Nonlocal Superposed Solutions II: Coupled Nonlinear 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 periodic as well as hyperbolic solutions of an Ablowitz-
Musslimani variant of the coupled nonlocal, nonlinear Schrödinger equation
(NLS) as well as a coupled nonlocal modified Korteweg-de Vries (mKdV) equation
which can be re-expressed as a linear superposition of the sum or the
difference of two hyperbolic or two periodic kink or pulse solutions. Besides,
we also discuss some of the other solutions admitted by these coupled
equations. These results demonstrate that the notion of the superposed
solutions extends to the coupled nonlocal nonlinear equations as well.

##  1 Introduction

Nonlinear theories are highly nontrivial because unlike the linear theories,
the nonlinear theories do not satisfy the superposition principle. However, by
now several nonlinear equations such as $\phi^{4}$, NLS, mKdV, KdV etc. have
been shown to satisfy a kind of superposition principle [1, 2]. In particular,
we showed that if a nonlinear equation admits a periodic pulse solution in
terms of the Jacobi elliptic functions [3] ${\rm dn}(x,m)$ and ${\rm
cn}(x,m)$, then the same equation will also admit a superposed periodic
solution ${\rm dn}(x,m)\pm\sqrt{m}{\rm cn}(x,m)$ where $m$ is the modulus of
the Jacobi elliptic function. Similarly, if a nonlinear equation admits a
solution in terms of ${\rm dn}^{2}(x,m)$ then it will also admit a solution in
terms of ${\rm dn}^{2}(x,m)\pm\sqrt{m}{\rm dn}(x,m){\rm cn}(x,m)$. Further, in
another publication [4] we showed that if a nonlinear equation satisfies
solutions in terms of ${\rm dn}(x,m)$ or ${\rm cn}(x,m)$, then the same
nonlinear equation will also admit a complex PT-invariant solution in terms of
${\rm dn}(x,m)\pm i\sqrt{m}{\rm sn}(x,m)$ or ${\rm cn}(x,m)\pm i{\rm
sn}(x,m)$. Similarly, if a nonlinear equation admits a periodic kink solution
in terms of ${\rm sn}(x,m)$, then the same nonlinear equation also admits a
complex PT-invariant solution in terms of $\sqrt{m}{\rm sn}(x,m)\pm i{\rm
dn}(x,m)$ as well as ${\rm sn}(x,m)\pm i{\rm cn}(x,m)$. We further showed that
if a nonlinear equation admits a solution in terms of ${\rm dn}^{2}(x,m)$ then
the same nonlinear equation will also admit complex PT-invariant periodic
solutions in terms of ${\rm dn}^{2}(x,m)\pm i\sqrt{m}{\rm dn}(x,m){\rm
sn}(x,m)$ or ${\rm dn}^{2}(x,m)\pm im{\rm sn}(x,m){\rm cn}(x,m)$.

Recently, inspired by the work of [5, 6] as well as earlier work in condensed
matter as well as field theory [7, 8, 9, 10], we have further extended the
notion of superposition for several nonlinear equations. In particular, we
showed that the symmetric and the asymmetric $\phi^{4}$ model, the NLS,
quadratic-cubic NLS as well as mKdV and mKdV-KdV equations admit novel
solutions which can be re-expressed as the superposition of the periodic kink
and the periodic pulse solutions [11]. Further, in some cases we also obtained
solutions which can be re-expressed as superposition of two hyperbolic kink
solutions. In another publication, we further extended the notion of the
superposition principle for the coupled nonlinear equations. In particular, we
showed that the coupled $\phi^{4}$, coupled NLS and coupled mKdV equations
admit novel solutions which can be re-expressed as the superposition of two
(hyperbolic) kink and two (hyperbolic) pulse solutions [12].

At this stage we recall that during the last few years the nonlocal, nonlinear
equations have received wide attention. The next obvious question is whether
this notion can also be extended to nonlocal nonlinear equations. As a first
step in that direction we recently showed [13] that two different nonlocal
versions of the NLS [14, 15], nonlocal mKdV [17] as well as the nonlocal
Hirota equation [18, 19] admit novel solutions which can be re-expressed as
the superposition of two kink or two pulse solutions. It is then natural to
enquire if the notion of superposition can also be extended to the coupled
nonlocal equations. The purpose of this paper is to partially answer this
question. In particular, we show that the Ablowitz-Musslimani variant [14, 16]
of the coupled nonlocal NLS equations [2] as well as the coupled nonlocal mKdV
equations admit novel solutions which can be re-expressed as the superposition
of two hyperbolic as well as two periodic kink and pulse solutions.

The plan of the paper is the following. In Sec. II, we consider a coupled
Ablowitz-Musslimani variant of the nonlocal NLS and show that it admits a
large number of novel solutions which can be re-expressed as the superposition
of two kink or two pulse solutions. In particular, while in Sec. IIA we
discuss the superposed hyperbolic kink or pulse solutions, in Sec. IIB we
discuss superposed periodic kink or the pulse solutions. In Sec. III we
discuss a coupled nonlocal mKdV model and show that this model also admits
novel solutions which can be re-expressed as superposition of either two kink
or two pulse solutions. Finally, in Sec. IV we summarize our main results and
point out some of the open problems.

In Appendix A we discuss those solutions of the coupled Ablowitz-Musslimani
variant of the nonlocal NLS equation where the two fields are proportional to
each other. In Appendix B we discuss those solutions of the coupled nonlocal
mKdV model where the two fields are proportional to each other.

##  2 A Coupled Ablowitz-Musslimani Variant of Nonlocal NLS Model

Let us consider the following coupled Ablowitz-Musslimani variant of the
nonlocal NLS which we had introduced in [2]

$iu_{x,t}+u_{xx}(x,t)+[g_{11}u(x,t)u^{*}(-x,t)+g_{12}v(x,t)v^{*}(-x,t)]u(x,t)=0\,,$
(1)
$iv_{x,t}+v_{xx}(x,t)+[g_{21}u(x,t)u^{*}(-x,t)+g_{22}v(x,t)v^{*}(-x,t)]v(x,t)=0\,.$
(2)

Note that for comparison with the standard version of the Manakov system [20],
we have changed the notation used in [2]. In particular, we have replaced $z$
with $t$ and also instead of the couplings $a,b,f,e$ as used in [2] we have
used the couplings $g_{11},g_{12},g_{21}$ and $g_{22}$, respectively. Notice
that the limit $g_{11}=g_{12}=g_{21}=g_{22}$ in the local coupled NLS case is
the Manakov system while $g_{11}=g_{21}=-g_{12}=-g_{22}$ in the local case is
the integrable MZS system [21, 22, 23].

In [2] we had obtained 34 solutions of the coupled equations, most of which
were periodic in terms of the Jacobi elliptic functions and a few in terms of
hyperbolic solutions. However, it turns out that we had missed several
interesting solutions of the coupled equations. The purpose of this paper is
to discuss these solutions and to show that they can be re-expressed as the
superposition of a kink and an antikink or two kink or two pulse solutions.
For this purpose we will make use of six identities for the Jacobi elliptic
functions which can be easily derived by using the well known addition
theorems for the Jacobi elliptic functions ${\rm sn}(x,m),{\rm cn}(x,m)$ and
${\rm dn}(x,m)$ [3]

${\rm sn}(a+b,m)=\frac{{\rm sn}(a,m){\rm cn}(b,m){\rm dn}(b,m)+{\rm
sn}(b,m){\rm cn}(a,m){\rm dn}(a,m)}{1-m{\rm sn}^{2}(a,m){\rm sn}^{2}(b,m)}\,,$
(3) ${\rm cn}(a+b,m)=\frac{{\rm cn}(a,m){\rm cn}(b,m)-{\rm sn}(a,m){\rm
dn}(a,m){\rm sn}(b,m){\rm dn}(b,m)}{1-m{\rm sn}^{2}(a,m){\rm sn}^{2}(b,m)}\,,$
(4) ${\rm dn}(a+b,m)=\frac{{\rm dn}(a,m){\rm dn}(b,m)-m{\rm sn}(a,m){\rm
cn}(a,m){\rm sn}(b,m){\rm cn}(b,m)}{1-m{\rm sn}^{2}(a,m){\rm sn}^{2}(b,m)}\,.$
(5)

In particular, on using Eq. (3) we obtain the identities for the sum of two
periodic kink solutions, i.e.

