is it
possible to describe all the above effects connected with these two equations
in a unified way, by means of one more general equation?

In order to answer this question we consider the following more general
nonlinear wave equation

$\frac{\partial^{2}U}{\partial t^{2}}-C\frac{\partial^{2}U}{\partial
z^{2}}=-\alpha_{0}^{2}\sin U-A(\frac{\partial U}{\partial
t})^{2}\frac{\partial^{2}U}{\partial z^{2}}-\sigma(\frac{\partial U}{\partial
z})^{2}\frac{\partial^{2}U}{\partial t^{2}}+B\frac{\partial U}{\partial
t}\frac{\partial U}{\partial z}\frac{\partial^{2}U}{\partial t\partial z},$
(1)

or in the dimensionless form

$\frac{\partial^{2}U}{\partial t^{2}}-\frac{\partial^{2}U}{\partial
z^{2}}=-\sin U-(\frac{\partial U}{\partial
t})^{2}\frac{\partial^{2}U}{\partial z^{2}}-(\frac{\partial U}{\partial
z})^{2}\frac{\partial^{2}U}{\partial t^{2}}+2\frac{\partial U}{\partial
t}\frac{\partial U}{\partial z}\frac{\partial^{2}U}{\partial t\partial z},$

where $U(z,t)$ is a real function of space coordinate $z$ and time $t$ and
represents the wave profile, while $\alpha_{0}^{2},\;$ $A,\;B,\;$ $C,$ and
$\sigma$ are the real constants.

Eq.(1) in particular cases goes into the Sine-Gordon equation and the Born-
Infeld equation. Indeed, when the condition $A=B=\sigma=0$ is fulfilled,
Eq.(1) is transformed into the Sine-Gordon equation

$\frac{\partial^{2}U}{\partial t^{2}}-C\frac{\partial^{2}U}{\partial
z^{2}}=-\alpha_{0}^{2}\sin U.$ (2)

This nonlinear equation was analyzed by means of inverse scattering transform
[2, 4], by which it is possible to obtain the complete solution of Eq.(2) in
the form of the nonlinear solitary waves.

Among nonlinear solitary waves, the nonlinear waves have a relatively low
amplitude for which

$U<<1$ (3)

are considered quite often. Under the condition Eq.(3), the Sine-Gordon
equation (2) is reduced to the nonlinear Klein-Gordon equation [1]

$\frac{\partial^{2}U}{\partial t^{2}}-C\frac{\partial^{2}U}{\partial
z^{2}}=-\alpha_{0}^{2}U+\frac{\alpha_{0}^{2}}{6}U^{3}-\mathcal{O}(U^{5}).$ (4)

In case when the condition $\alpha_{0}^{2}=0$ is fulfilled, Eq.(1) is reduced
to the Born-Infeld equation

$\frac{\partial^{2}U}{\partial t^{2}}-C\frac{\partial^{2}U}{\partial
z^{2}}=-A(\frac{\partial U}{\partial t})^{2}\frac{\partial^{2}U}{\partial
z^{2}}-\sigma(\frac{\partial U}{\partial z})^{2}\frac{\partial^{2}U}{\partial
t^{2}}+B\frac{\partial U}{\partial t}\frac{\partial U}{\partial
z}\frac{\partial^{2}U}{\partial t\partial z}.$ (5)

This is a nonlinear modification of the Maxwell wave equation, which includes
the so-called Born-Infeld nonlinearity.

Sometimes $2+1$ (two space coordinates and time) dimensional version of the
Born-Infeld equation also has been investigated [15].

Under the condition Eq.(3), the nonlinear wave equation (1) is reduced to the
form

$\frac{\partial^{2}U}{\partial t^{2}}-C\frac{\partial^{2}U}{\partial
z^{2}}=-A(\frac{\partial U}{\partial t})^{2}\frac{\partial^{2}U}{\partial
z^{2}}-\sigma(\frac{\partial U}{\partial z})^{2}\frac{\partial^{2}U}{\partial
t^{2}}+B\frac{\partial U}{\partial t}\frac{\partial U}{\partial
z}\frac{\partial^{2}U}{\partial t\partial
z}-\alpha_{0}^{2}U+\frac{\alpha_{0}^{2}}{6}U^{3}-\mathcal{O}(U^{5}).$ (6)

