# Local evidence for collective spin excitations in a distorted kagome
antiferromagnet:Pr3BWO9

K. Y. Zeng1,2 F. Y. Song3 Z. M. Tian3<EMAIL_ADDRESS>L. S. Ling1 W.
Tong1 Long Ma1<EMAIL_ADDRESS>Li Pi1,2 1Anhui Province Key Laboratory of
Condensed Matter Physics at Extreme Conditions, High Magnetic Field
Laboratory, Chinese Academy of Sciences, Hefei 230031, China
2 Hefei National Laboratory for Physical Sciences at the Microscale,
University of Science and Technology of China, Hefei 230026, China
3 School of Physics and Wuhan National High Magnetic Field Center, Huazhong
University of Science and Technology, Wuhan 430074, PR China

###### Abstract

We report the local probe investigation of a frustrated antiferromagnet
Pr3BWO9 with the distorted kagome lattice. Absence of magnetic order or spin
freezing is indicated by the spectral analysis with the temperatures down to
0.3 K. The Knight shifts show an upturn behavior with the sample cooling down,
which is further suppressed by external field. For the spin dynamics, gapped
spin excitation is observed from the temperature dependence of spin-lattice
relaxation rates, with the gap size proportional to the applied magnetic field
intensity. Comparatively, an unexpected sharp peak is observed in the nuclear
spin-spin relaxation rate data at $T^{*}\sim 4-5$ K. These results indicate an
unconventional persistent fluctuating paramagnetic ground state with
antiferromagnetic collective spin excitations in the strongly frustrated spin
system.

Geometrical frustration has triggered enormous research interest in recent
years, not only for the possible relationship with high-$T_{c}$
superconductivity but also the interesting ground states and exotic quantum
excitations in its own rightBalents (2010). For the antiferromagnetically
coupled spins located at triangles, the anti-parallel configuration favored by
the nearest interaction cannot be simultaneously fulfilled, leading to very
large degeneracy of ground states as well as strong quantum fluctuations. One
of the most intriguing ground state is the quantum spin liquid (QSL)Balents
(2010), where the spins dynamically entangled with each other but never order
even at zero temperature. Among various types of QSL proposed by theory, the
common feature of telling QSL from other states is the deconfined spinon
excitations with fractional quantum number. As a result, research into the
low-energy spin excitation property is of great importance in the study of
frustrated antiferromagnets.

Compared with the edge-shared triangular lattice, the corner-shared triangles
(kagome lattice) with low coordination and weak second-neighbor interaction
possess stronger magnetic frustration effect, thus is more attractive. In the
near decade, ZnCu3(OH)6Cl2 is the intensively studied promising kagome lattice
realizing the QSL stateNorman (2016), where observed is the continuous spin
excitationHan et al. (2012), fingerprints for the fractional spinonsBalents
(2010). However, two inherent drawbacks exist in ZnCu3(OH)6Cl2. One is the
natural mixing of Cu2+ and Zn2+ due to their similar ion radiusImai et al.
(2011), complicating the investigation of the intrinsic properties of QSL
state. The other one is the almost impossibility of chemical substitution at
the magnetic sites, prohibiting further exploration of other novel states and
excitations in this system.

For magnetic lanthanide ions, the local crystal electric field (CEF) splits
the spin-orbit entangled $J$ momentum space into $2J+1$ states, and finally
determine the single ion anisotropy with effective $J_{eff}$. By locating
different lanthanide ions to kagome lattice, various novel ground states and
spin excitations can be realized, relying on different spin anisotropy, spin-
orbit coupling and exchange and dipolar interaction. The rare-earth based
Ln3M2Sb3O14 (Ln= lanthanide, M= Mg or Zn) with kagome lattice is a precise
example for testing this judgementDun et al. (2016). For Dy3Mg2Sb3O14 with
anisotropic Ising spins, the kagome spin ice with ”two-in-one-out” or ”one-in-
two-out” magnetic structure is identified by neutron scatteringPaddison et al.
(2016). While for the Ln=Ho case, a new quantum state with both characteristic
of classical spin ice and quantum fluctuation and tunneling is foundDun et al.
(2020). For the Ln=Gd case with isotropic large Heisenberg large spins, the
dipolar interaction may result in the observed magnetic ordered state with 120
degree structureDun et al. (2016); Wellm et al. (2020). Recently, possible QSL
state is proposed based on the persistent strong low-energy spin excitations
down to $T=50$ mK by muon spin rotation ($\mu$SR) and the magnetic
contribution to the linear temperature dependence of heat capacityDing et al.
(2018). Controversially, the random mixing of Tm3+ and Zn2+ is observed very
recentlyMa et al. (2020), which may lead to the mimicry of QSL. Thus, the
rare-earth based kagome lattice has supplied a new playground for exploring
exotic frustrated quantum states.

To overcome the possible anti-site disorder, some of us have synthesized a new
family of rare-earth based antiferromagnets RE3BWO9 (RE=Pr, Nd, Gd-Ho) with
the lanthanide ions locating on the distorted kagome structure in $ab$-plane
and stacking in an $AB$-type fashion along $c$-axisAshtar et al. (2020). $Dc$
magnetization analysis suggests similar magnetic behaviors to other rare-earth
based frustrated antiferromagnetsAshtar et al. (2020). However, no further
study on the ground state and spin excitations is reported up to now.

In this letter, we employ nuclear magnetic resonance (NMR) as a local probe to
study the spin excitations in the lanthanide frustrated antiferromagnet
Pr3BWO9 with a distorted kagome lattice. With temperatures down to 0.3 K, the
paramagnetic state persists without any magnetic order or spin freezing as
evidenced by 11B spectral analysis. The Knight shifts measuring spin
correlations at $\bf{q}=0$ show an upturn behavior with the sample cooling
down, which is further suppressed by external field. For the spin dynamics,
thermally activated behavior is observed in the temperature dependence of
spin-lattice relaxation rates, indicating gapped spin excitations. The gap
size is proportional to the field intensity for both field directions, but
with a very different slope, demonstrating the spin anisotropy. Comparatively,
an unexpected sharp peak is observed in the nuclear spin-spin relaxation rate
data at $T^{*}\sim 4-5$ K, which is contributed by the hyperfine field
fluctuation parallel to the applied direction. These results indicate an
unconventional persistent fluctuating paramagnetic ground state with
collective spin excitations in the strongly frustrated spin system.

Single crystals of Pr3BWO9 were synthesized by the conventional flux method.
For NMR studies, crystals with typical dimensions of $1.5\times 1.5\times 0.2$
mm3 are used. Our NMR measurements are conducted with 11B nuclei
($\gamma_{n}=13.655$ MHz/T, $I=3/2$) with a phase-coherent NMR spectrometer.
The spectra are obtained by summing up or integrating the spectral weight by
sweeping the frequency under fixed magnetic fields. The measurement of spin-
lattice relaxation rates is performed by the inversion-recovery method. The
spin-spin relaxation rates are obtained by measuring the nuclear transverse
dephasing with the two-pulse Hahn echo sequence.

Figure 1: (color online) (a) The crystalline structure of Pr3BWO9 as seen
against $c$-axis. The distorted kagome lattice is marked by the bold red
lines. (b) Typical 11B NMR spectra with the magnetic field applied
perpendicular or parallel to the $c$-axis of the single crystal. The central
and satellite transitions are respectively marked by the solid blue and dotted
green arrows.

The crystal structure of Pr3BWO9 is built up by blocks of PrO8, WO6 and BO3
polyhedra in the corner- or edge-sharing mannerAshtar et al. (2020). There is
only one Wyckoff position for Pr, W, and B sites. The BO3 trigons share edges
with the PrO8 polyhedra, leading to a very strong hyperfine coupling between
11B nuclei and magnetic moments. Top view of the crystal structure along
$c$-axis is shown in Fig.1(a). The magnetic Pr3+ sites form a distorted kagome
structure in the crystalline $ab$-plane, which is intensionally marked with
bold red lines.

In Fig.1 (b), we show typical 11B NMR spectra at $T=60$ K with a $10$ tesla
field along different directions. Both spectrum are composed by three peaks,
one at the frequency center and the other two locate symmetrically at both
sides. For nuclei with spin $I=3/2$ in non-zero local electric field gradient
(EFG) under high magnetic fields, the first order correction to the Zeeman
energy term due to quadruple interactions splits the single NMR transition
into three, respectively named as the central transition and
satellitesSlichter (1990); Abragam (1961). The frequency of satellites
strongly depend on the angle $\theta$ between applied field and the main axis
of EFG. For the present sample without $ab$-plane EFG anisotropy (confirmed by
in-plane angle rotation, data not shown), the correction to the frequency can
be written as $\nu_{m}^{(1)}=\nu_{Q}(m-1/2)(3\cos^{2}\theta-1)/2$, where
$\nu_{Q}$ and $m$ respectively denote the quadruple frequency and nuclear
magnetic quantum number. The $\nu_{Q}$ is calculated to be $\sim 1.325$ MHz at
$T=60$ K, and the main axis of EFG is along the crystalline $c$-axis. These
observations are fully consistent with the crystal structure symmetry.