${\rm sn}(y+\Delta,m)+{\rm sn}(y-\Delta,m)=\frac{2{\rm sn}(y,m){\rm
cn}(\Delta,m)}{{\rm dn}(\Delta,m)[1+B{\rm cn}^{2}(y,m)]}\,,~{}~{}B=\frac{m{\rm
sn}^{2}(\Delta,m)}{{\rm dn}^{2}(\Delta,m)}\,,$ (6)

or the sum of a periodic kink and an antikink solution, i.e.

${\rm sn}(y+\Delta,m)-{\rm sn}(y-\Delta,m)=\frac{2{\rm cn}(y,m){\rm
dn}(y,m){\rm sn}(\Delta,m)}{{\rm dn}^{2}(\Delta,m)[1+B{\rm
cn}^{2}(y,m)]}\,.~{}~{}B=\frac{m{\rm sn}^{2}(\Delta,m)}{{\rm
dn}^{2}(\Delta,m)}\,.$ (7)

On the other hand, by using Eq. (4) we obtain the identities for the sum of
the two periodic pulse solutions, i.e.

${\rm cn}(y+\Delta,m)+{\rm cn}(y-\Delta,m)=\frac{2{\rm cn}(y,m){\rm
cn}(\Delta,m)}{{\rm dn}^{2}(\Delta,m)[1+B{\rm
cn}^{2}(y,m)]}\,,~{}~{}B=\frac{m{\rm sn}^{2}(\Delta,m)}{{\rm
dn}^{2}(\Delta,m)}\,,$ (8) ${\rm cn}(y-\Delta,m)-{\rm
cn}(y+\Delta,m)=\frac{2{\rm sn}(y,m){\rm dn}(y,m){\rm sn}(\Delta,m)}{{\rm
dn}(\Delta,m)[1+B{\rm cn}^{2}(y,m)]}\,.~{}~{}B=\frac{m{\rm
sn}^{2}(\Delta,m)}{{\rm dn}^{2}(\Delta,m)}\,.$ (9)

Finally, on using the addition theorem (5) we obtain the identities for the
sum of the two periodic pulse solutions, i.e.

${\rm dn}(y+\Delta,m)+{\rm dn}(y-\Delta,m)=\frac{2{\rm dn}(y,m)}{{\rm
dn}(\Delta,m)[1+B{\rm cn}^{2}(y,m)]}\,,~{}~{}B=\frac{m{\rm
sn}^{2}(\Delta,m)}{{\rm dn}^{2}(\Delta,m)}\,,$ (10) $\displaystyle{\rm
dn}(y-\Delta,m)-{\rm dn}(y+\Delta,m)=\frac{2m{\rm sn}(y,m){\rm cn}(y,m){\rm
sn}(\Delta,m){\rm cn}(\Delta,m)}{{\rm dn}^{2}(\Delta,m)[1+B{\rm
cn}^{2}(y,m)]}\,,$ $\displaystyle B=\frac{m{\rm sn}^{2}(\Delta,m)}{{\rm
dn}^{2}(\Delta,m)}\,.$ (11)

In the limit $m=1$, from the above identities we can deduce the corresponding
four hyperbolic identities. In particular, in the limit $m=1$, from Eq. (6) we
obtain an identity for the sum of two (hyperbolic) kink solutions, i.e.

$\tanh(y+\Delta)+\tanh(y-\Delta)=\frac{2\sinh(y)\cosh(y)}{[B+\cosh^{2}(y)]}\,,~{}~{}B=\sinh^{2}(\Delta)\,,$
(12)

while from Eq. (7) in the limit $m=1$ we obtain an identity for the sum of a
(hyperbolic) kink and an antikink solution, i.e.

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

On the other hand, from Eqs. (8) and (9) (or (10) and (2)), in the $m=1$
limit, we obtain the identities for the sum of two (hyperbolic) pulse
solutions

${\rm sech}(y+\Delta)+{\rm
sech}(y-\Delta)=\frac{2\cosh(y)\cosh(\Delta)}{[B+\cosh^{2}(y)]}\,,~{}~{}B=\sinh^{2}(\Delta)\,,$
(14) ${\rm sech}(y-\Delta)-{\rm
sech}(y+\Delta)=\frac{2\sinh(y)\sinh(\Delta)}{[B+\cosh^{2}(y)]}\,,~{}~{}B=\sinh^{2}(\Delta)\,.$
(15)

In the next two subsections, we obtain 7 hyperbolic and 19 periodic solutions,
respectively, of the coupled Eqs. (1) and (2) which can be re-expressed as the
superposition of either a kink and an antikink or two kink or two pulse
solutions.

###  2.1 Superposed Hyperbolic kink and Pulse Solutions

There are two different kind of solutions to the coupled Eqs. (1) and (2)
depending on whether $v(x,t)\propto u(x,t)$ or not. In this section we only
discuss those solutions where the two fields $v(x,t)$ and $u(x,t)$ are
distinct (and not proportional to each other) while in Appendix A we discuss
those solutions where $v(x,t)\propto u(x,t)$. As we show there, when the two
fields $u$ and $v$ are proportional to each other, effectively we only need to
solve the uncoupled Ablowitz-Musslimani nonlocal NLS equation.

We start from Eqs. (1) and (2) and take the ansatz

$u(x,t)=e^{i\omega_{1}(t+t_{0})}u(x)\,,~{}~{}v(x,t)=e^{i\omega_{2}(t+t_{0})}v(x)\,.$
(16)

On substituting this ansatz in Eqs. (1) and (2) we obtain

$u_{xx}(x)=\omega_{1}u(x)-[g_{11}u(x)u^{*}(-x)+g_{12}v(x)v^{*}(-x)]u(x)\,,$
(17)
$v_{xx}(x)=\omega_{2}v(x)-[g_{21}u(x)u^{*}(-x)+g_{22}v(x)v^{*}(-x)]v(x)\,.$
(18)

We now show that the coupled Eqs. (17) and (18) and hence Eqs. (1) and (2)
admit 7 novel hyperbolic solutions which can be re-expressed as a
superposition of either two kink or two pulse solutions.

One major difference between the local and the nonlocal case is that, unlike
the local case, the solutions of the nonlocal NLS Eqs. (1) and (2) are not
invariant with respect to the shift in $x$. However, a shift in $t$ is
allowed. For simplicity, from now onward we will take $t_{0}$ to be zero even
though such a shift is always allowed.

Superposed Solution I

It is easy to check that the coupled Eqs. (17) and (18) (and hence Eqs. (1)
and (2)) admit the hyperbolic solution

$u(x,t)=e^{i\omega_{1}t}\frac{A}{B+\cosh^{2}(\beta
x)}\,,~{}~{}v(x,t)=e^{i\omega_{2}t}\frac{D\cosh(\beta x)\sinh(\beta
x)}{B+\cosh^{2}(\beta x)}\,,~{}~{}B>0\,,$ (19)

provided

$\displaystyle\omega_{1}=-2\beta^{2}\,,~{}~{}g_{11}A^{2}=-2B(2B+1)\beta^{2}\,,~{}~{}g_{12}D^{2}=6\beta^{2}\,,$
$\displaystyle\omega_{2}=-2\beta^{2}\,,~{}~{}g_{21}A^{2}=-6B(B+1)\beta^{2}\,,~{}~{}g_{22}D^{2}=2\beta^{2}\,.$
(20)

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

On comparing the solution (19) with the identities (12) and (13), the solution
(19) can be re-expressed as

$\displaystyle
u(x,t)=e^{i\omega_{1}t}\frac{\beta}{\sqrt{2|g_{11}|}}[\tanh(\beta
x+\Delta)-\tanh(\beta x-\Delta)]\,,$ $\displaystyle
v(x,t)=e^{i\omega_{2}t}\frac{\sqrt{3}\beta}{\sqrt{2g_{12}}}[\tanh(\beta
x+\Delta)+\tanh(\beta x-\Delta)]\,,$ (21)

where $\sinh^{2}(\Delta)=B$. Note that here while $u(x,t)$ corresponds to a
superposition of a kink and an antikink solution, $v(x,t)$ is a superposition
of two kink solutions.