When considering nonlinear solitary waves, there are two different types of
them: single-component (scalar) and two-component (vector) solitary nonlinear
waves. These waves can be formed both in different and in the same physical
system. Their properties and formation conditions are different. A single-
component wave propagates in a medium without changing its profile. Completely
different is a two-component nonlinear wave, which is a bound state of two
wave packets having the same speeds and directions of propagation, and in
general cases, different oscillation frequencies. These wave packets can have
both parallel polarizations or mutually perpendicular polarizations, for
example, for waveguide modes. Such a two-component wave is a vector breather
or a two-breather molecule. In the process of propagation, the breathers that
make up the molecule interact with each other and exchange energy.

Some effects cannot be described using single-component nonlinear waves and it
becomes necessary to use the concept of a breather molecule [16-18]. The most
striking example is the effect of self-induced transparency, one of the main
pulses of which is precisely a special type of the two-breather molecule,
which consists of two connected breathers with the same polarizations, one of
the breathers oscillates with the sum, and the second with the difference in
frequencies and wave numbers. Such a two-component solitary nonlinear wave was
first considered at the nonlinear coherent interaction of the optical wave
with the system of the resonance atoms and is called a vector $0\pi$ pulse of
self-induced transparency [19-22].

Various mathematical approaches are used to analyze the properties of solitary
nonlinear waves. Among them, the perturbative reduction method for analysis of
the single-component nonlinear waves very often meet [23, 24]. This method
uses one complex auxiliary function and two real parameters, which is enough
to study single-component nonlinear waves but not enough to study two-
component nonlinear waves. To study two-component nonlinear waves, a
generalized perturbative reduction method was developed (see, for instance [9,
19-22] and references therein) which uses two complex functions and eight real
parameters which are enough for the analysis of two-component nonlinear waves.

In the theory of self-induced transparency, the role of the second derivatives
of the strength of the electric field of the wave with respect to the spatial
coordinate and time in the Maxwell wave equation was studied using various
mathematical methods. In particular, using the perturbative reduction method
it was found that the second derivatives lead to only small quantitative
corrections to the values of the parameters of the pulses of self-induced
transparency (see, for instance [25, 26]).

Using the generalized perturbative reduction method, in the theory of self-
induced transparency qualitatively new result was obtained. Namely, thanks to
this method it was shown that in the Maxwell wave equation the second
derivatives of the strength of the electric field of the pulse with respect to
the spatial coordinate and time the determining role play for an adequate
study of the effect of the self-induced transparency. To take into account the
terms of the second derivative in the Maxwell wave equation was allowed to
prove that one of the main pulses of self-induced transparency is the vector
$0\pi$ pulse (two-breather molecule), but not the single-component scalar
$0\pi$ pulse, as this was previously believed. The reason for this is that in
the study of self-induced transparency in the Maxwell wave equation, the
second derivatives of the strength of the electric field of the pulse with
respect to the spatial coordinate and time were neglected, or their influence
in the process of the formation of the pulses of self-induced transparency was
investigated by using the standard perturbative reduction method, or by means
of some other mathematical methods which intended for analysis of the single-
component waves [19-22, 25-34].

Although the vector $0\pi$ pulse of self-induced transparency, which is a
special type of two-breather molecule, was first noticed for optical nonlinear
waves, later, using the generalized perturbative reduction method, the same
two-breather molecules were discovered for waves of a different nature
described by other nonlinear partial differential equations, such as are the
Boussinesq-type equation, the Benjamin-Bona-Mahony equation, and the Hirota
equation [7, 9, 35-37].

Taking into account that a two-breather molecule in which one breather
oscillates with the sum, and the second with the difference in frequencies and
wave numbers, qualitatively changes the physical picture of self-induced
transparency that cannot be described within the framework of single-component
nonlinear waves, as well as the fact that such a two-breather molecule arises
in completely different areas of physics for a function of a different nature,
allows concluding that existence of this type of a two-breather molecule, i.e.
type of the vector $0\pi$ pulse of self-induced transparency, expresses the
rather general property of matter. Thus, its further study for new equation
(6) is relevant and reasonable.