The full NMR spectra at different temperatures with the magnetic field applied
parallel or perpendicular to the $c$-axis are shown in Fig.2 (a) and (b). For
both field directions, all the three peaks shift to lower frequency side and
broaden gradually with the sample cooling down. For $T=0.3$ K far below the
reported Curie-Weiss temperature $|\theta_{cw}|=6.88$ K, the main feature of
the spectrum is maintained as that at much higher temperatures except the
increased line width. No further line splitting or amplitude modulation are
observed in the present sample. We note that the hyperfine coupling between
${}^{11}B$ nuclei and the magnetic sites is very strong as evidenced by the
enormous Knight shift and ultra-fast spin-lattice relaxation (shown latter).
Any magnetic ordering of commensurate or incommensurate type or spin freezing,
if present, is very unlikely to be missed in the low-temperature spectra for
both field directions. Thus, our data support a paramagnetic state in Pr3BWO9
with the temperature down to $0.05\theta_{cw}$, demonstrating the strong
magnetic frustration.

Figure 2: (color online) The frequency-swept 11B NMR spectra at different
temperatures with a 3.5 Tesla field applied along (a) or perpendicular to
$c$-axis(b). (c) and (d): The temperature dependence of 11B Knight shift
${}^{11}K$ under different magnetic fields. The dotted lines are fits to the
Curie-Weiss function. The obtained Curie-Weiss temperature ($\theta_{cw}$) as
a function of field intensity is plotted in the inset.

The intrinsic local spin susceptibility can be measured by the Knight shift,
suffering less from impurity effect than bulk probes. The temperature
dependence of Knight shift for different magnetic fields is shown in Fig.2 (c)
and (d). The Knight shifts (${}^{11}K$) are calculated as the relative line
shift of the central transition with respect to the Larmor frequency, after
removing the second order correction rising from quadruple interaction. With
the sample cooling down, all the Knight shifts share the similar Curie-Weiss
upturn behavior with $dc$ susceptibility. Fittings with the equation
${}^{11}K\propto 1/(T+\theta_{cw})$ to the low temperature data yield a nearly
field-independent $\theta_{cw}$ value (shown in the inset). The $\theta_{cw}$
is calculated to be $\sim 6$ K for $\mu_{0}H\perp c$ and $\sim 8$ K for
$\mu_{0}H||c$, again demonstrating a moderate antiferromagnetic coupling
strength. With the increasing field intensity, the Knight shift tends to level
off at low temperatures. This results from the saturated paramagnetic Pr3+
magnetic moments, which is also observed in other frustratedZeng et al.
(2020); Baenitz and et al. (2018); Bordelon et al. (2019) or Kitaev
antiferromagnetsZheng et al. (2017).

Figure 3: (color online) (a)(b): Spin-lattice relaxation rates
($1/^{11}T_{1}$) as a function of temperature under different magnetic fields
with $\mu_{0}H||c$ or $\mu_{0}H\perp c$. The solid red lines are
representative fits to $1/T_{1}\propto\exp(-\Delta/T)$ (See the text). (c)(d):
Spin-spin relaxation rates ($1/^{11}T_{2}$) as a function of temperature under
different magnetic fields with $\mu_{0}H||c$ or $\mu_{0}H\perp c$.

To further explore the spin dynamics in Pr3BWO9, we have measured both nuclear
spin-lattice and spin-spin relaxation rates ($1/^{11}T_{1}$ and
$1/^{11}T_{2}$), and show their temperature dependence under different
magnetic fields in Fig.3. Both measurements are made at the central peak. The
$1/^{11}T_{1}$ is obtained by fitting the nuclear magnetization to the
function, $M(t)/M(\infty)=1-0.4\exp(-t/T_{1})-0.6\exp(-6t/T_{1})$. An
additional stretching factor $\beta$ is needed for temperatures below $T\sim
40$ K for the observable distribution of the relaxation. For $1/^{11}T_{2}$,
the echo intensity decay can be well reproduced by the function,
$M(2\tau)=M(0)\exp(-2\tau/T_{2})$ ($\tau$ is the time interval between $\pi/2$
and $\pi$ pulse), for the studied temperature range. This demonstrates the
transverse dephasing is mainly contributed by the spin fluctuations via
Redfield mechanismRedfield (1955) instead of indirect and dipole coupling
between nuclei, which will result in a gaussian componentM. Takigawa and
Uchida (1998). For the high temperature region of $T>20$ K, the NMR relaxation
rates $1/^{11}T_{1}$ and $1/^{11}T_{2}$ share a very similar temperature
dependence, first increase mildly, then tend to level off and finally bend
over with the sample cooling down. A wide hump peaked at $T\sim 30$ K is
observed in both relaxation rates.

The high temperature hump is contributed by the crystal electric field
(CEF)-related excitations. The magnetic behavior of rare-earth ions strongly
depends on the local CEF where they occupy. The $(2J+1)$-fold degenerated
states of $4f$-electrons are lifted by a certain CEF. For the non-Kramers ion
Pr3+with $J=4$, the crystal field interactions lead to nine singlet states
with possible accidental- or near-degeneraciesScheie et al. (2018). The mild
increasing $1/^{11}T_{1}$ and $1/^{11}T_{2}$ at high temperatures directly
connect with the slowing down of spin fluctuations of electrons occupying
thermally activated CEF levelsLumata et al. (2010); Zeng et al. (2020). The
wide hump results from the reduced thermal population at low temperatures,
consistent with the slope change observed in $dc$-susceptibilityAshtar et al.
(2020).

Figure 4: (color online) The Arrhenius plot of $1/^{11}T_{1}$ versus inverse
temperature $1/T$ for magnetic fields along (a) or perpendicular (b) to
$c$-axis. Solid lines are linear fits to the data. (c): Field dependence of
the spin excitation gap in Kelvin. Solid lines are fits to
$\Delta(H)=\Delta_{0}+g\mu_{B}H$.

More intriguing is the low temperature spin dynamics dominated by the CEF
ground state of Pr3+. With the sample further cooling from $T=10$ K, the
$1/^{11}T_{1}$ drops steeply with temperature for all the applied fields (See
Fig.3(a) and (b)). Actually, the temperature dependence of $1/^{11}T_{1}$ can
be well described by the thermally activated equation
$1/T_{1}\propto\exp(-\Delta/T)$, with a finite spin gap $\Delta$. The gapped
spin excitations under magnetic fields are better demonstrated by the
Arrhenius plot, i.e., $1/^{11}T_{1}$ versus inverse temperature on a
semilogarithmic scale as shown in Fig.4 (a) and (b). The field dependence of
the gap size is shown in Fig.4(c) for both field orientations. In sharp
contrast, the temperature dependence of $1/^{11}T_{2}$ at $T<10$ K completely
deviates from that of $1/^{11}T_{1}$. A prominent peak around $T^{*}\sim 4$ K
is observed, which is unusual in strongly correlated electron systems. When
the magnetic field strength increases, the peak is suppressed, and shifts to a
little higher temperature, which is more pronounced for fields along $c$-axis.

We discuss the spin excitation property in Pr3BWO9 implied by our data. The
low-temperature Knight shifts show a Curie-Weiss upturn behavior, while a
contrasting gapped behavior is observed in $1/^{11}T_{1}$ with the sample
cooling down. By roughly extrapolating the field dependence of the gap size
(See Fig.4(c)), a non-zero spin gap should also exist at zero field. In
another word, this excitation gap is not field-induced, but inherent. The
Knight shift,expressed as $K\propto A_{hf}\chi(\bf{q}=0)$, measures the spin
excitation at the wave-vector $\bf{q}=0$Slichter (1990); Abragam (1961). The
hyperfine coupling constant $A_{hf}$ is negative in the present sample.
However, the $1/^{11}T_{1}$ is contributed by the dynamic spin susceptibility
summed over the whole reciprocal $\bf{q}$-space, which can be formulated as
$1/T_{1}\propto
T\sum_{\bf{q}}|A(\bf{q})|^{2}[\chi^{{}^{\prime\prime}}(\bf{q},\omega_{L})]/\omega_{L}$Slichter
(1990); Abragam (1961). The $\omega_{L}$ denotes the Larmor frequency, which
can be treated as zero in condensed matter community. Thus, the contrasting
low temperature behavior of $1/^{11}T_{1}$ indicates the strongly
$\bf{q}$-dependent spin excitation, and the gap opens at a non-zero wave-
vector. In such an antiferromagnet with moderate coupling strength, this
result actually suggests the spin excitation gaps at the antiferromagnetic
wave-vector.

The obvious $\bf{q}$-dependent spin excitations in Pr3BWO9 is very different
with the mostly investigated spin liquid candidate, ZnCu3(OH)6Cl2. Fractional
excitation continuum is observed by inelastic neutron scattering in the later
compoundHan et al. (2012). By the recent NMR resultsFu et al. (2015), the spin
gapped behavior is observed in both Knight shifts and spin-lattice
relaxations, which further indicates a gapped spin liquid state in
ZnCu3(OH)6Cl2. Additionally, the spin gap is suppressed by the applied
magnetic field due to Zeeman effect, whose slope corresponds to spinons with
$S=1/2$. Similar results are observed in its counterpart Cu3Zn(OH)6FBrFeng et
al. (2017). Suppose spin liquid state do exist in these materials, the spin
gap originates from the majority of short-ranged valence bonds.