Superposed Solution II

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

$u(x,t)=e^{i\omega_{1}t}\frac{A}{B+\cosh^{2}(\beta
x)}\,,~{}~{}v(x,t)=e^{i\omega_{2}t}\frac{D\sinh(\beta x)}{B+\cosh^{2}(\beta
x)}\,,~{}~{}B>0\,,$ (22)

provided

$\displaystyle\omega_{1}=4\beta^{2}\,,~{}~{}g_{11}A^{2}=2(B+1)(2B+3)\beta^{2}\,,~{}~{}g_{12}D^{2}=-6(2B+1)\beta^{2}\,,$
$\displaystyle\omega_{2}=\beta^{2}\,,~{}~{}g_{21}A^{2}=6(B+1)\beta^{2}\,,~{}~{}g_{22}D^{2}=-2(3+4B)\beta^{2}\,.$
(23)

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

On comparing the solution (22) with the identities (13) and (15), the solution
(22) can be re-expressed as

$\displaystyle
u(x,t)=e^{i\omega_{1}t}\frac{\sqrt{3}\beta}{\sqrt{2g_{21}}\sinh(\Delta)}[\tanh(\beta
x+\Delta)-\tanh(\beta x-\Delta)]\,,$ $\displaystyle
v(x,t)=e^{i\omega_{2}t}\frac{\sqrt{3\cosh(2\Delta)}\beta}{\sqrt{2|g_{12}|}\sinh(\Delta)}[{\rm
sech}(\beta x-\Delta)-{\rm sech}(\beta x+\Delta)]\,,$ (24)

where $\sinh^{2}(\Delta)=B$. Note that here while $u(x,t)$ corresponds to a
superposition of a kink and an antikink solution, $v(x,t)$ is a superposition
of two pulse solutions.

Superposed Solution III

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

$u(x,t)=e^{i\omega_{1}t}\frac{A}{B+\cosh^{2}(\beta
x)}\,,~{}~{}v(x,t)=e^{i\omega_{2}t}\frac{D\cosh(\beta x)}{B+\cosh^{2}(\beta
x)}\,,~{}~{}B>0\,,$ (25)

provided

$\displaystyle\omega_{1}=4\beta^{2}\,,~{}~{}g_{12}D^{2}=6(1+2B)\beta^{2}\,,~{}~{}g_{11}A^{2}=2B(2B-1)\beta^{2}\,,$
$\displaystyle\omega_{2}=\beta^{2}\,,~{}~{}g_{21}A^{2}=-6B\beta^{2}\,,~{}~{}g_{22}D^{2}=2(1+4B)\beta^{2}\,.$
(26)

Thus for this solution while $g_{12}$, $g_{22}$, $\omega_{1}$, $\omega_{2}>0$,
$g_{21}<0$, $g_{11}\geq 0$ if either $B\geq 1/2$ or $B\leq 0$, and $g_{11}<0$
if $0<B<1/2$.

On comparing the solution (25) with the identities (13) and (14), the solution
(25) can be re-expressed as

$\displaystyle
u(x,t)=e^{i\omega_{1}t}\frac{\sqrt{3}\beta}{\sqrt{2|g_{21}|}\cosh(\Delta)}[\tanh(\beta
x+\Delta)-\tanh(\beta x-\Delta)]\,,$ $\displaystyle
v(x,t)=e^{i\omega_{2}t}\frac{\sqrt{3}\tanh(\Delta)\beta}{\sqrt{2|g_{21}|}}[{\rm
sech}(\beta x+\Delta)+{\rm sech}(\beta x-\Delta)]\,,$ (27)

where $\sinh^{2}(\Delta)=B$. Note that here while $u(x,t)$ corresponds to a
superposition of a kink and an antikink solution, $v(x,t)$ is a superposition
of two pulse solutions.

Superposed Solution IV

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

$u(x,t)=e^{i\omega_{1}t}\frac{A\sinh(\beta x)}{B+\cosh^{2}(\beta
x)}\,,~{}~{}v(x,t)=e^{i\omega_{2}t}\frac{D\sinh(\beta x)\cosh(\beta
x)}{B+\cosh^{2}(\beta x)}\,,~{}~{}B>0\,,$ (28)

provided

$\displaystyle(B+1)\omega_{1}=-(5-B)\beta^{2}\,,~{}~{}(1+B)g_{11}A^{2}=-2B(1+4B)\beta^{2}\,,$
$\displaystyle(1+B)g_{12}D^{2}=6\beta^{2}\,,~{}~{}(1+B)\omega_{2}=2(2B-1)\beta^{2}\,,$
$\displaystyle(B+1)g_{21}A^{2}=-6B(2B+1)\beta^{2}\,,~{}~{}(B+1)g_{22}D^{2}=-2(2B-1)\beta^{2}\,.~{}~{}~{}$
(29)

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

On making use of the identities (12) and (15), one can then re-express the
solution (28) as

$\displaystyle u(x,t)=e^{i\omega_{1}t}\frac{\beta}{\sqrt{2|g_{11}|}}[{\rm
sech}(\beta x-\Delta)-{\rm sech}(\beta x+\Delta)]\,,$ $\displaystyle
v(x,t)=e^{i\omega_{2}t}\frac{\sqrt{3}\beta}{\sqrt{2g_{12}}\cosh(\Delta)}[\tanh(\beta
x+\Delta)+\tanh(\beta x-\Delta)]\,,$ (30)

where $\sinh^{2}(\Delta)=B$. Note that here while $u(x,t)$ corresponds to a
superposition of two pulse solutions, $v(x,t)$ is a superposition of two kink
solutions.

Superposed Solution V

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

$u(x,t)=e^{i\omega_{1}t}\frac{A\cosh(\beta x)}{B+\cosh^{2}(\beta
x)}\,,~{}~{}v(x,t)=e^{i\omega_{2}t}\frac{D\sinh(\beta x)\cosh(\beta
x)}{B+\cosh^{2}(\beta x)}\,,~{}~{}B>0\,,$ (31)

provided

$\displaystyle
B\omega_{1}=(6+B)\beta^{2}\,,~{}~{}Bg_{11}A^{2}=2(B+1)(4B+3)\beta^{2}\,,$
$\displaystyle
Bg_{12}D^{2}=-6\beta^{2}\,,~{}~{}B\omega_{2}=2(2B+3)\beta^{2}\,,$
$\displaystyle
Bg_{21}A^{2}=6(B+1)(2B+1)\beta^{2}\,,~{}~{}Bg_{22}D^{2}=-2(3+2B)\beta^{2}\,.$
(32)

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

On making use of the identities (12) and (14), one can then re-express the
solution (2.2) as

$\displaystyle
u(x,t)=e^{i\omega_{1}t}\frac{\sqrt{3\cosh(2\Delta)}\beta}{\sqrt{2g_{21}}\sinh(\Delta)}[{\rm
sech}(\beta x+\Delta)+{\rm sech}(\beta x-\Delta)]\,,$ (33) $\displaystyle
v(x,t)=e^{i\omega_{2}t}\frac{\sqrt{3}\beta}{\sqrt{2|g_{12}|}\sinh(\Delta)}[\tanh(\beta
x+\Delta)+\tanh(\beta x-\Delta)]\,,$

where $\sinh^{2}(\Delta)=B$. Note that here while $u(x,t)$ corresponds to a
superposition of two pulse solutions, $v(x,t)$ is a superposition of two kink
solutions.