The main purpose of this work is to use the generalized perturbative reduction
method to obtain a solution to the nonlinear wave equation (6) in the form of
the two-breather molecule, in which one breather oscillates with the sum and
the second with the difference in frequencies and wave numbers.

The rest of this paper is organized as follows: In Section II, we consider the
nonlinear wave equation (6) for slowly varying complex envelope functions.
Using the generalized perturbative reduction method, we transformed the
obtained equation into coupled nonlinear Schrödinger equations for auxiliary
functions. In Section III, we present the explicit analytical solution of a
nonlinear wave equation (6) in the form of a two-breather molecule Eq.(III).
Finally, in Section IV, we discuss the obtained results.

##  II The generalized perturbative reduction method

We considered a solitary nonlinear wave with the carrier frequency $\omega$
and wave number $k$ propagating along the positive $z$-axis. To study solitary
nonlinear waves, the value of the width of pulse $T$ is of great importance.
We considered pulses whose width satisfies the condition $T>>1/\omega$, i.e.
the pulse duration is much longer than the inverse frequency of the carrier
wave. For such a wave, we can use the slowly varying envelope approximation,
and for real function $U$, use the expansion [5, 38]

$U(z,t)=\sum_{l=\pm 1}\hat{u}_{l}(z,t)Z_{l},$ (7)

where $Z_{l}=e^{il(kz-\omega t)}$ is the exponential fast oscillating
function, and $\hat{u}_{l}$ is the slowly varying complex envelope function,
which satisfies inequalities

$\left|\frac{\partial\hat{u}_{l}}{\partial
t}\right|\ll\omega|\hat{u}_{l}|,\;\;\;\;\;\;\;\;\;\;\;\left|\frac{\partial\hat{u}_{l}}{\partial
z}\right|\ll k|\hat{u}_{l}|.$

For the reality of $U$, it is supposed that the expression
$\hat{u}_{+1}=\hat{u}^{*}_{-1}$ is valid.

The nonlinear wave equation (6) can be presented as

$\frac{\partial^{2}U}{\partial t^{2}}-C\frac{\partial^{2}U}{\partial
z^{2}}=-\alpha_{0}^{2}U+N(U),$ (8)

where

$N(U)=-A(\frac{\partial U}{\partial t})^{2}\frac{\partial^{2}U}{\partial
z^{2}}-\sigma(\frac{\partial U}{\partial z})^{2}\frac{\partial^{2}U}{\partial
t^{2}}+B\frac{\partial U}{\partial t}\frac{\partial U}{\partial
z}\frac{\partial^{2}U}{\partial t\partial
z}+\frac{\alpha_{0}^{2}}{6}U^{3}-\mathcal{O}(U^{5})$ (9)

is the nonlinear term of the equation.

Substituting Eq.(7) into (8), we obtained the following dispersion relation

${\omega}^{2}=Ck^{2}+\alpha_{0}^{2},$ (10)

and the rest of the equation (8) in the form

$\sum_{l=\pm 1}Z_{l}(-2il\omega\frac{\partial\hat{u}_{l}}{\partial
t}-2ilkC\frac{\partial\hat{u}_{l}}{\partial
z}+\frac{\partial^{2}\hat{u}_{l}}{\partial
t^{2}}-C\frac{\partial^{2}\hat{u}_{l}}{\partial z^{2}})=N(\hat{u}_{l}).$ (11)

where $N(\hat{u}_{l})$ is the nonlinear part of Eq.(9), expressed by means of
the complex envelope function $\hat{u}_{l}$.

In order to consider the two-component nonlinear wave solution of Eq.(6), we
used the generalized perturbative reduction method developed in Refs.[9,
19-22], which made it possible to transform the nonlinear wave equation (11)
for the function $\hat{u}_{l}$ to the coupled nonlinear Schrödinger equations
for auxiliary functions.