There exist short-ranged collective spin excitations in the present sample. In
contrast to the gapped behavior observed in low temperature $1/^{11}T_{1}$, a
sharp peak is observed in the nuclear spin-spin relaxation rate
($1/^{11}T_{2}$) at $T^{*}\sim 4$ K. The spin fluctuation contribution to
($1/^{11}T_{2}$) through the Redfield mechanism can be expressed as
$1/T_{2}^{Redfield}=1/T_{2}^{\prime}+1/(2T_{1})$Slichter (1990). The first
term describes the in-plane decoherence contributed by the fluctuations of
longitudinal hyperfine field. The later one has the same origination as
$1/^{11}T_{1}$, both from the transverse hyperfine field fluctuations. Thus,
the different behavior at low temperatures apparently results from the
anisotropic fluctuating hyperfine field.

Understanding how longitudinal fluctuations contribute $1/T_{2}^{\prime}$
supplies the key ingredient for revealing the spin excitation property. In the
two-pulse Hahn echo experiment, dephasing of the transverse nuclear
magnetization during the timing interval between pulses is refocused by
applying the second $\pi$-pulse after the equal time interval. If there exists
some kinds of dynamically inhomogeneous longitudinal field with the typical
frequency of kilohertz, the changed local magnetic field will give rise to the
failure of refocusing, also the decreased echo intensity. One of typical
examples for this effect is the well-known vortex in type-II superconductors
under magnetic fieldSuh et al. (1993). Measurements of $1/T_{2}$ have supplied
another novel approach to study the vortex dynamics in both cuprates and iron
based high-$T_{C}$ superconductorsRecchia et al. (1997); Bossoni et al.
(2012); Rigamonti et al. (1998). The vortex core movement contributes a very
similar enhanced relaxation as observed here. Another example is fluids in a
pore space. The spin-spin relaxation is obviously faster than the pure fluids,
which is enhanced by the interactions between nuclei and pore walls or
paramagnetic centersKorringa et al. (1962). Thus, the enhanced $1/T_{2}$
should be related with excitations of typical collective characteristic.

In another distorted kagome system Nd3Ga5SiO14 and Pr3Ga5SiO14, electron spin
resonance study have revealed the collective spin excitations instead of the
long-range continuumGhosh et al. (2013, 2014). In Pr3Ga5SiO14, Dynamical
short-range ordering is suggested from diffuse neutron-scattering
observationsLumata et al. (2010) . $\mu$SR experiments indicate a fluctuating
collective paramagnetic ground state in Nd3Ga5SiO14Zorko et al. (2008). From
the NMR experiments, a wipeout of the Ga NMR signal below $T=25$ K is observed
in Nd3Ga5SiO14Zorko et al. (2008), resulting from the enhanced $1/T_{2}$. In
Pr3Ga5SiO14, The enhanced $1/T_{2}$ at low temperatures is also observed by
NMRLumata et al. (2010), although the physical connection between them is not
pointed out. In the present Pr3BWO9, our observations have provide unambiguous
local evidence for the short-ranged collective excitations. We stress that the
short-range correlations still is dynamic in the timescale of NMR for the
studied magnetic field range, as no abrupt change is observed in the
temperature dependence of NMR spectra. This is in sharp contrast with what
occurs in Nd3Ga5SiO14, where field-induced partial magnetic order is
identified by neutron scattering and heat transportZhou et al. (2007); Li et
al. (2012).

We close our discussion by comments on the single ion anisotropy in this rare-
earth kagome spin system. With applied field, the spin excitation gap
increases linearly due to Zeeman effect as shown in Fig.4(c). Two anisotropic
$g$-factors of $g_{c}=3.047$ and $g_{ab}=1.167$ are obtained from the slope of
linear fits. The magnetic moment along crystalline $c$-axis is close to the
$3.58\mu_{B}$ of free Pr3+ ions. Thus, the coupling between Pr3+ non-zero
orbital moments and CEF make the magnetic moment energetically favorable to be
along $c$-axis.

In conclusion, we have performed detailed NMR study on the newly-synthesized
distorted kagome spin system Pr3BWO9. Quantum paramagnetic state persists down
to $T=0.3$ K far below its Curie-Weiss temperature, yielding strong
frustration effect. The unconventional short-ranged collective spin
excitations is identified by comparatively studying the temperature dependence
of Knight shifts, spin-lattice relaxation rates and spin-spin relaxation
rates. A spin gap opened at the antiferromagnetic wave-vector is suggested
from our data, whose size further shows linear field dependence. This short-
ranged collective spin excitations with a distinct $\bf{q}$-dependent spin gap
is totally different from the short-ranged valence bonds proposed in the spin
liquid state. Our experiments reveal a highly unconventional spin excitation
mode in frustrated antiferromagnets. Moreover, we have supplied a new approach
to detect collective spin excitations in magnetic materials.

This research was supported by the National Key Research and Development
Program of China (Grant No. 2016YFA0401802), the National Natural Science
Foundation of China (Grants No. 11874057, 11874158, U1732273 and 21927814) and
the Users with Excellence Program of Hefei Science center CAS (Grant No.
2019HSC-UE008). A portion of this work was supported by the High Magnetic
Field Laboratory of Anhui Province.

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Superposed Hyperbolic Kink and Pulse Solutions of Coupled $\phi^{4}$, NLS and
MKdV Equations

Avinash Khare

Physics Department, Savitribai Phule Pune University

Pune 411007, India

Avadh Saxena

Theoretical Division and Center for Nonlinear Studies, Los Alamos National
Laboratory, Los Alamos, New Mexico 87545, USA

Abstract:

We obtain novel solutions of a coupled $\phi^{4}$, a coupled nonlinear
Schrödinger (NLS) and a coupled modified Korteweg de Vries (MKdV) model which
can be re-expressed as a linear superposition of either the sum or the
difference of two hyperbolic kink or two hyperbolic pulse solutions. These
results demonstrate that the notion of superposed solutions extends to coupled
nonlinear equations as well.

##  1 Introduction

Superposition principle is the hallmark of linear theories and it has helped
us in understanding various features of such theories. In contrast, because of
the nonlinear term, the nonlinear theories lack such a superposition
principle. This we believe is one of the reasons why until now we only have a
limited knowledge of such theories. Undoubtedly, during the last five decades
substantial progress has been made in understanding several novel features of
a number of nonlinear models but even at present new features of nonlinear
theories are being uncovered. In a recent paper [1] we have shown that a large
number of nonlinear equations, including both symmetric and asymmetric
$\phi^{4}$, MKdV, NLS and several other nonlinear equations, many of which
have found wide application in physics, admit superposed periodic kink and
periodic pulse solutions.

By superposed here we mean that a solution is a linear combination of two kink
solutions or two pulse solutions. However, only one hyperbolic superposed
solution was obtained by us in some of these models as the difference of two
kink solutions. In fact, even before us a number of previous researchers [2,
3, 4, 5, 6, 7] had obtained such a superposed kink solution in a variety of
physical models. However, to the best of our knowledge, to date no one has
been able to obtain solutions which can be re-expressed either as the sum of
two (hyperbolic) kink solutions or the sum or difference of two pulse
solutions. The purpose of this paper is to partially fill this gap.

We present three well known coupled models, i.e. a coupled $\phi^{4}$, a
coupled NLS and a coupled MKdV, which have found widespread applications in
ferroelectric [8] and multiferroic [9] phase transiitons, signal propagation
in optical fibers [10], etc. We then obtain novel solutions which can be re-
expressed either as the sum or the difference of two (hyperbolic) kink or two
pulse solutions. Needless to say that these models also admit novel solutions
which can be re-expressed as the sum or the difference of two periodic kink or
two periodic pulse solutions. However, in this paper we only confine ourselves
to the superposition of hyperbolic kink and pulse solutions. In a subsequent
paper we hope to address the question of the superposition of periodic kink
and pulse solutions in these models.

The plan of the paper is the following. In Sec. II, we discuss a coupled
$\phi^{4}$ model which has received attention previously [11] and obtain seven
solutions which can be re-expressed as either the sum or the difference of two
kink or two pulse solutions. In Sec. III, we discuss a coupled NLS model, also
popularly known as the Manakov model [12] and show that it admits seven
solutions (which are similar to the corresponding solutions of the coupled
$\phi^{4}$ model) which can be re-expressed as the sum or the difference of
two kink or two pulse solutions. In Sec. IV we discuss a coupled MKdV model
[13] and show that this model admits four solutions which can be re-expressed
as the sum or the difference of two kink solutions. Finally, in Sec. V we
summarize our main results and point out some of the open problems.

##  2 Novel Superposed Solutions of a Coupled $\phi^{4}$ Model

Let us consider the following coupled $\phi^{4}$ equations [11]

$\phi_{1xx}=a_{1}\phi_{1}+[b_{1}\phi_{1}^{2}+d_{1}\phi_{2}^{2}]\phi_{1}\,,$
(1)
$\phi_{2xx}=a_{2}\phi_{2}+[b_{2}\phi_{1}^{2}+d_{2}\phi_{2}^{2}]\phi_{2}\,.$
(2)

Before we discuss the superposed kink and pulse solutions, let us note that
the coupled Eqs. (1) and (2) admit the following kink and pulse solutions.