Superposed Solution VI

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

$u(x,t)=e^{i\omega_{1}t}\frac{A\sinh(\beta x)}{B+\cosh^{2}(\beta
x)}\,,~{}~{}v(x,t)=e^{i\omega_{2}t}\frac{D\cosh(\beta x)}{B+\cosh^{2}(\beta
x)}\,,~{}~{}B>0\,,$ (34)

provided

$\displaystyle\omega_{1}=\beta^{2}\,,~{}~{}g_{11}A^{2}=-2B\beta^{2}\,,~{}~{}g_{12}D^{2}=6(B+1)\beta^{2}\,,$
$\displaystyle\omega_{2}=\beta^{2}\,,~{}~{}g_{21}A^{2}=-6B\beta^{2}\,,~{}~{}g_{22}D^{2}=2(B+1)\beta^{2}\,.$
(35)

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

On making use of the identities (14) and (15), one can then re-express the
solution (34) as

$\displaystyle u(x,t)=e^{i\omega_{1}t}\frac{\beta}{\sqrt{2|g_{11}|}}[{\rm
sech}(\beta x-\Delta)-{\rm sech}(\beta x+\Delta)]\,,$ $\displaystyle
v(x,t)=e^{i\omega_{2}t}\frac{\beta}{\sqrt{2g_{22}}}[{\rm sech}(\beta
x+\Delta)+{\rm sech}(\beta x-\Delta)]\,,$ (36)

where $\sinh^{2}(\Delta)=B$. Note that here both $u(x,t),v(x,t)$ correspond to
the superposition of two pulse solutions.

Superposed Solution VII

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

$u(x,t)=e^{i\omega_{1}t}[1-\frac{A}{B+\cosh^{2}(\beta
x)}]\,,~{}~{}v(x,t)=e^{i\omega_{2}t}\frac{D}{B+\cosh^{2}(\beta x)}\,,$ (37)

where $A,B,D>0$ provided

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

Thus for this solution $g_{11}$, $g_{12}$, $\omega_{1}$, $g_{21}<0$, while
$\omega_{2}$, $g_{22}>0$.

On comparing the solution (37) with the identity (15), the solution (37) can
be re-expressed as

$\displaystyle
u(x,t)=e^{i\omega_{1}t}\left[1-\frac{K_{1}\beta}{\sinh(2\Delta)\sqrt{2|g_{11}|}}\right][\tanh(\beta
x+\Delta)-\tanh(\beta x-\Delta)]\,,$ $\displaystyle
v(x,t)=e^{i\omega_{2}t}\frac{\sqrt{3K_{1}K_{3}}\beta}{2\sinh(2\Delta)\sqrt{2|g_{12}|}}[\tanh(\beta
x+\Delta)-\tanh(\beta x-\Delta)]\,,$ (39)

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

$K_{1}=\left[3\cosh(2\Delta)+\sqrt{8+\cosh^{2}(2\Delta)}\right]\,,$ (40)
$K_{3}=\left[\sqrt{8+\cosh^{2}(2\Delta)-\cosh(2\Delta)}\right]\,.$ (41)

Note that here both $u(x,t),v(x,t)$ correspond to the superposition of a kink
and an antikink solution.

###  2.2 Superposed Periodic Kink and Pulse Solutions

We now show that in case $u$ and $v$ are distinct (i.e. not proportional to
each other) then by starting from the ansatz (16) and hence using the coupled
Eqs. (17) and (18), and thus the coupled Eqs. (1) and (2) admit not only the 7
superposed hyperbolic but also 15 superposed periodic kink (i.e. ${\rm
sn}(x,m)$) and periodic pulse (i.e. ${\rm dn}(x,m)$ as well as ${\rm
cn}(x,m)$) solutions. We now present these 15 solutions one by one.

Solution VIII

It is easy to check that

$u(x,t)=e^{i\omega_{1}t}\frac{A{\rm cn}(\beta x,m)}{1+B{\rm cn}^{2}(\beta
x,m)}\,,~{}~{}~{}v(x,t)=e^{i\omega_{2}t}\frac{D{\rm sn}(\beta x,m){\rm
dn}(\beta x,m)}{1+B{\rm cn}^{2}(\beta x,m)}\,,$ (42)

where $B>0$, is an exact solution of the coupled Eqs. (1) and (2) provided

$\displaystyle\omega_{1}=\omega_{2}=(2m-1)\beta^{2}\,,~{}~{}g_{12}D^{2}=3g_{22}D^{2}=-6B\beta^{2}\,,$
$\displaystyle g_{21}A^{2}=3g_{11}A^{2}=6(B+1)[m-(1-m)B]\beta^{2}\,.$ (43)

On using the identities (8) and (9), the coupled solution (42) can be re-
expressed as

$\displaystyle u(x,t)=e^{i\omega_{1}t}\sqrt{\frac{m}{2g_{21}}}\beta[{\rm
cn}(\beta x+\Delta,m)+{\rm cn}(\beta x-\Delta,m)]\,,$ $\displaystyle
v(x,t)=e^{i\omega_{2}t}\sqrt{\frac{m}{2|g_{12}|}}\beta[{\rm cn}(\beta
x-\Delta,m)-{\rm cn}(\beta x+\Delta,m)]\,,$ (44)

where $B={m{\rm sn}^{2}(\Delta,m)}/{{\rm dn}^{2}(\Delta,m)}$.

Solution IX

It is easy to check that

$u(x,t)=e^{i\omega_{1}t}\frac{A{\rm dn}(\beta x,m)}{1+B{\rm cn}^{2}(\beta
x,m)}\,,~{}~{}v(x,t)=e^{i\omega_{2}t}\frac{D{\rm sn}(\beta x,m){\rm cn}(\beta
x,m)}{1+B{\rm cn}^{2}(\beta x,m)}\,,$ (45)

where $B>0$, is an exact solution of the coupled Eqs. (1) and (2) provided

$\displaystyle\omega_{1}=\omega_{2}=(2-m)\beta^{2}\,,~{}~{}g_{21}D^{2}=3g_{22}D^{2}=-6B[m-(1-m)B]\beta^{2}\,,$
$\displaystyle g_{21}A^{2}=3g_{11}A^{2}=6(1+B)\beta^{2}\,.$ (46)

On using the identities (10) and (2), the coupled solution (45) can be re-
expressed as

$\displaystyle u(x,t)=e^{i\omega_{1}t}\frac{\beta}{\sqrt{2g_{11}}}[{\rm
dn}(\beta x+\Delta,m)+{\rm dn}(\beta x-\Delta,m)]\,,$ $\displaystyle
v(x,t)=e^{i\omega_{2}t}\frac{\beta}{\sqrt{2|g_{22}|}}[{\rm dn}(\beta
x-\Delta,m)-{\rm dn}(\beta x+\Delta,m)]\,,$ (47)

where $B={m{\rm sn}^{2}(\Delta,m)}/{{\rm dn}^{2}(\Delta,m)}$.

Solution X

It is straightforward to check that

$u(x,t)=e^{i\omega_{1}t}\frac{A{\rm cn}(\beta x,m)}{1+B{\rm cn}^{2}(\beta
x,m)}\,,~{}~{}v(x,t)=e^{i\omega_{2}t}\frac{D{\rm cn}(\beta x,m){\rm sn}(\beta
x,m)}{1+B{\rm cn}^{2}(\beta x,m)}\,,$ (48)

where $B>0$, is an exact solution to the coupled Eqs. (1) and (2) provided

$\displaystyle\omega_{1}=[(2m-1)-6(1-m)B]\beta^{2}\,,~{}~{}g_{12}D^{2}=-6[m+(1-m)B^{2}]B\beta^{2}\,,$
$\displaystyle g_{11}A^{2}=-2[3(1-m)B^{3}+7(1-m)B^{2}+B(4-5m)-m]\beta^{2}\,,$
$\displaystyle\omega_{2}=[(5m-4)-6(1-m)B]\beta^{2}\,,$ $\displaystyle
g_{22}D^{2}=-2[3(1-m)B^{2}+2(1-m)B+m]B\beta^{2}\,,$ $\displaystyle
b_{21}A^{2}=-6[(1-m)B^{3}+3(1-m)B^{2}+(2-3m)B-m]\beta^{2}\,.$ (49)

On using the identities (8) and (2), the coupled solution (48) can be re-
expressed as

$u(x,t)=e^{i\omega_{1}t}\frac{A{\rm dn}^{2}(\Delta,m)}{2{\rm
cn}(\Delta,m)}[{\rm cn}(\beta x+\Delta,m)+{\rm cn}(\beta x-\Delta,m)]\,,$ (50)
$v(x,t)=e^{i\omega_{2}t}\frac{D{\rm dn}^{2}(\Delta)}{2m{\rm sn}(\Delta){\rm
cn}(\Delta)}[{\rm dn}(\beta x-\Delta,m)-{\rm dn}(\beta x+\Delta,m)]\,,$ (51)

where $A,D$ are as given by Eq. (2.2) while $B={m{\rm sn}^{2}(\Delta,m)}/{{\rm
dn}^{2}(\Delta,m)}$.