Following this method, the complex envelope function $\hat{u}_{l}$ can be
represented as

$\hat{u}_{l}(z,t)=\sum_{\alpha=1}^{\infty}\sum_{n=-\infty}^{+\infty}\varepsilon^{\alpha}Y_{l,n}f_{l,n}^{(\alpha)}(\zeta_{l,n},\tau),$
(12)

where $\varepsilon$ is a small parameter,

$Y_{l,n}=e^{in(Q_{l,n}z-\Omega_{l,n}t)},\;\;\;\zeta_{l,n}=\varepsilon
Q_{l,n}(z-v_{{g;}_{l,n}}t),$
$\tau=\varepsilon^{2}t,\;\;\;v_{{g;}_{l,n}}=\frac{\partial\Omega_{l,n}}{\partial
Q_{l,n}}.$

It is assumed that the complex auxiliary functions $f_{l,n}^{(\alpha)}$ and
oscillating real parameters $\Omega_{l,n}$ and $Q_{l,n}$ satisfy the
inequalities for any $l$ and $n$

$\\\ \omega\gg\Omega_{l,n},\;\;k\gg Q_{l,n},\;\;\;$ $\left|\frac{\partial
f_{l,n}^{(\alpha)}}{\partial
t}\right|\ll\Omega_{l,n}\left|f_{l,n}^{(\alpha)}\right|,\;\;\left|\frac{\partial
f_{l,n}^{(\alpha)}}{\partial\eta}\right|\ll
Q_{l,n}\left|f_{l,n}^{(\alpha)}\right|.$

Substituting Eq.(12) into Eq.(11), we obtained the nonlinear wave equation for
auxiliary function $f_{l,n}^{(\alpha)}$ in the following form:

$\sum_{l=\pm 1}\sum_{\alpha=1}^{\infty}\sum_{n=\pm
1}\varepsilon^{\alpha}Z_{l}Y_{l,n}[W_{l,n}+\varepsilon
J_{l,n}-\varepsilon^{2}ilh_{l,n}\frac{\partial}{\partial\tau}-\varepsilon^{2}Q^{2}H_{l,n}\frac{\partial^{2}}{\partial\zeta^{2}}+O(\varepsilon^{3})]f_{l,n}^{(\alpha)}=N(\hat{u}_{l}),$
(13)

where

$W_{l,n}=-2nl\omega\Omega_{l,n}+2nlkQ_{l,n}C-\Omega^{2}_{l,n}+CQ_{l,n}^{2},$
$J_{l,n}=2iQ_{l,n}[l\omega
v_{{g;}_{l,n}}-lkC+n\Omega_{l,n}v_{{g;}_{l,n}}-CnQ_{l,n}],$
$h_{l,n}=2(\omega+ln\Omega_{l,n}),$ $H_{l,n}=C-v_{{g;}_{l,n}}^{2}.$ (14)

Following the standard procedure and equating to zero, the terms with the same
powers of $\varepsilon$, from Eq.(13), we obtained a series of equations. In
the first order of $\varepsilon$, we found a connection between the parameters
$\Omega_{l,n}$ and $Q_{l,n}$. When

$2(CkQ_{\pm 1,\pm 1}-\omega\Omega_{\pm 1,\pm 1})-\Omega^{2}_{\pm 1,\pm
1}+CQ_{\pm 1,\pm 1}^{2}=0,$ (15)

then $f_{\pm 1,\pm 1}^{(1)}\neq 0$ and when

$2(CkQ_{\pm 1,\mp 1}-\omega\Omega_{\pm 1,\mp 1})+\Omega^{2}_{\pm 1,\mp
1}-CQ_{\pm 1,\mp 1}^{2}=0,$ (16)

then $f_{\pm 1,\mp 1}^{(1)}\neq 0$.

From Eq.(14), in the second order of $\varepsilon$, we obtained the equation
$J_{\pm 1,\pm 1}=J_{\pm 1,\mp 1}=0$ and the expression

$v_{{g;}_{l,n}}=C\frac{k+lnQ_{l,n}}{\omega+ln\Omega_{l,n}}.$ (17)

Next, we considered the nonlinear term $N(\hat{u}_{l})$ of the nonlinear wave
equation (13). Substituting Eqs.(12) and (7) into Eq.(9) for the nonlinear
term of the nonlinear wave equation, we obtained