It is well known that Eqs. (1) and (2) admit a kink solution in both
$\phi_{1}$ and $\phi_{2}$, i.e.

$\phi_{1}(x)=A_{1}\tanh(\beta x)\,,~{}~{}\phi_{2}=A_{2}\tanh(\beta x)\,,$ (3)

provided

$a_{1}=a_{2}=-2\beta^{2}\,,~{}~{}b_{1}A_{1}^{2}+d_{1}A_{2}^{2}=b_{2}A_{1}^{2}+d_{2}A_{2}^{2}=2\beta^{2}\,.$
(4)

It is also known that Eqs. (1) and (2) admit a pulse solution in both
$\phi_{1}$ and $\phi_{2}$, i.e.

$\phi_{1}(x)=A_{1}{\rm sech}(\beta x)\,,~{}~{}\phi_{2}=A_{2}{\rm sech}(\beta
x)\,,$ (5)

provided

$a_{1}=a_{2}=\beta^{2}\,,~{}~{}b_{1}A_{1}^{2}+d_{1}A_{2}^{2}=b_{2}A_{1}^{2}+d_{2}A_{2}^{2}=-2\beta^{2}\,.$
(6)

Finally, Eqs. (1) and (2) are also known to admit a mixed kink-pulse solution,
i.e. say a kink solution in $\phi_{1}$ and a pulse solution in $\phi_{2}$ or
vice versa. In particular, they admit

$\phi_{1}(x)=A_{1}\tanh(\beta x)\,,~{}~{}\phi_{2}=A_{2}{\rm sech}(\beta x)\,,$
(7)

provided

$\displaystyle
b_{1}A_{1}^{2}-d_{1}A_{2}^{2}=b_{2}A_{1}^{2}-d_{2}A_{2}^{2}=2\beta^{2}\,,$
$\displaystyle
b_{2}A_{1}^{2}=\beta^{2}-a_{2}\,,~{}~{}d_{1}A_{2}^{2}=-a_{1}-2\beta^{2}\,.$
(8)

We now show that the coupled Eqs. (1) and (2) in fact also admit novel
solutions which can be re-expressed as the sum or the difference of the above
kink and pulse solutions.

Let us first note that in case $\phi_{2}(x)=\alpha\phi_{1}(x)$, where $\alpha$
is a real number, then Eq. (2) is identical to Eq. (1) provided

$a_{2}=a_{1}\,,~{}~{}b_{1}+\alpha^{2}d_{1}=b_{2}+\alpha^{2}d_{2}\,,$ (9)

and in this case we only need to solve Eq. (1) which as we have recently shown
[1] does not admit superposed hyperbolic solutions.

We now show that instead when $\phi_{2}$ and $\phi_{1}$ are not proportional
to each other then the coupled Eqs. (1) and (2) admit seven distinct solutions
which can be re-expressed as the sum or the difference of either two kink or
two pulse solutions.

Superposed Solution I

It is easy to check that the coupled Eqs. (1) and (2) admit the superposed
hyperbolic solution

$\phi_{1}(x)=\frac{A}{B+\cosh^{2}(\beta
x)}\,,~{}~{}\phi_{2}(x)=\frac{D\cosh(\beta x)\sinh(\beta x)}{B+\cosh^{2}(\beta
x)}\,,~{}~{}B>0\,,$ (10)

provided

$\displaystyle
a_{1}=a_{2}=2\beta^{2}\,,~{}~{}b_{1}A^{2}=-2B(B+1)\beta^{2}\,,~{}~{}d_{1}D^{2}=-6\beta^{2}\,,$
$\displaystyle b_{2}A^{2}=-6B(B+1)\beta^{2}\,,~{}~{}d_{2}D^{2}=-2\beta^{2}\,.$
(11)

Thus for this solution $b_{1},d_{1},b_{2},d_{2}$ are all negative while
$a_{1},a_{2}$ are positive.

Now on making use of the addition theorem for $\tanh(a+b)$, one can derive two
novel identities

$\tanh(x+\Delta)-\tanh(x-\Delta)=\frac{\sinh(2\Delta)}{B+\cosh^{2}(x)}\,,~{}~{}B=\sinh^{2}(\Delta)\,,$
(12)
$\tanh(x+\Delta)+\tanh(x-\Delta)=\frac{2\sinh(x)\cosh(x)}{B+\cosh^{2}(x)}\,,~{}~{}B=\sinh^{2}(\Delta)\,.$
(13)

On comparing the solution (10) with the identities (12) and (13), solution I
as given by Eq. (10) can be re-expressed as

$\displaystyle\phi_{1}(x)=\frac{\beta}{\sqrt{2|b_{1}|}}[\tanh(\beta
x+\Delta)-\tanh(\beta x-\Delta)]\,,$
$\displaystyle\phi_{2}(x)=\frac{\sqrt{3}\beta}{\sqrt{2|d_{1}|}}[\tanh(\beta
x+\Delta)+\tanh(\beta x-\Delta)]\,,$ (14)

where $\sinh^{2}(\Delta)=B$.

Superposed Solution II

It is easy to check that the coupled Eqs. (1) and (2) admit the superposed
hyperbolic solution

$\phi_{1}(x)=\frac{A}{B+\cosh^{2}(\beta
x)}\,,~{}~{}\phi_{2}(x)=\frac{D\sinh(\beta x)}{B+\cosh^{2}(\beta
x)}\,,~{}~{}B>0\,,$ (15)

provided

$\displaystyle
a_{1}=-4\beta^{2}\,,~{}~{}b_{1}A^{2}=2(2B^{2}+5B+3)\beta^{2}\,,~{}~{}d_{1}D^{2}=6(2B+1)\beta^{2}\,,$
$\displaystyle
a_{2}=-\beta^{2}\,,~{}~{}b_{2}A^{2}=6(B+1)\beta^{2}\,,~{}~{}d_{2}\beta^{2}=2(3+4B)\beta^{2}\,.$
(16)

Thus for this solution $b_{1},d_{1},b_{2},d_{2}$ are all positive while
$a_{1},a_{2}$ are negative.

Now on making use of the addition theorem for ${\rm sech}(a+b)$, one can
derive two novel identities

${\rm sech}(x-\Delta)-{\rm
sech}(x+\Delta)=\frac{2\sinh(x)\sinh(\Delta)}{B+\cosh^{2}(x)}\,,~{}~{}B=\sinh^{2}(\Delta)\,,$
(17) ${\rm sech}(x+\Delta)+{\rm
sech}(x-\Delta)=\frac{2\cosh(\Delta)\cosh(x)}{B+\cosh^{2}(x)}\,,~{}~{}B=\sinh^{2}(\Delta)\,.$
(18)

On comparing the solution (15) with the identities (12) and (17), solution II
as given by Eq. (15) can be re-expressed as

$\displaystyle\phi_{1}(x)=\frac{\sqrt{3}\beta}{\sqrt{2b_{2}}\sinh(\Delta)}[\tanh(\beta
x+\Delta)-\tanh(\beta x-\Delta)]\,,$
$\displaystyle\phi_{2}(x)=\frac{\sqrt{3\cosh(2\Delta)}\beta}{\sqrt{2d_{1}}\sinh(\Delta)}[{\rm
sech}(\beta x-\Delta)-{\rm sech}(\beta x+\Delta)]\,,$ (19)

where $B=\sinh^{2}(\Delta)$.

Superposed Solution III

It is easy to check that the coupled Eqs. (1) and (2) admit the superposed
hyperbolic solution

$\phi_{1}(x)=\frac{A}{B+\cosh^{2}(\beta
x)}\,,~{}~{}\phi_{2}(x)=\frac{D\cosh(\beta x)}{B+\cosh^{2}(\beta
x)}\,,~{}~{}B>0\,,$ (20)

provided

$\displaystyle
a_{1}=-4\beta^{2}\,,~{}~{}b_{1}A^{2}=2B(2B-1)\beta^{2}\,,~{}~{}d_{1}D^{2}=6(2B+1)\beta^{2}\,,$
$\displaystyle
a_{2}=-\beta^{2}\,,~{}~{}b_{2}A^{2}=-6B\beta^{2}\,,~{}~{}d_{2}\beta^{2}=2(1+4B)\beta^{2}\,.$
(21)

Thus for this solution while $d_{1},d_{2}<0$, $a_{1},a_{2},b_{2}<0$ while
$b_{1}>$ $(<)$ 0 depending on if $B>$ $(<)$ 1/2 while $b_{1}=0$ at $B=1/2$.

On comparing the solution (2) with the identities (12) and (18), solution III
as given by Eq. (20) can be re-expressed as

$\displaystyle\phi_{1}(x)=\frac{\sqrt{3}\beta}{\sqrt{2|b_{2}|}\cosh(\Delta)}[\tanh(\beta
x+\Delta)-\tanh(\beta x-\Delta)]\,,$
$\displaystyle\phi_{2}(x)=\frac{\sqrt{3\cosh(2\Delta)}\beta}{\sqrt{2d_{1}}\cosh(\Delta)}[{\rm
sech}(\beta x+\Delta)+{\rm sech}(\beta x-\Delta)]\,,$ (22)

where $B=\sinh^{2}(\Delta)$.