Solution XI

It is easy to check that

$u(x,t)=e^{i\omega_{1}t}\frac{A{\rm dn}(\beta x,m)}{1+B{\rm cn}^{2}(\beta
x,m)}\,,~{}~{}v(x,t)=e^{i\omega_{2}t}\frac{D{\rm sn}(\beta x,m){\rm dn}(\beta
x,m)}{1+B{\rm cn}^{2}(\beta x,m)}\,,$ (52)

where $B>0$, is an exact solution of the coupled Eqs. (1) and (2) provided

$\displaystyle[m-(1-m)B]\omega_{1}=[(1-m)(4+m)B+m(2-m)]\beta^{2}\,,$
$\displaystyle[m-(1-m)B]g_{11}A^{2}=2(B+1)[3(1-m)B^{2}+2(1-m)B+m]\,,$
$\displaystyle m[m-(1-m)B]g_{12}D^{2}=-6B[m-2(1-m)B-(1-m)B^{2}]\beta^{2}\,,$
$\displaystyle[m-(1-m)B]\omega_{2}=[(1-m)(4m+1)B+m(5-4m)]\beta^{2}\,,$
$\displaystyle
m[m-(1-m)B]g_{22}D^{2}=-B[2m^{2}-5m(1-m)B-3(1-m^{2})B^{2}]\beta^{2}\,,$
$\displaystyle[m-(1-m)B]g_{21}A^{2}=[6(1-m)B^{3}+6(1-m)B^{2}+(m^{2}+4m+1)B+6m]\beta^{2}\,.$
(53)

On using the identities (9) and (10), the coupled solution (52) can be re-
expressed as

$u(x,t)=e^{i\omega_{1}t}\frac{A{\rm dn}(\Delta,m)}{2}[{\rm dn}(\beta
x+\Delta,m)+{\rm dn}(\beta x-\Delta,m)]\,,$ (54)
$v(x,t)=e^{i\omega_{2}t}\frac{D{\rm dn}(\Delta,m)}{2{\rm sn}(\Delta,m)}[{\rm
cn}(\beta x-\Delta,m)-{\rm cn}(\beta x+\Delta,m)]\,,$ (55)

where $A,D$ are given by Eq. (2.2) while $B={m{\rm sn}^{2}(\Delta,m)}/{{\rm
dn}^{2}(\Delta,m)}$.

Hyperbolic Limit

In the limit $m=1$, all the four solutions VIII to XI go over to the
hyperbolic solution VI, i.e. in this limit all four solutions can be re-
expressed as superposition of two pulse solutions, i.e. ${\rm sech}(\beta
x-\Delta)\pm{\rm sech}(\beta x+\Delta)$.

Solution XII

It is straightforward to check that

$u(x,t)=e^{i\omega_{1}t}\frac{A{\rm cn}(\beta x,m)}{1+B{\rm cn}^{2}(\beta
x,m)}\,,~{}~{}v(x,t)=e^{i\omega_{2}t}\frac{D{\rm sn}(\beta x,m)}{1+B{\rm
cn}^{2}(\beta x,m)}\,,$ (56)

where $B>0$, is an exact solution of the coupled Eqs. (1) and (2) provided

$\displaystyle
B\omega_{1}=[(2m-1)B+6m]\beta^{2}\,,~{}~{}Bg_{12}D^{2}=-6[m+(1-m)B]\beta^{2}\,,$
$\displaystyle
Bg_{11}A^{2}=2[3m+7mB+(5m-1)B^{2}-(1-m)B^{3}]B\omega_{2}=[(5m-1)B+6m]\beta^{2}\,,$
$\displaystyle
Bg_{22}D^{2}=-2[3m+2mB+(1-m)B^{2}]\beta^{2}\,,~{}~{}Bg_{21}A^{2}=6(B+1)[m+2mB-(1-m)B^{2}]\,.$
(57)

On using the identities (6) and (8), the coupled solution (56) can be re-
expressed as

$u(x,t)=e^{i\omega_{1}t}\frac{A{\rm dn}^{2}(\Delta,m)}{2{\rm
cn}(\Delta,m)}[{\rm cn}(\beta x+\Delta,m)+{\rm cn}(\beta x-\Delta,m)]\,,$ (58)
$v(x,t)=e^{i\omega_{2}t}\frac{D{\rm dn}(\Delta,m)}{2{\rm cn}(\Delta,m)}[{\rm
sn}(\beta x+\Delta,m)+{\rm sn}(\beta x-\Delta,m)]\,,$ (59)

where $A,D$ are given by Eq. (2.2) while $B={m{\rm sn}^{2}(\Delta,m)}/{{\rm
dn}^{2}(\Delta,m)}$.

Solution XIII

It is easy to check that

$u(x,t)=e^{i\omega_{1}t}\frac{A{\rm dn}(\beta x,m)}{1+B{\rm cn}^{2}(\beta
x,m)}\,,~{}~{}~{}v(x,t)=e^{i\omega_{2}t}\frac{D{\rm sn}(\beta x,m)}{1+B{\rm
cn}^{2}(\beta x,m)}\,,$ (60)

where $B>0$, is an exact solution of the coupled Eqs. (1) and (2) provided

$\displaystyle
B\omega_{1}=[(5m-4)B+6]\beta^{2}\,,~{}~{}B\omega_{2}=[((5m-1)B+6m]\beta^{2}\,,$
$\displaystyle
Bg_{12}D^{2}=-6[(1-m)^{2}B^{3}+(1-m)(2-3m)B^{2}-3m(1-m)B+m^{2}]\beta^{2}\,,$
$\displaystyle
Bg_{11}A^{2}=2(B+1)[3m+2(3m-1)B-3(1-m)B^{2}]\beta^{2}\,,\beta^{2}\,,$
$\displaystyle Bg_{21}A^{2}=6(B+1)[m+2mB-(1-m)B^{2}]\beta^{2}\,,$
$\displaystyle
Bg_{22}D^{2}=-2[3m^{2}+m(9m-7)B+(4-9m)(1-m)B^{2}+3(1-m)^{2}]\,.$ (61)

On using the identities (6) and (10), the coupled solution (60) can be re-
expressed as

$u(x,t)=e^{i\omega_{1}t}\frac{A{\rm dn}(\Delta,m)}{2}[{\rm dn}(\beta
x+\Delta,m)+{\rm dn}(\beta x-\Delta,m)]\,,$ (62)
$v(x,t)=e^{i\omega_{2}t}\frac{D{\rm dn}(\Delta,m)}{2{\rm cn}(\Delta,m)}[{\rm
sn}(\beta x+\Delta,m)+{\rm sn}(\beta x-\Delta,m)]\,,$ (63)

where $A,D$ are given by Eq. (2.2) while $B={m{\rm sn}^{2}(\Delta,m)}/{{\rm
dn}^{2}(\Delta,m)}$.

Hyperbolic Limit

In the limit $m=1$, the two solutions XII and XIII go over to the hyperbolic
solution V, i.e. in this limit both the solutions can be re-expressed as
superposition of pulse (i.e. ${\rm sech}(\beta x+\Delta)+{\rm sech}(\beta
x-\Delta)$) and kink (i.e. $\tanh(\beta x+\Delta)+\tanh(\beta x-\Delta)$)
solutions, respectively.