$\varepsilon^{3}\;Z_{+1}[(\tilde{q}_{+}|f_{+1,+1}^{(1)}|^{2}+\tilde{r}_{+}|f_{+1,-1}^{(1)}|^{2})f_{+1,+1}^{(1)}Y_{+1,+1}+(\tilde{q}_{-}|f_{+1,-1}^{(1)}|^{2}+\tilde{r}_{-}|f_{+1,+1}^{(1)}|^{2})Y_{+1,-1}f_{+1,-1}^{(1)}]$
(18)

and plus terms proportional to $Z_{-1}$. In Eq.(18) we use the notations

$\tilde{q}_{\pm}=\frac{\alpha_{0}^{2}}{2}+\mathfrak{q}_{\pm},$
$\tilde{r}_{\pm}=\alpha_{0}^{2}+\mathfrak{r}_{\pm},$ (19)

where

$\mathfrak{q}_{\pm}=(A+\sigma-B)(\omega\pm\Omega_{\pm})^{2}(k\pm
Q_{\pm})^{2},$ $\mathfrak{r}_{\pm}=2[A(k\pm
Q_{\pm})^{2}(\omega\mp\Omega_{\mp})^{2}+\sigma(\omega\pm\Omega_{\pm})^{2}(k\mp
Q_{\mp})^{2}-B(\omega+\Omega_{+})(\omega-\Omega_{-})(k+Q_{+})(k-Q_{-})].$

From Eqs.(13) and (18), in the third order of $\varepsilon$, we obtained the
system of nonlinear equations

$i\frac{\partial
f_{+1,+1}^{(1)}}{\partial\tau}+Q_{+}^{2}\frac{H_{+1,+1}}{h_{+1,+1}}\frac{\partial^{2}f_{+1,+1}^{(1)}}{\partial\zeta_{+1,+1}^{2}}+(\frac{\tilde{q}_{+}}{h_{+1,+1}}|f_{+1,+1}^{(1)}|^{2}+\frac{\tilde{r}_{+}}{h_{+1,+1}}|f_{+1,-1}^{(1)}|^{2})f_{+1,+1}^{(1)}=0,$
$i\frac{\partial
f_{+1,-1}^{(1)}}{\partial\tau}+Q_{-}^{2}\frac{H_{+1,-1}}{h_{+1,-1}}\frac{\partial^{2}f_{+1,-1}^{(1)}}{\partial\zeta_{+1,-1}^{2}}+(\frac{\tilde{q}_{-}}{h_{+1,-1}}|f_{+1,-1}^{(1)}|^{2}+\frac{\tilde{r}_{-}}{h_{+1,-1}}|f_{+1,+1}^{(1)}|^{2})f_{+1,-1}^{(1)}=0.$
(20)

##  III Two-breather molecule of the nonlinear wave equation

After transforming back to the space coordinate $z$ and time $t$, from
Eqs.(20), we obtained the coupled nonlinear Schrödinger equations for the
auxiliary functions $\Lambda_{\pm}=\varepsilon f_{+1,\pm 1}^{(1)}$ in the
following form:

$i(\frac{\partial\Lambda_{\pm}}{\partial
t}+v_{\pm}\frac{\partial\Lambda_{\pm}}{\partial
z})+p_{\pm}\frac{\partial^{2}\Lambda_{\pm}}{\partial
z^{2}}+q_{\pm}|\Lambda_{\pm}|^{2}\Lambda_{\pm}+r_{\pm}|\Lambda_{\mp}|^{2}\Lambda_{\pm}=0,$
(21)

where

$v_{\pm}=v_{g;_{+1,\pm 1}}=C\frac{k\pm
Q_{\pm}}{\omega\pm\Omega_{\pm}},,\;\;\;\;\;\;\;\;\;\;\;\;\;\;p_{\pm}=\frac{C-v_{\pm}^{2}}{2(\omega\pm\Omega_{\pm})},$
$q_{\pm}=\frac{\tilde{q}_{\pm}}{2(\omega\pm\Omega_{\pm})},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;r_{\pm}=\frac{\tilde{r}_{\pm}}{2(\omega\pm\Omega_{\pm})},$
$\Omega_{+}=\Omega_{+1,+1}=\Omega_{-1,-1},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\Omega_{-}=\Omega_{+1,-1}=\Omega_{-1,+1},$
$Q_{+}=Q_{+1,+1}=Q_{-1,-1},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Q_{-}=Q_{+1,-1}=Q_{-1,+1}.$
(22)