Superposed Solution IV

It is easy to check that the coupled Eqs. (1) and (2) admit the superposed
hyperbolic solution

$\phi_{1}(x)=\frac{A\sinh(\beta x)}{B+\cosh^{2}(\beta
x)}\,,~{}~{}\phi_{2}(x)=\frac{D\sinh(\beta x)\cosh(\beta x)}{B+\cosh^{2}(\beta
x)}\,,~{}~{}B>0\,,$ (23)

provided

$\displaystyle
a_{1}=\frac{(5-B)\beta^{2}}{(1+B)}\,,~{}~{}b_{1}A^{2}=2(2B^{2}+5B+3)\beta^{2}\,,~{}~{}d_{1}D^{2}=6(2B+1)\beta^{2}\,,$
$\displaystyle
a_{2}=-\beta^{2}\,,~{}~{}b_{2}A^{2}=6(B+1)\beta^{2}\,,~{}~{}d_{2}\beta^{2}=2(3+4B)\beta^{2}\,.$
(24)

Thus for this solution $b_{1},d_{1},b_{2},d_{2}$ are all positive while
$a_{1},a_{2}$ are negative.

On making use of the novel identities (13) and (17), one can then re-express
solution IV as given by Eq. (23) as

$\displaystyle\phi_{1}(x)=\frac{\sqrt{3}\beta}{\sqrt{2b_{2}}\coth(\Delta)}[{\rm
sech}(\beta x-\Delta)-{\rm sech}(\beta x+\Delta)]\,,$
$\displaystyle\phi_{2}(x)=\frac{\sqrt{3\cosh(2\Delta)}\beta}{\sqrt{2d_{1}}}[\tanh(\beta
x+\Delta)+\tanh(\beta x-\Delta)]\,,$ (25)

where $B=\sinh^{2}(\Delta)$.

Superposed Solution V

It is easy to check that the coupled Eqs. (1) and (2) admit the superposed
hyperbolic solution

$\phi_{1}(x)=\frac{A\cosh(\beta x)}{B+\cosh^{2}(\beta
x)}\,,~{}~{}\phi_{2}(x)=\frac{D\sinh(\beta x)\cosh(\beta x)}{B+\cosh^{2}(\beta
x)}\,,~{}~{}B>0\,,$ (26)

provided

$\displaystyle
a_{1}=-\frac{(6+B)\beta^{2}}{B}\,,~{}~{}b_{1}A^{2}=\frac{2(B+1)(4B+3)\beta^{2}}{B}\,,$
(27) $\displaystyle
d_{1}D^{2}=\frac{6\beta^{2}}{B}\,,~{}~{}a_{2}=-\frac{2(2B+3)beta^{2}}{B}\,,$
$\displaystyle
b_{2}A^{2}=\frac{2(2B^{2}+13B+3)\beta^{2}}{B}\,,~{}~{}d_{2}=2(3+4B)\beta^{2}\,.$

Thus for this solution $b_{1},d_{1},b_{2},d_{2}$ are all positive while
$a_{1},a_{2}$ are negative.

On making use of the novel identities (13) and (18), one can then re-express
solution V as given by Eq. (26) re-expressed as

$\displaystyle\phi_{1}(x)=\frac{\sqrt{4\sinh^{2}(\Delta)+3}\beta}{\sqrt{2b_{1}}\sinh(\Delta)}[{\rm
sech}(\beta x+\Delta)+{\rm sech}(\beta x-\Delta)]\,,$
$\displaystyle\phi_{2}(x)=\frac{\sqrt{4\sinh^{2}(\Delta)+3}\beta}{\sqrt{2d_{2}}\sinh(\Delta)}[\tanh(\beta
x+\Delta)+\tanh(\beta x-\Delta)]\,,$ (28)

where $B=\sinh^{2}(\Delta)$.

Superposed Solution VI

It is easy to check that the coupled Eqs. (1) and (2) admit the superposed
hyperbolic solution

$\phi_{1}(x)=\frac{A\sinh(\beta x)}{B+\cosh^{2}(\beta
x)}\,,~{}~{}\phi_{2}(x)=\frac{D\cosh(\beta x)}{B+\cosh^{2}(\beta
x)}\,,~{}~{}B>0\,,$ (29)

provided

$\displaystyle
a_{1}=-\beta^{2}\,,~{}~{}b_{1}A^{2}=2B\beta^{2}\,,~{}~{}d_{1}D^{2}=6(B+1)\beta^{2}\,,$
$\displaystyle
a_{2}=-\beta^{2}\,,~{}~{}b_{2}A^{2}=6B\beta^{2}\,,~{}~{}d_{2}\beta^{2}=2(B+1)\beta^{2}\,.$
(30)

Thus for this solution $b_{1},d_{1},b_{2},d_{2}$ are all positive while
$a_{1},a_{2}$ are negative.

On making use of the novel identities (17) and (18), one can then re-express
solution VI as given by Eq. (29) as

$\displaystyle\phi_{1}(x)=\frac{\beta}{\sqrt{2b_{1}}}[{\rm sech}(\beta
x-\Delta)-{\rm sech}(\beta x+\Delta)]\,,$
$\displaystyle\phi_{2}(x)=\frac{\beta}{\sqrt{2d_{2}}}[{\rm sech}(\beta
x+\Delta)+{\rm sech}(\beta x-\Delta)]\,,$ (31)

where $B=\sinh^{2}(\Delta)$.

Superposed Solution VII

It is easy to check that the coupled Eqs. (1) and (2) admit the superposed
hyperbolic solution

$\phi_{1}(x)=1-\frac{A}{B+\cosh^{2}(\beta
x)}\,,~{}~{}\phi_{2}(x)=\frac{D}{B+\cosh^{2}(\beta x)}\,,~{}~{}A,B,D>0\,,$
(32)

provided

$\displaystyle
b_{1}=-a_{1}=2\beta^{2}\,,~{}~{}A=\frac{3(1+2B)+\sqrt{8+(2B+1)^{2}}}{4}\,,$
$\displaystyle
d_{1}D^{2}=[\sqrt{8+(2B+1)^{2}}-(2B+1)]\frac{3A\beta^{2}}{2}\,,~{}~{}b_{2}+a_{2}=4\beta^{2}\,,$
$\displaystyle
b_{2}A=3(2B+1)\beta^{2}\,,~{}~{}b_{2}A^{2}+d_{2}D^{2}=8B(B+1)\beta^{2}\,.$
(33)

Thus for this solution $b_{1},d_{1},a_{2},b_{2}$ are all positive, $a_{1}<0$
while the sign of $d_{2}$ depends on the value of $B$.

On comparing the solution (32) with the identity (12), solution VII as given
by Eq. (32) can be re-expressed as

$\displaystyle\phi_{1}(x)=1-\frac{\beta}{\sqrt{2|b_{1}|}}[\tanh(\beta
x+\Delta)-\tanh(\beta x-\Delta)]\,,$
$\displaystyle\phi_{2}(x)=\frac{\sqrt{3}\beta}{2\sqrt{d_{1}}}K[\tanh(\beta
x+\Delta)-\tanh(\beta x-\Delta)]\,,$ (34)

where $\sinh^{2}(\Delta)=B$ and

$K=\bigg{[}\cosh(2\Delta)\sqrt{8+\cosh^{2}(2\Delta)}-[2+\cosh^{2}(2\Delta)]\bigg{]}^{1/2}\,.$
(35)

##  3 Hyperbolic Superposed Solutions of the Coupled NLS Model

Let us consider the following coupled NLS equations [12]

$iu_{1t}+u_{1xx}+[g_{11}|u_{1}|^{2}+g_{12}|u_{2}|^{2}]u_{1}\,,$ (36)
$iu_{2t}+u_{2xx}+[g_{21}|u_{1}|^{2}+g_{22}|u_{2}|^{2}]u_{2}\,.$ (37)

Before we discuss the superposed kink and pulse solutions, let us note that
the coupled Eqs. (36) and (37) admit the following kink and pulse solutions.
In particular, it is well known that Eqs. (36) and (37) admit a kink solution
in both $u_{1}$ and $u_{2}$, i.e.

$u_{1}(x,t)=A_{1}e^{i\omega_{1}t}\tanh(\beta
x)\,,~{}~{}u_{2}(x,t)=A_{2}e^{i\omega t}\tanh(\beta x)\,,$ (38)

provided

$\omega_{1}=\omega_{2}=-2\beta^{2}\,,~{}~{}g_{11}A_{1}^{2}+g_{12}A_{2}^{2}=g_{21}A_{1}^{2}+g_{22}A_{2}^{2}=2\beta^{2}\,.$
(39)

It is also known that Eqs. (36) and (37) admit a pulse solution in both
$u_{1}$ and $u_{2}$, i.e.