Solution XIV

It is straightforward to check that

$u(x,t)=e^{i\omega_{1}t}\frac{A{\rm cn}(\beta x,m)}{1+B{\rm cn}^{2}(\beta
x,m)}\,,~{}~{}v(x,t)=e^{i\omega_{2}t}\frac{D{\rm cn}(\beta x,m){\rm dn}(\beta
x,m)}{1+B{\rm cn}^{2}(\beta x,m)}\,,$ (64)

where $B>0$, is an exact solution of the coupled Eqs. (1) and (2) provided

$\displaystyle\omega_{1}=[(2m-1)-6(1-m)B]\beta^{2}\,,~{}~{}mg_{12}D^{2}=-6B[m-(1-m)B^{2}]\beta^{2}\,,$
$\displaystyle mg_{11}A^{2}=2[m^{2}+m(5m-1)B-7m(1-m)B^{2}-3(1-m)^{2}B^{3}]\,,$
$\displaystyle\omega_{2}=[((5m-1)-6(1-m)B]\beta^{2}\,,~{}~{}mg_{22}D^{2}=6[m+2mB+3(1-m)B^{2}]\beta^{2}\,,$
$\displaystyle
mg_{21}A^{2}=2B[-m^{2}+(3m-1)mB-3m(1-m)B^{2}+(1-m)^{2}B^{3}]\beta^{2}\,.$ (65)

On using the identities (7) and (8), the coupled solution (64) can be re-
expressed as

$u(x,t)=e^{i\omega_{1}t}\frac{A{\rm dn}^{2}(\Delta,m)}{2{\rm
cn}(\Delta,m)}[{\rm cn}(\beta x+\Delta,m)+{\rm cn}(\beta x-\Delta,m)]\,,$ (66)
$v(x,t)=e^{i\omega_{2}t}\frac{D{\rm dn}^{2}(\Delta,m)}{2{\rm
sn}(\Delta,m)}[{\rm sn}(\beta x+\Delta,m)-{\rm sn}(\beta x-\Delta,m)]\,,$ (67)

where $A,D$ are given by Eq. (2.2) while $B={m{\rm sn}^{2}(\Delta,m)}/{{\rm
dn}^{2}(\Delta,m)}$.

Solution XV

It is easy to check that

$u(x,t)=e^{i\omega_{1}t}\frac{A{\rm dn}(\beta x,m)}{1+B{\rm cn}^{2}(\beta
x,m)}\,,~{}~{}v(x,t)=e^{i\omega_{2}t}\frac{D{\rm cn}(\beta x,m){\rm dn}(\beta
x,m)}{1+B{\rm cn}^{2}(\beta x,m)}\,,$ (68)

where $B>0$, is an exact solution of the coupled Eqs. (1) and (2) provided

$\displaystyle[m-(1-m)B]\omega_{1}=[m(2-m)+(1-m)(4+m)B]\beta^{2}\,,$
$\displaystyle[m-(1-m)B]g_{11}A^{2}=2[m+2(1+m)B-(1-m)B^{2}]\beta^{2}\,,$
$\displaystyle[m-(1-m)B]g_{12}D^{2}=-B[6m^{2}-12m(1-m)B+4(1-m)(2-m)B^{2}]\beta^{2}\,,$
$\displaystyle[m-(1-m)B]\omega_{2}=[(5-m)m+(1-m^{2})B]\beta^{2}\,,$
$\displaystyle[m-(1-m)B]g_{22}A^{2}=6[m+2mB-(1-m)B^{2}]\beta^{2}\,,$
$\displaystyle[m-(1-m)B]g_{21}D^{2}=-2B[m-2(2-m)B-2(1-m)B^{2}]\beta^{2}\,.$
(69)

On using the identities (7) and (10), the coupled solution (68) can be re-
expressed as

$u(x,t)=e^{i\omega_{1}t}\frac{A{\rm dn}(\Delta,m)}{2}[{\rm dn}(\beta
x+\Delta,m)+{\rm dn}(\beta x-\Delta,m)]\,,$ (70)
$v(x,t)=e^{i\omega_{2}t}\frac{D{\rm dn}^{2}(\Delta,m)}{2{\rm
sn}(\Delta,m)}[{\rm sn}(\beta x+\Delta,m)-{\rm sn}(\beta x-\Delta,m)]\,,$ (71)

where $A,D$ are given by Eq. (2.2) while $B={m{\rm sn}^{2}(\Delta,m)}/{{\rm
dn}^{2}(\Delta,m)}$.

Hyperbolic Limit

In the limit $m=1$, the two solutions XIV and XV go over to the hyperbolic
solution III, i.e. in this limit both the solutions can be re-expressed as
superposition of one pulse (i.e. ${\rm sech}(\beta x+\Delta)+{\rm sech}(\beta
x-\Delta)$) and one kink (i.e. $\tanh(\beta x+\Delta)-\tanh(\beta x-\Delta)$)
solution.

Solution XVI

It is easy to check that

$u(x,t)=e^{i\omega_{1}t}\frac{A{\rm cn}(\beta x,m){\rm sn}(\beta x,m)}{1+B{\rm
cn}^{2}(\beta x,m)}\,,~{}~{}v(x,t)=e^{i\omega_{2}t}\frac{D{\rm cn}(\beta
x,m){\rm dn}(\beta x,m)}{1+B{\rm cn}^{2}(\beta x,m)}\,,$ (72)

where $B>0$, is an exact solution of the coupled Eqs. (1) and (2) provided

$\displaystyle\omega_{1}=[(5m-4)-6(1-m)B]\beta^{2}\,,$ $\displaystyle
g_{12}D^{2}=6(B+1)[m-2(1-m)B+(1-m)B^{2}]\beta^{2}\,,$ $\displaystyle
g_{11}A^{2}=-2[3m^{2}+(9m-5)mB-(1-m)(9m-2)B^{2}+3(1-m)^{2}B^{3}]\beta^{2}\,,$
$\displaystyle\omega_{2}=[((5m-1)-6(1-m)B]\beta^{2}\,,~{}~{}g_{22}D^{2}=2m(B+1)[3+2(3m-2)B-3(1-m)^{2}]\beta^{2}\,,$
$\displaystyle
g_{21}A^{2}=-6[m^{2}+m(3m-1)B-3m(1-m)B^{2}+(1-m)^{3}B^{3}]\beta^{2}\,.$ (73)

On using the identities (7) and (2), the coupled solution (72) can be re-
expressed as

$u(x,t)=e^{i\omega_{1}t}\frac{A{\rm dn}^{2}(\Delta,m)}{2m{\rm
cn}(\Delta,m){\rm sn}(\Delta,m)}[{\rm dn}(\beta x+\Delta,m)-{\rm dn}(\beta
x-\Delta,m)]\,,$ (74) $v(x,t)=e^{i\omega_{2}t}\frac{D{\rm
dn}^{2}(\Delta,m)}{2{\rm sn}(\Delta,m)}[{\rm sn}(\beta x+\Delta,m)-{\rm
sn}(\beta x-\Delta,m)]\,,$ (75)

where $A,D$ are given by Eq. (2.2) while $B={m{\rm sn}^{2}(\Delta,m)}/{{\rm
dn}^{2}(\Delta,m)}$.

Solution XVII

It is easy to check that

$u(x,t)=e^{i\omega_{1}t}\frac{A{\rm cn}(\beta x,m){\rm dn}(\beta x,m)}{1+B{\rm
cn}^{2}(\beta x,m)}\,,~{}~{}v(x,t)=e^{i\omega_{2}t}\frac{D{\rm sn}(\beta
x,m){\rm dn}(\beta x,m)}{1+B{\rm cn}^{2}(\beta x,m)}\,,$ (76)

where $B>0$, is an exact solution of the coupled Eqs. (1) and (2) provided

$\displaystyle[m-(1-m)B]\omega_{1}=[(5-m)m+(1-m^{2})B]\beta^{2}\,,$
$\displaystyle[m-(1-m)B]\omega_{2}=[(1-m)(1+4m)B+m(5-4m)]\beta^{2}\,,$
$\displaystyle[m-(1-m)B]g_{12}D^{2}=-6[m+2mB-(1-m)B^{2}]\beta^{2}\,,~{}~{}0<m<1\,,$
$\displaystyle[m-(1-m)B]g_{11}A^{2}=[6m+2(1+4m)B+2(1+m)mB^{2}+(1-m)B^{3}]\beta^{2}\,,$
$\displaystyle[m-(1-m)B]g_{22}D^{2}=-2[3m+4mB-(1-m)B^{2}]\beta^{2}\,,~{}~{}[m-(1-m)B]g_{21}A^{2}$
$\displaystyle=[6m+6mB-2(1-m)(8m-1)B^{2}-4(1-m)(2m-1)B^{3}]\beta^{2}\,.$ (77)

On using the identities (7) and (9), the coupled solution (76) can be re-
expressed as

$u(x,t)=e^{i\omega_{1}t}\frac{A{\rm dn}^{2}(\Delta,m)}{2{\rm
sn}(\Delta,m)}[{\rm sn}(\beta x+\Delta,m)-{\rm sn}(\beta x-\Delta,m)]\,,$ (78)
$v(x,t)=e^{i\omega_{2}t}\frac{D{\rm dn}(\Delta,m)}{2{\rm sn}(\Delta,m)}[{\rm
cn}(\beta x-\Delta,m)-{\rm cn}(\beta x+\Delta,m)]\,,$ (79)

where $A,D$ are given by Eq. (2.2) while $B={m{\rm sn}^{2}(\Delta,m)}/{{\rm
dn}^{2}(\Delta,m)}$.