The solution of Eq.(21) is given by [19, 20, 22]

$\Lambda_{\pm}=\frac{A_{\pm}}{bT}Sech(\frac{t-\frac{z}{V_{0}}}{T})e^{i(k_{\pm}z-\omega_{\pm}t)},$
(23)

where $A_{\pm},\;k_{\pm}$ and $\omega_{\pm}$ are the real constants, and
$V_{0}$ is the velocity of the nonlinear wave. We assume that
$k_{\pm}<<Q_{\pm}$ and $\omega_{\pm}<<\Omega_{\pm}.$

Combining Eqs.(7), (12) and (23), we obtained the two-breather molecule
solution to the nonlinear wave equation (6) in the following form:

$U(z,t)=\mathfrak{A}Sech(\frac{t-\frac{z}{V_{0}}}{T})\\{\cos[(k+Q_{+}+k_{+})z-(\omega+\Omega_{+}+\omega_{+})t]$
$+(\frac{p_{-}q_{+}-p_{+}r_{-}}{p_{+}q_{-}-p_{-}r_{+}})^{\frac{1}{2}}\cos[(k-Q_{-}+k_{-})z-(\omega-\Omega_{-}+\omega_{-})t]\\},$
(24)

where $\mathfrak{A}$ is the amplitude of the nonlinear pulse. The expressions
for the parameters $k_{\pm}$ and $\omega_{\pm}$ are given by

$k_{\pm}=\frac{V_{0}-v_{\pm}}{2p_{\pm}},$
$\omega_{+}=\frac{p_{+}}{p_{-}}\omega_{-}+\frac{V^{2}_{0}(p_{-}^{2}-p_{+}^{2})+v_{-}^{2}p_{+}^{2}-v_{+}^{2}p_{-}^{2}}{4p_{+}p_{-}^{2}}.$
(25)

##  IV Conclusion

We investigated the two-breather molecule solution of the nonlinear wave
equation (6) when the slowly varying envelope approximation Eq. (7) was valid.
The nonlinear pulse with the width $T>>\Omega_{\pm}^{-1}>>\omega^{-1}$ was
considered.

Using the generalized perturbation reduction method Eq.(12), Eq.(6) is
transformed to the coupled nonlinear Schrödinger equations (21) for the
function $\Lambda_{\pm 1}$. As a result, the two-component nonlinear pulse
oscillating with the sum $\omega+\Omega_{+}$ $(k+Q_{+})$ and difference
$\omega-\Omega_{-}$ ($k-Q_{-}$) of the frequencies (wave numbers) Eq.(24)
(two-breather molecule) was obtained. The dispersion relation and the
relations between oscillating parameters $\Omega_{\pm}$ and $Q_{\pm}$ were
determined from Eqs.(10), (15) and (16). The parameters of the nonlinear pulse
from Eqs.(17), (19), (22) and (25) were determined.

Eq.(6) can describe all effects, which can be considered by means of the
nonlinear Klein-Gordon equation Eq.(4) and the Born-Infeld equation Eq.(5)
separately. Eqs.(24) are also solutions of these equations, and all obtained
results are valid for these two equations.

In the special case when $\alpha_{0}^{2}=0$, Eq.(6) is reduced to the Born-
Infeld equation Eq.(5). The dispersion law Eq.(10) for the Born-Infeld
equation is reduced to the form ${\omega}^{2}=Ck^{2},$ and the parameters of
the wave have the form $\tilde{q}_{\pm}=\mathfrak{q}_{\pm}$ and
$\tilde{r}_{\pm}=\mathfrak{r}_{\pm}.$

In the second special case when the condition $A=B=\sigma=0$ is fulfilled,
Eq.(6) is reduced to the nonlinear Klein-Gordon equation Eq.(4). The
dispersion law Eq.(10) is valid, and the parameters of the wave have the form
$\tilde{q}_{\pm}=\frac{\alpha_{0}^{2}}{2}$ and
$\tilde{r}_{\pm}=\alpha_{0}^{2}$.