$u_{1}(x,t)=A_{1}e^{i\omega_{1}t}{\rm sech}(\beta
x)\,,~{}~{}u_{2}(x,t)=A_{2}e^{i\omega_{2}t}{\rm sech}(\beta x)\,,$ (40)

provided

$\omega_{1}=\omega_{2}=\beta^{2}\,,~{}~{}g_{11}A_{1}^{2}+g_{12}A_{2}^{2}=g_{21}A_{1}^{2}+g_{22}A_{2}^{2}=-2\beta^{2}\,.$
(41)

Finally, Eqs. (36) and (37) are also known to admit a mixed kink-pulse
solution, i.e. say a kink solution in $u_{1}$ and a pulse solution in $u_{2}$
or vice versa. In particular, they admit

$u_{1}(x,t)=A_{1}e^{i\omega_{1}t}\tanh(\beta
x)\,,~{}~{}u_{2}(x,t)=A_{2}e^{i\omega_{2}t}{\rm sech}(\beta x)\,,$ (42)

provided

$\displaystyle
g_{11}A_{1}^{2}-g_{12}A_{2}^{2}=g_{21}A_{1}^{2}-g_{22}A_{2}^{2}=2\beta^{2}\,,$
$\displaystyle
g_{21}A_{1}^{2}=\beta^{2}-\omega_{2}\,,~{}~{}g_{12}A_{2}^{2}=-\omega_{1}-2\beta^{2}\,.$
(43)

We now show that the coupled Eqs. (36) and (37) in fact also admit novel
solutions which can be re-expressed as the sum or the difference of the above
kink and pulse solutions.

Let us first note that in case $u_{2}(x,t)=\alpha u_{1}(x,t)$, where $\alpha$
is a real number, then Eq. (37) is identical to Eq. (36) provided

$g_{11}+\alpha^{2}g_{12}=g_{21}+\alpha^{2}g_{22}\,,$ (44)

and in this case we only need to solve Eq. (36) which as we have recently
shown [1] does not admit superposed hyperbolic solutions.

We now show that instead when $u_{2}$ and $u_{1}$ are not proportional to each
other then the coupled Eqs. (36) and (37) admit seven distinct solutions which
can be re-expressed as the sum or the difference of either two kink or two
pulse solutions.

Superposed Solution I

It is easy to check that the coupled Eqs. (36) and (37) admit the superposed
hyperbolic solution

$u_{1}(x,t)=e^{i\omega_{1}t}\frac{A}{B+\cosh^{2}(\beta
x)}\,,~{}~{}u_{2}(x,t)=e^{i\omega_{2}t}\frac{D\cosh(\beta x)\sinh(\beta
x)}{B+\cosh^{2}(\beta x)}\,,~{}~{}B>0\,,$ (45)

provided

$\displaystyle\omega_{1}=\omega_{2}=2\beta^{2}\,,~{}~{}g_{11}A^{2}=-2B(B+1)\beta^{2}\,,~{}~{}g_{12}D^{2}=-6\beta^{2}\,,$
(46) $\displaystyle
g_{21}A^{2}=-6B(B+1)\beta^{2}\,,~{}~{}g_{22}D^{2}=-2\beta^{2}\,.$

Thus for this solution $g_{11},g_{12},g_{21},g_{22}$ are all negative while
$\omega_{1}=\omega_{2}$ are positive.

On comparing the solution (45) with the identities (12) and (13), the solution
I as given by Eq. (45) can be re-expressed as

$\displaystyle
u_{1}(x,t)=e^{i\omega_{1}t}\frac{\beta}{\sqrt{2|g_{11}|}}[\tanh(\beta
x+\Delta)-\tanh(\beta x-\Delta)]\,,$ $\displaystyle
u_{2}(x,t)=e^{i\omega_{2}t}\frac{\sqrt{3}\beta}{\sqrt{2|g_{12}|}}[\tanh(\beta
x+\Delta)+\tanh(\beta x-\Delta)]\,,$ (47)

where $\sinh^{2}(\Delta)=B$.

Superposed Solution II

It is easy to check that the coupled Eqs. (36) and (37) admit the superposed
hyperbolic solution

$u_{1}(x,t)=e^{i\omega t}\frac{A}{B+\cosh^{2}(\beta
x)}\,,~{}~{}u_{2}(x,t)=e^{i\omega_{2}t}\frac{D\sinh(\beta
x)}{B+\cosh^{2}(\beta x)}\,,~{}~{}B>0\,,$ (48)

provided

$\displaystyle\omega_{1}=-4\beta^{2}\,,~{}~{}g_{11}A^{2}=2(2B^{2}+5B+3)\beta^{2}\,,~{}~{}g_{12}D^{2}=6(2B+1)\beta^{2}\,,$
$\displaystyle\omega_{2}=-\beta^{2}\,,~{}~{}g_{21}A^{2}=6(B+1)\beta^{2}\,,~{}~{}g_{22}\beta^{2}=2(3+4B)\beta^{2}\,.$
(49)

Thus for this solution $g_{11},g_{12},g_{21},g_{22}$ are all positive while
$\omega_{1},\omega_{2}$ are negative.

On comparing the solution (48) with the identities (12) and (17), solution II
as given by Eq. (48) can be re-expressed as

$\displaystyle
u_{1}(x,t)=e^{i\omega_{1}t}\frac{\sqrt{3}\beta}{\sqrt{2g_{21}}\sinh(\Delta)}[\tanh(\beta
x+\Delta)-\tanh(\beta x-\Delta)]\,,$ $\displaystyle
u_{2}(x,t)=e^{i\omega_{2}t}\frac{\sqrt{3\cosh(2\Delta)}\beta}{\sqrt{2g_{12}}\sinh(\Delta)}[{\rm
sech}(\beta x-\Delta)-{\rm sech}(\beta x+\Delta)]\,,$ (50)

where $B=\sinh^{2}(\Delta)$.

Superposed Solution III

It is easy to check that the coupled Eqs. (36) and (37) admit the superposed
hyperbolic solution

$u_{1}(x,t)=e^{i\omega_{1}t}\frac{A}{B+\cosh^{2}(\beta
x)}\,,~{}~{}u_{2}(x,t)=e^{i\omega_{2}t}\frac{D\cosh(\beta
x)}{B+\cosh^{2}(\beta x)}\,,~{}~{}B>0\,,$ (51)

provided

$\displaystyle\omega_{1}=-4\beta^{2}\,,~{}~{}g_{11}A^{2}=2B(2B-1)\beta^{2}\,,~{}~{}g_{12}D^{2}=6(2B+1)\beta^{2}\,,$
$\displaystyle\omega_{2}=-\beta^{2}\,,~{}~{}g_{21}A^{2}=-6B\beta^{2}\,,~{}~{}g_{22}\beta^{2}=2(1+4B)\beta^{2}\,.$
(52)

Thus for this solution while $g_{12},g_{22}<0$,
$\omega_{1},\omega_{2},g_{21}<0$ while $g_{11}>$ $(<)$ 0 depending on if $B>$
$(<)$ 1/2 while $g_{11}=0$ at $B=1/2$.

On comparing the solution (3) with the identities (12) and (18), solution III
as given by Eq. (51) can be re-expressed as

$\displaystyle
u_{1}(x,t)=e^{i\omega_{1}t}\frac{\sqrt{3}\beta}{\sqrt{2|g_{21}|}\cosh(\Delta)}[\tanh(\beta
x+\Delta)-\tanh(\beta x-\Delta)]\,,$ $\displaystyle
u_{2}(x,t)=e^{i\omega_{2}t}\frac{\sqrt{3\cosh(2\Delta)}\beta}{\sqrt{2g_{12}}\cosh(\Delta)}[{\rm
sech}(\beta x+\Delta)+{\rm sech}(\beta x-\Delta)]\,,$ (53)

where $B=\sinh^{2}(\Delta)$.

Superposed Solution IV

It is easy to check that the coupled Eqs. (36) and (37) admit the superposed
hyperbolic solution

$u_{1}(x,t)=e^{i\omega_{1}t}\frac{A\sinh(\beta x)}{B+\cosh^{2}(\beta
x)}\,,~{}~{}u_{2}(x,t)=e^{i\omega_{2}t}\frac{D\sinh(\beta x)\cosh(\beta
x)}{B+\cosh^{2}(\beta x)}\,,~{}~{}B>0\,,$ (54)

provided

$\displaystyle\omega_{1}=\frac{(5-B)\beta^{2}}{(1+B)}\,,~{}~{}g_{11}A^{2}=2(2B^{2}+5B+3)\beta^{2}\,,~{}~{}g_{12}D^{2}=6(2B+1)\beta^{2}\,,$
$\displaystyle\omega_{2}=-\beta^{2}\,,~{}~{}g_{21}A^{2}=6(B+1)\beta^{2}\,,~{}~{}g_{22}\beta^{2}=2(3+4B)\beta^{2}\,.$
(55)

Thus for this solution $g_{11},g_{21},g_{21},g_{22}$ are all positive while
$\omega_{1},\omega_{2}$ are negative.

On making use of the novel identities (13) and (17), one can then re-express
solution IV as given by Eq. (54) as

$\displaystyle
u_{1}(x,t)=e^{i\omega_{1}t}\frac{\sqrt{3}\beta}{\sqrt{2g_{21}}\coth(\Delta)}[{\rm
sech}(\beta x-\Delta)-{\rm sech}(\beta x+\Delta)]\,,$ $\displaystyle
u_{2}(x,t)=e^{i\omega_{2}t}\frac{\sqrt{3\cosh(2\Delta)}\beta}{\sqrt{2g_{12}}}[\tanh(\beta
x+\Delta)+\tanh(\beta x-\Delta)]\,,$ (56)

where $B=\sinh^{2}(\Delta)$.