Hyperbolic Limit

In the limit $m=1$, the two solutions XVI and XVII go over to the hyperbolic
solution II, i.e. in this limit both the solutions can be re-expressed as
superposition of one pulse (i.e. ${\rm sech}(\beta x-\Delta)-{\rm sech}(\beta
x+\Delta)$) and one kink (i.e. $\tanh(\beta x+\Delta)-\tanh(\beta x-\Delta)$)
solution.

Solution XVIII

It is straightforward to check that

$u(x,t)=e^{i\omega_{1}t}\frac{A{\rm sn}(\beta x,m)}{1+B{\rm cn}^{2}(\beta
x,m)}\,,~{}~{}v(x,t)=e^{i\omega_{2}t}\frac{D{\rm sn}(\beta x,m){\rm dn}(\beta
x,m)}{1+B{\rm cn}^{2}(\beta x,m)}\,,$ (80)

where $B>0$, is an exact solution of the coupled Eqs. (1) and (2) provided

$\displaystyle(1+B)\omega_{1}=[(5-m)B-(1+m)]\beta^{2}\,,~{}~{}m(1+B)g_{12}D^{2}=-6B[m+2mB-(1-m)B^{2}]\beta^{2}\,,$
$\displaystyle
m(1+B)g_{11}A^{2}=2[m^{2}-m(m+1)B+5m(1-m)B^{2}-3(1-m)^{2}B^{3}]\beta^{2}\,,$
$\displaystyle(1+B)\omega_{2}=[(5-4m)B-(4m+1)]\beta^{2}\,,~{}~{}m(1+B)g_{22}D^{2}=2[m+4mB-3(1-m)B^{2}]B\beta^{2}\,,$
$\displaystyle
m(1+B)g_{21}A^{2}=6[m^{2}-m(1-m)B+m(1-m)B^{2}-(1-m)^{2}B^{3}]\beta^{2}\,.$
(81)

On using the identities (6) and (9), the coupled solution (80) can be re-
expressed as

$u(x,t)=e^{i\omega_{1}t}\frac{A{\rm dn}(\Delta,m)}{2{\rm cn}(\Delta,m)}[{\rm
sn}(\beta x+\Delta,m)+{\rm sn}(\beta x-\Delta,m)]\,,$ (82)
$v(x,t)=e^{i\omega_{2}t}\frac{D{\rm dn}(\Delta,m)}{2{\rm sn}(\Delta,m)}[{\rm
cn}(\beta x-\Delta,m)-{\rm cn}(\beta x+\Delta,m)]\,,$ (83)

where $A,D$ are given by Eq. (2.2) while $B={m{\rm sn}^{2}(\Delta,m)}/{{\rm
dn}^{2}(\Delta,m)}$.

Solution XIX

It is easy to check that

$u(x,t)=e^{i\omega_{1}t}\frac{A{\rm sn}(\beta x,m)}{1+B{\rm cn}^{2}(\beta
x,m)}\,,~{}~{}v(x,t)=e^{i\omega_{2}t}\frac{D{\rm sn}(\beta x,m){\rm cn}(\beta
x,m)}{1+B{\rm cn}^{2}(\beta x,m)}\,,$ (84)

where $B>0$ is an exact solution of the coupled Eqs. (1) and (2) provided

$\displaystyle(1+B)\omega_{1}=[(5-m)B-(1+m)]\beta^{2}\,,$
$\displaystyle(1+B)g_{12}D^{2}=-6B[m+2mB-(1-m)B^{2}]\beta^{2}\,,$
$\displaystyle(1+B)g_{11}A^{2}=2[m-2(2-m)B-(1-m)B^{2}]\beta^{2}\,,$
$\displaystyle(1+B)g_{21}A^{2}=6[m-2(1-m)B-(1-m)B^{2}]\beta^{2}\,,$
$\displaystyle(1+B)\omega_{2}=[(2-m)B-(4+m)]\beta^{2}\,,$
$\displaystyle(1+B)g_{22}D^{2}=-2B[m+2(1+m)B-(1-m)B^{2}]\beta^{2}\,.$ (85)

On using the identities (6) and (2), the coupled solution (84) can be re-
expressed as

$u(x,t)=e^{i\omega_{1}t}\frac{A{\rm dn}(\Delta,m)}{2{\rm cn}(\Delta,m)}[{\rm
sn}(\beta x+\Delta,m)+{\rm sn}(\beta x-\Delta,m)]\,,$ (86)
$v(x,t)=e^{i\omega_{2}t}\frac{D{\rm dn}^{2}(\Delta,m)}{2m{\rm
cn}(\Delta,m){\rm sn}(\Delta,m)}[{\rm dn}(\beta x-\Delta,m)-{\rm dn}(\beta
x+\Delta,m)]\,,$ (87)

where $A$, $D$ are as given by Eq. (2.2) while $B={m{\rm
sn}^{2}(\Delta,m)}/{{\rm dn}^{2}(\Delta,m)}$.

Hyperbolic Limit

In the limit $m=1$, the two solutions XVIII and XIX go over to the hyperbolic
solution IV, i.e. in this limit both the solutions can be re-expressed as
superposition of one pulse (i.e. ${\rm sech}(\beta x-\Delta)-{\rm sech}(\beta
x+\Delta)$) and one kink (i.e. $\tanh(\beta x+\Delta)+\tanh(\beta x-\Delta)$)
solution.

Solution XX

It is strightforward to check that

$u(x,t)=e^{i\omega_{1}t}\frac{A{\rm sn}(\beta x,m)}{1+B{\rm cn}^{2}(\beta
x,m)}\,,~{}~{}v(x,t)=e^{i\omega_{2}t}\frac{D{\rm cn}(\beta x,m){\rm dn}(\beta
x,m)}{1+B{\rm cn}^{2}(\beta x,m)}\,,~{}~{}B>0\,,$ (88)

is an exact solution of the coupled Eqs. (1) and (2) provided

$\displaystyle\omega_{1}=\omega_{2}=-(1+m)\beta^{2}\,,~{}~{}g_{12}D^{2}=3g_{22}D^{2}=-6B(B+1)\beta^{2}\,,$
$\displaystyle g_{21}A^{2}=3g_{11}A^{2}=6[m-(1-m)B]\beta^{2}\,.$ (89)

On using the identities (6) and (7), the coupled solution (88) can be re-
expressed as

$u(x,t)=e^{i\omega_{1}t}\frac{\sqrt{m}\beta}{\sqrt{2g_{11}}}[{\rm sn}(\beta
x+\Delta,m)+{\rm sn}(\beta x-\Delta,m)]\,,$ (90)
$v(x,t)=e^{i\omega_{2}t}\frac{\sqrt{m}\beta}{\sqrt{2|g_{22}|}}[{\rm sn}(\beta
x+\Delta,m)-{\rm sn}(\beta x-\Delta,m)]\,,$ (91)

where $B={m{\rm sn}^{2}(\Delta,m)}/{{\rm dn}^{2}(\Delta,m)}$.