Although the Sine-Gordon equation (2) and the nonlinear Klein-Gordon equation
(4) arise in a number of physical fields and applied mathematics, there is one
more very important effect of self-induced transparency, the simplified
version of which is based on the Sine-Gordon equation, where the function $U$
is the optical pulse envelope [38]. Indeed, the nonlinear coherent interaction
of an optical pulse with resonant optical atoms is governed by the Bloch-
Maxwell equations [27, 38]. When the Rabi frequency of the wave is real and
the longitudinal and transverse relaxations are ignored, these equations are
reduced to the Sine-Gordon equation [1-4], and for small amplitude waves when
$U<<1$, the Sine-Gordon equation is reduced to the nonlinear Klein-Gordon
equation (4).

Ref.[19] shows that, under the conditions of self-induced transparency, Eq.(4)
has a two-component vector $0\pi$ pulse solution. One component oscillates
with the sum $w+\Omega_{+}$ ($\kappa+Q_{+}$), and the other with the
difference $w-\Omega_{-}$ ($\kappa-Q_{-})$ between the frequencies (wave
numbers) in the region of the parameters $w$ and $\kappa$ which that are two
or three orders lower than the carrier wave frequency $\omega$ and wave number
$k$. In other words, the expressions $\omega/w$ and $k/\kappa$ are of orders
$10^{2}\div 10^{3}$. In this case, the conditions of the nonlinear wave
existence

$\omega>>w>>\Omega_{\pm}>>T^{-1},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;k>>\kappa>>Q_{\pm}>>(V_{0}T)^{-1}$
(26)

are fulfilled. Here $w,$ $\kappa$, $\Omega_{\pm}$ and $Q_{\pm}$ are the
different oscillating parameters. The phase modulation is neglected.

Taking into account the phase modulation, the character of the wave process
changes significantly. As shown in Refs.[20-22], the two-component vector
$0\pi$ pulse solution of the Bloch-Maxwell equations oscillates with the sum
and difference in frequencies in the region of the carrier wave frequency and
wave number of the optical pulse. In this case, the conditions of formation of
the two-component vector $0\pi$ pulse have a form

$\omega>>\Omega_{\pm}>>T^{-1},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;k>>Q_{\pm}>>(V_{0}T)^{-1},$
(27)

that is considerably weaker than Eq.(26), as shown in Ref.[19].

From Eq.(6), in the special case when the condition $A=B=\sigma=0$ is
fulfilled, Eq.(6) is transformed to the nonlinear Klein-Gordon equation (4),
which has a solution in the form of Eq.(24) under the condition Eq.(27). When
function $U$ is the area of the optical pulse envelope, the two-breather
molecule Eq.(24) coincides with the vector $0\pi$ pulse of self-induced
transparency [19-22].

We found that Eq.(24) is a solution of Eqs.(6) and (4) under the condition
Eq.(27), as was obtained earlier in the case of the Bloch-Maxwell equations
but without the phase modulation. In other words, in the present work, we
obtained the solution in the form of the two-breather molecule or the vector
$0\pi$ pulse of self-induced transparency Eq.(24) for the Eq.(4) under the
less stringent conditions Eq.(27), compared to the solution of this equation
obtained earlier in Ref.[19] under the condition Eq.(26).

The considered solution in the form of a two-breather molecule - one of the
breathers oscillates with the sum and the second with the difference in
frequencies Eq.(24) - shows that this nonlinear wave occurs in many completely
different areas of research for various nonlinear partial differential
equations. Therefore, it can be concluded that the existence of such a
nonlinear solitary wave expresses a rather general property of matter, as well
as this takes place for the soliton or the breather.

We considered the nonlinear wave equation (1) (Eq.(6)), which unified the
Sine-Gordon equation (2) (the nonlinear Klein-Gordon equation (4)) and the
Born-Infeld equation (5). A similar approach used earlier for the Hirota
equation, which in one particular case reduced to a scalar nonlinear
Schrödinger equation, and in another particular case, it transformed into a
complex modified Korteweg de Vries equation. Notable, there was a difference;
unlike the nonlinear wave equations (1) and (6), where the function $U(z,t)$
satisfying these equations was a real function, in the cases of the Hirota
equation, the scalar nonlinear Schrödinger equation and the complex modified
Korteweg de Vries equation, the solutions of these equations were found to be
complex functions [7, 39, 40].

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