Superposed Solution V

It is easy to check that the coupled Eqs. (36) and (37) admit the superposed
hyperbolic solution

$u_{1}(x,t)=e^{i\omega_{1}t}\frac{A\cosh(\beta x)}{B+\cosh^{2}(\beta
x)}\,,~{}~{}u_{2}(x,t)=e^{i\omega_{2}t}\frac{D\sinh(\beta x)\cosh(\beta
x)}{B+\cosh^{2}(\beta x)}\,,~{}~{}B>0\,,$ (57)

provided

$\displaystyle\omega_{1}=-\frac{(6+B)\beta^{2}}{B}\,,~{}~{}g_{11}A^{2}=\frac{2(B+1)(4B+3)\beta^{2}}{B}\,,$
$\displaystyle
g_{12}D^{2}=\frac{6\beta^{2}}{B}\,,~{}~{}\omega_{2}=-\frac{2(2B+3)beta^{2}}{B}\,,$
$\displaystyle
g_{21}A^{2}=\frac{2(2B^{2}+13B+3)\beta^{2}}{B}\,,~{}~{}g_{22}=2(3+4B)\beta^{2}\,.$
(58)

Thus for this solution $g_{11},g_{12},g_{21},g_{22}$ are all positive while
$\omega_{1},\omega_{2}$ are negative.

On making use of the novel identities (13) and (18), one can then re-express
solution V as given by Eq. (57) as

$\displaystyle
u_{1}(x,t)=\frac{\sqrt{4\sinh^{2}(\Delta)+3}\beta}{\sqrt{2g_{11}}\sinh(\Delta)}[{\rm
sech}(\beta x+\Delta)+{\rm sech}(\beta x-\Delta)]\,,$ $\displaystyle
u_{2}(x,t)=\frac{\sqrt{4\sinh^{2}(\Delta)+3}\beta}{\sqrt{2g_{22}}\sinh(\Delta)}[\tanh(\beta
x+\Delta)+\tanh(\beta x-\Delta)]\,,$ (59)

where $B=\sinh^{2}(\Delta)$.

Superposed Solution VI

It is easy to check that the coupled Eqs. (36) and (37) admit the superposed
hyperbolic solution

$u_{1}(x,t)=e^{i\omega_{1}t}\frac{A\sinh(\beta x)}{B+\cosh^{2}(\beta
x)}\,,~{}~{}u_{2}(x,t)=e^{i\omega_{2}t}\frac{D\cosh(\beta
x)}{B+\cosh^{2}(\beta x)}\,,~{}~{}B>0\,,$ (60)

provided

$\displaystyle\omega_{1}=-\beta^{2}\,,~{}~{}g_{11}A^{2}=2B\beta^{2}\,,~{}~{}g_{12}D^{2}=6(B+1)\beta^{2}\,,$
$\displaystyle\omega_{2}=-\beta^{2}\,,~{}~{}g_{21}A^{2}=6B\beta^{2}\,,~{}~{}g_{22}\beta^{2}=2(B+1)\beta^{2}\,.$
(61)

Thus for this solution $g_{11},g_{12},g_{21},g_{22}$ are all positive while
$\omega_{1},\omega_{2}$ are negative.

On making use of the novel identities (17) and (18), one can then re-express
solution VI as given by Eq. (60) as

$\displaystyle u_{1}(x,t)=\frac{\beta}{\sqrt{2g_{11}}}[{\rm sech}(\beta
x-\Delta)-{\rm sech}(\beta x+\Delta)]\,,$ $\displaystyle
u_{2}(x,t)=\frac{\beta}{\sqrt{2g_{22}}}[{\rm sech}(\beta x+\Delta)+{\rm
sech}(\beta x-\Delta)]\,,$ (62)

where $B=\sinh^{2}(\Delta)$.

Superposed Solution VII

It is easy to check that the coupled Eqs. (1) and (2) admit the superposed
hyperbolic solution

$u_{1}(x,t)=e^{i\omega_{1}t}[1-\frac{A}{B+\cosh^{2}(\beta
x)}]\,,~{}~{}\phi_{2}(x)=e^{i\omega_{2}t}\frac{D}{B+\cosh^{2}(\beta
x)}\,,~{}~{}A,B,D>0\,,$ (63)

provided

$\displaystyle
g_{11}=-\omega_{1}=2\beta^{2}\,,~{}~{}A=\frac{3(1+2B)+\sqrt{8+(2B+1)^{2}}}{4}\,,$
$\displaystyle
g_{12}D^{2}=[\sqrt{8+(2B+1)^{2}}-(2B+1)]\frac{3A\beta^{2}}{2}\,,~{}~{}g_{21}+\omega_{2}=4\beta^{2}\,,$
(64) $\displaystyle
g_{21}A=3(2B+1)\beta^{2}\,,~{}~{}g_{21}A^{2}+g_{22}D^{2}=8B(B+1)\beta^{2}\,.$

Thus for this solution $g_{11},g_{12},\omega_{2},g_{21}$ are all positive,
$\omega_{1}<0$ while the sign of $g_{22}$ depends on the value of $B$.

On comparing the solution (63) with the identity (12), solution VII as given
by Eq. (63) can be re-expressed as

$\displaystyle
u_{1}(x,t)=e^{i\omega_{1}t}[1-\frac{\beta}{\sqrt{2|g_{11}|}}[\tanh(\beta
x+\Delta)-\tanh(\beta x-\Delta)]]\,,$ $\displaystyle
u_{2}(x,t)=\frac{\sqrt{3}\beta}{2\sqrt{g_{12}}}K[\tanh(\beta
x+\Delta)-\tanh(\beta x-\Delta)]\,,$ (65)

where $\sinh^{2}(\Delta)=B$ and

$K=\bigg{[}\cosh(2\Delta)\sqrt{8+\cosh^{2}(2\Delta)}-[2+\cosh^{2}(2\Delta)]\bigg{]}^{1/2}\,.$
(66)

##  4 Hyperbolic Superposed Solutions of the Coupled MKdV Model

Let us consider the following coupled MKdV equations [13]

$u_{1t}+u_{1xxx}+6[g_{11}u_{1}^{2}+g_{12}u_{2}^{2}]u_{1x}=0\,,$ (67)
$u_{2t}+u_{2xxx}+6[g_{21}u_{1}^{2}+g_{22}u_{2}^{2}]u_{2x}=0\,,$ (68)

Before we discuss the superposed kink and pulse solutions, let us note that
the coupled Eqs. (67) and (68) admit the following kink and pulse solutions.
In particular, it is well known that Eqs. (67) and (68) admit a kink solution
in both $u_{1}$ and $u_{2}$, i.e.

$u_{1}(x,t)=A_{1}\tanh(\xi)\,,~{}~{}u_{2}(x,t)=A_{2}\tanh(\xi)\,,~{}~{}\xi=\beta(x-vt)\,,$
(69)

provided

$v=-2\beta^{2}\,,~{}~{}g_{11}A_{1}^{2}+g_{12}A_{2}^{2}=g_{21}A_{1}^{2}+g_{22}A_{2}^{2}=-\beta^{2}\,.$
(70)

It is also known that Eqs. (67) and (68) admit a pulse solution in both
$u_{1}$ and $u_{2}$, i.e.

$u_{1}(x,t)=A_{1}{\rm sech}(\xi)\,,~{}~{}u_{2}(x,t)=A_{2}{\rm sech}(\xi)\,,$
(71)

provided

$v=\beta^{2}\,,~{}~{}g_{11}A_{1}^{2}+g_{12}A_{2}^{2}=g_{21}A_{1}^{2}+g_{22}A_{2}^{2}=2\beta^{2}\,.$
(72)

Finally, Eqs. (67) and (68) are also known to admit a mixed kink-pulse
solution, i.e. say a kink solution in $u_{1}$ and a pulse solution in $u_{2}$
or vice versa. In particular, they admit

$u_{1}(x,t)=A_{1}\tanh(\xi)\,,~{}~{}u_{2}(x,t)=A_{2}{\rm sech}(\xi)\,,$ (73)

provided

$\displaystyle
g_{12}A_{2}^{2}-g_{11}A_{1}^{2}=g_{22}A_{2}^{2}-g_{21}A_{1}^{2}=\beta^{2}\,,$
$\displaystyle v=6g_{21}A_{1}^{2}+\beta^{2}=6g_{12}A_{2}^{2}-2\beta^{2}\,.$
(74)

We now show that the coupled Eqs. (67) and (68) admit four novel solutions
which can be re-expressed as the sum or the difference of the above kink
solution.

We start with the ansatz

$u_{1}(x,t)=\phi_{1}(\xi)\,,~{}~{}u_{2}(x,t)=\phi_{2}(\xi)\,,~{}~{}\xi=\beta(x-vt)\,.$
(75)

On substituting the ansatz (75) in the coupled Eqs. (67) and (68) we get

$\beta^{2}\phi_{1\xi\xi\xi}=v\phi_{1\xi}-6[g_{11}\phi_{1}^{2}+g_{12}\phi_{2}^{2}]\phi_{1x}\,,$
(76)
$\beta^{2}\phi_{2\xi\xi\xi}=v\phi_{2\xi}-6[g_{21}\phi_{1}^{2}+g_{22}\phi_{2}^{2}]\phi_{2x}\,.$
(77)

We now show that the coupled Eqs. (76) and (77) and hence coupled Eqs. (67)
and (68) admit four hyperbolic superposed solutions in terms of
$\tanh(x+\Delta)\pm\tanh(x-\Delta)$. Out of these four solutions, while the
first solution is obtained when $u_{2}(x,t)\propto u_{1}(x,t)$, for the other
three solutions, $u_{1}(x,t)$ and $u_{2}(x,t)$ are not proportional to each
other.

Solution I

It is easy to check that

$u_{1}(x,t)=1-\frac{A}{B+\cosh^{2}(\xi)}\,,~{}~{}u_{2}(x,t)=\alpha
u_{1}(x,t)\,,~{}~{}A,B>0\,,$ (78)

is an exact solution to the coupled Eqs. (67) and (68), where $\alpha$ is any
real number provided

$\displaystyle
g_{11}+\alpha^{2}g_{12}=g_{21}+g_{22}\alpha^{2}=-\frac{(2B+1)^{2}}{4B(B+1)}\beta^{2}\,,$
$\displaystyle
A=\frac{4B(B+1)}{2B+1}\,,~{}~{}v=-\frac{(4B^{2}+4B+3)}{2B(B+1)}\beta^{2}\,.$
(79)

On comparing it with the identity (12), we can re-express solution (78) as

$u_{1}(x,t)=1-\frac{\beta\tanh(2\Delta)}{\sqrt{|g_{11}+\alpha^{2}g_{12}|}}[\tanh(\xi+\Delta)-\tanh(\xi-\Delta)]\,,~{}~{}B=\sinh^{2}(\Delta)\,,$
(80)

with $\xi=\beta(x-vt)$ and $u_{2}(x,t)=\alpha u_{1}(x,t)$ .

Solution II

Yet another solution to the coupled Eqs. (67) and (68) is

$u_{1}(x,t)=1-\frac{A}{B+\cosh^{2}(\xi)}\,,~{}~{}u_{2}(x,t)=\frac{D\cosh(\xi)\sinh(\xi)}{B+\cosh^{2}(\xi)}\,,~{}~{}A,B,D>0\,,$
(81)

provided

$\displaystyle
v=4\beta^{2}+6g_{11}+6g_{12}D^{2}\,,~{}~{}g_{11}+g_{12}D^{2}=g_{21}+g_{22}D^{2}\,,$
$\displaystyle
g_{11}A^{2}-B(2+B)g_{12}D^{2}=4B(B+1)\beta^{2}\,,~{}~{}(2B+1)g_{11}A^{2}+2B(B+1)Ag_{11}$
$\displaystyle=2B(B+1)(2B+1)\beta^{2}\,,g_{22}D^{2}=-\frac{(10B^{2}+12B+3)}{(2B+1)^{2}}\beta^{2}\,,$
$\displaystyle
g_{21}A^{2}=-\frac{B(B+1)(14B^{2}+12B+3)}{(2B+1)^{2}}\,,~{}~{}A=\frac{B(14B^{2}+12B+3)}{(2B+1)(B+1)}\beta^{2}\,.$
(82)

Note that for this solution $g_{21},g_{22},g_{12}<0$ while $g_{11}>0$.

On comparing it with the novel identities (12) and (13) we can re-express
solution (81) as the superposed solution

$u_{1}(x,t)=1-\frac{\beta[\tanh(\xi+\Delta)-\tanh(\xi-\Delta)]}{2\sinh(\Delta)\sqrt{|g_{21}\cosh(2\Delta)}}\,,$
(83)
$u_{2}(x,t)=\frac{\beta\sqrt{\cosh(\Delta)}}{\sqrt{|g_{22}|}\cosh(2\delta)}[\tanh(\xi+\Delta)+\tanh(\xi-\Delta)]\,,$
(84)

where $B=\sinh^{2}(\Delta)$.

Solution III

Yet another hyperbolic superposed solution is

$\phi_{1}(\xi)=1-\frac{A}{B+\cosh^{2}(\xi)}\,,~{}~{}\phi_{2}(\xi)=\frac{\alpha
A}{B+\cosh^{2}(\xi)}\,,A,B>0\,,$ (85)

provided

$\displaystyle
v=4\beta^{2}+6g_{11}\,,~{}~{}g_{21}=g_{11}<0\,,~{}~{}g_{22}=g_{12}\,,~{}~{}Ag_{11}=-(2B+1)\beta^{2}\,,$
$\displaystyle(g_{11}+\alpha^{2}g_{12})A^{2}=-4B(B+1)\beta^{2}\,.$ (86)

On comparing it with the identity (12), we can re-express solution (85) as

$u_{1}(x,t)=1-\frac{\beta}{\sqrt{|g_{11}+\alpha^{2}g_{22}|}}[\tanh(\xi+\Delta)-\tanh(\xi-\Delta)]\,,$
(87)

and

$u_{2}(x,t)=\frac{\alpha\beta}{\sqrt{|g_{11}+\alpha^{2}g_{22}|}}[\tanh(\xi+\Delta)-\tanh(\xi-\Delta)]\,,~{}~{}B=\sinh^{2}(\Delta)\,,$
(88)

where $B=\sinh^{2}(\Delta)$.

Solution IV

Yet another hyperbolic superposed solution to the coupled Eqs. (67) and (68)
is

$u_{1}(x,t)=1-\frac{A}{B+\cosh^{2}(\xi)}\,,~{}~{}\phi_{2}(\xi)=-b-\frac{\alpha
A}{B+\cosh^{2}(\xi)}\,,~{}~{}A,B,b>0$ (89)

provided

$\displaystyle v-4\beta^{2}=g_{12}+b^{2}g_{22}=g_{11}+b^{2}g_{12}\,,$
$\displaystyle A(g_{11}-g_{12}\alpha b)==A(g_{21}-g_{22}\alpha
b)=-(2B+1)\beta^{2}\,,$
$\displaystyle(g_{11}+\alpha^{2}g_{12})A^{2}==(g_{21}+\alpha^{2}g_{22})=-4B(B+1)\beta^{2}\,.$
(90)

On comparing it with the novel identity, we can re-express solution (88) as

$u_{1}(x,t)=1-\frac{\beta}{\sqrt{|g_{11}+\alpha^{2}g_{22}|}}[\tanh(\xi+\Delta)-\tanh(\xi-\Delta)]\,,$
(91)

and

$u_{2}(x,t)=-b-\frac{\alpha\beta}{\sqrt{|g_{11}+\alpha^{2}g_{22}|}}[\tanh(\xi+\Delta)-\tanh(\xi-\Delta)]\,,$
(92)

where $B=\sinh^{2}(\Delta)$.

##  5 Conclusion and Open Problems

In this paper we have considered a coupled $\phi^{4}$ [11], a coupled NLS [12]
and a coupled MKdV [13] model and demonstrated that all of them not only admit
single kink and single pulse solutions but also admit novel solutions which
can be re-expressed in terms of the sum or the difference of two kink or two
pulse solutions. For the coupled $\phi^{4}$ and the coupled NLS models, we
have obtained six superposed solutions essentially covering all possible cases
among $\tanh(x+\Delta)\pm\tanh(x-\Delta)$ and ${\rm sech}(x-\Delta)\pm{\rm
sech}(x+\Delta)$ with the two coupled members (or fields) being distinct. On
the other hand, for the coupled MKdV we have only one such solution involving
$\tanh(x+\Delta)\pm\tanh(x-\Delta)$. Besides, we have obtained one superposed
solution each of the coupled $\phi^{4}$ and the coupled NLS models and three
superposed solutions of the coupled MKdV model where both the coupled members
(or fields) involve only $\tanh(x+\Delta)-\tanh(x-\Delta)$.

This paper raises several questions some of which are:

  1. 1.

How many of these superposed solutions are stable in each coupled model? If
stable, where can one look for such structures experimentally, e.g. in
photonics [10]? What is the connection of such superposed solutions vis a vis
a single kink or a single pulse solution?

  2. 2.

So far, except for $\tanh(x+\Delta)-\tanh(x-\Delta)$, no one has been able to
find superposed hyperbolic solutions of the form
$\tanh(x+\Delta)+\tanh(x-\Delta)$ or of the form ${\rm sech}(x+\Delta)\pm{\rm
sech}(x-\Delta)$ in any uncoupled model. It would be worthwhile finding such
solutions in uncoupled models. This will enable the physical interpretation of
such superposed solutions in comparison to the corresponding single kink or
single pulse solution.

  3. 3.

Presumably there are other coupled systems, e.g. coupled KdV arising in two-
layer fluids [14], which might admit solutions similar to those presented in
this paper. It would be desirable to find such models and the corresponding
superposed solutions.

  4. 4.

The three coupled models considered in this paper will obviously also admit
superposed periodic kink and pulse solutions in terms of Jacobi elliptic
functions. It would be interesting to find such solutions and see how many of
them smoothly go over to the hyperbolic superposed solutions obtained in this
paper. We hope to address this issue in the near future.

Acknowledgment

One of us (AK) is grateful to Indian National Science Academy (INSA) for the
award of INSA Senior Scientist position at Savitribai Phule Pune University.
The work at Los Alamos National Laboratory was carried out under the auspices
of the U.S. DOE and NNSA under Contract No. DEAC52-06NA25396.

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