Hyperbolic Limit

In the limit $m=1$, the solution XX goes over to the hyperbolic solution I,
i.e. in this limit the solution can be re-expressed as superposition of two
kink solutions (i.e. $\tanh(\beta x+\Delta)\pm\tanh(\beta x-\Delta)$).

Solution XXI

It is easy to check that

$u(x,t)=e^{i\omega_{1}t}\frac{A{\rm cn}(\beta x,m)}{1+B{\rm cn}^{2}(\beta
x,m)}\,,~{}~{}v(x,t)=e^{i\omega_{2}t}\frac{D{\rm dn}(\beta x,m)}{1+B{\rm
cn}^{2}(\beta x,m)}\,,$ (92)

where $B>0$, is an exact solution of the coupled Eqs. (1) and (2) provided

$\displaystyle\omega_{1}=[(2m-1)+\frac{6m}{B}]\beta^{2}\,,~{}~{}(1-m)g_{12}D^{2}=-6[(1-m)B]+\frac{m}{B}]\beta^{2}\,,$
$\displaystyle
g_{11}A^{2}=2\bigg{[}(1-m)^{2}B^{2}-(4m-3)(1-m)B-7m(1-m)B+\frac{3m^{2}}{B}\bigg{]}\beta^{2}\,,$
$\displaystyle\omega_{2}=[(5m-4)+\frac{6m}{B}]\beta^{2}\,,~{}~{}(1-m)g_{22}D^{2}=2[2(1-m)-(1-m)B-\frac{3m}{B}]\beta^{2}\,,$
$\displaystyle(1-m)g_{21}A^{2}=\frac{6\beta^{2}}{B}[m-(1-m)B][1-(1-m)(B+1)^{2}]\,.$
(93)

Note that this solution is only valid if $0<m<1$.

On using the identities (8) and (10), the coupled solution (92) can be re-
expressed as

$u(x,t)=e^{i\omega_{1}t}\frac{A{\rm dn}^{2}(\Delta,m)}{2{\rm
cn}(\Delta,m)}[{\rm cn}(\beta x+\Delta,m)+{\rm cn}(\beta x-\Delta,m)]\,,$ (94)
$v(x,t)=e^{i\omega_{2}t}\frac{D{\rm dn}(\Delta,m)}{2}[{\rm dn}(\beta
x+\Delta,m)+{\rm dn}(\beta x-\Delta,m)]\,,$ (95)

where $A,D$ are as given by Eq. (2.2) while $B={m{\rm sn}^{2}(\Delta,m)}/{{\rm
dn}^{2}(\Delta,m)}$.

Solution XXII

It is straightforward to check that

$u(x,t)=e^{i\omega_{1}t}\frac{A{\rm cn}(\beta x,m){\rm sn}(\beta x,m)}{1+B{\rm
cn}^{2}(\beta x,m)}\,,~{}~{}v(x,t)=e^{i\omega_{2}t}\frac{D{\rm sn}(\beta
x,m){\rm dn}(\beta x,m)}{1+B{\rm cn}^{2}(\beta x,m)}\,,$ (96)

where $B>0$, is an exact solution of the coupled Eqs. (1) and (2) provided

$\displaystyle(B+1)\omega_{1}=[(2-m)B-(4+m)]\beta^{2}\,,~{}~{}(1-m)(B+1)g_{11}A^{2}$
$\displaystyle=-[(1-m)(2-m)B^{3}+(1-m)(5m-4)B^{2}+12m(1-m)B-6m^{2}]\beta^{2}\,,$
$\displaystyle(1-m)(B+1)g_{12}D^{2}=-6[1-(1-m)(1+B)^{2}]\beta^{2}\,,$
$\displaystyle(B+1)\omega_{2}=[(5-4m)B-(4m+1)]\beta^{2}\,,$
$\displaystyle(1-m)(B+1)g_{21}A^{2}=[1-15m+20m^{2}-(1-m)(5+2m)B$
$\displaystyle+(1-m)(2m-1)B^{2}+(1-m)(2m-1)B^{3}]\beta^{2}\,,$
$\displaystyle(1-m)(B+1)g_{22}D^{2}=-2[3-8(1-m)B-(1-m)B^{2}]\beta^{2}\,.$ (97)

Note that this solution is only valid for $0<m<1$.

On using the identities (9) and (2), the coupled solution (96) can be re-
expressed as

$u(x,t)=e^{i\omega_{1}t}\frac{A{\rm dn}^{2}(\Delta,m)}{2m{\rm
cn}(\Delta,m){\rm sn}(\Delta,m)}[{\rm dn}(\beta x-\Delta,m)-{\rm dn}(\beta
x+\Delta,m)]\,,$ (98) $v(x,t)=e^{i\omega_{2}t}\frac{D{\rm dn}(\Delta,m)}{2{\rm
sn}(\Delta,m)}[{\rm cn}(\beta x-\Delta,m)-{\rm cn}(\beta x-\Delta,m)]\,,$ (99)

where $A,D$ are as given by Eq. (2.2) while $B={m{\rm sn}^{2}(\Delta,m)}/{{\rm
dn}^{2}(\Delta,m)}$.

##  3 Superposed Solutions of a Coupled Nonlocal mKdV Model

Let us consider the following model of coupled nonlocal mKdV equations

$u_{t}(x,t)+u_{xxx}+6[g_{11}u(x,t)u(-x,-t)+g_{12}v(x,t)v(-x,-t)]u_{x}(x,t)=0\,,$
(100)
$v_{t}(x,t)+v_{xxx}+6[g_{21}u(x,t)u(-x,-t)+g_{22}v(x,t)v(-x,-t)]v_{x}(x,t)=0\,.$
(101)

We now show that these coupled nonlocal equations admit a large number of
exact solutions. In particular, there are two types of solutions to these
coupled equations depending on if $v(x,t)$ is proportional to $u(x,t)$ or not
which we discuss one by one. In this section we only discuss those solutions
where the two coupled fields $u$ and $v$ are distinct (and not proportional to
each other) while in Appendix B we discuss those solutions of the nonlocal
coupled mKdV model in which the two fields $v$ and $u$ are proportional to
each other.

One major difference between the local and the nonlocal case is that, unlike
the local case, the solutions of the nonlocal mKdV Eqs. (100) and (101) are
not invariant with respect to shifts in $x$ and $t$.

###  3.1 Solutions When $u(x,t)$ and $v(x,t)$ are Distinct

It turns out that in case $v(x,t)$ and $u(x,t)$ are distinct then the coupled
Eqs. (100) and (101) admit a number of exact solutions which we now present
one by one.

Solution I

It is easy to show that

$u(x,t)=A{\rm dn}(\xi,m)\,,~{}~{}v(x,t)=B\sqrt{m}{\rm
sn}(\xi,m)\,,~{}~{}\xi=\beta(x-ct)\,,$ (102)

is an exact solution to the coupled Eqs. (100) and (101) provided

$g_{11}A^{2}+g_{12}B^{2}=g_{21}A^{2}+g_{22}B^{2}=\beta^{2}\,,$ (103)
$c=(2-m)\beta^{2}-6g_{12}B^{2}\,,$ (104) $2(g_{22}-g_{12})B^{2}=\beta^{2}\,.$
(105)

Solution II

It is straightforward to show that

$u(x,t)=A\sqrt{m}{\rm cn}(\xi,m)\,,~{}~{}v(x,t)=B\sqrt{m}{\rm
sn}(\xi,m)\,,~{}~{}\xi=\beta(x-ct)\,,$ (106)

is an exact solution to the coupled Eqs. (100) and (101) provided Eqs. (103)
and (105) are satisfied and if further

$c=(5-4m)\beta^{2}-6[g_{12}B^{2}+(1-m)g_{11}A^{2}]\,.$ (107)

In the limit $m=1$, both the solutions XXIII and XXIV go over to the
hyperbolic solution

$u(x,t)=A{\rm
sech}(\xi)\,,~{}~{}v(x,t)=B\tanh(\xi)\,,~{}~{}\xi=\beta(x-ct)\,,$ (108)

provided Eqs. (103) and (105) are satisfied and if further

$c=\beta^{2}-6[g_{12}B^{2}+(1-m)g_{11}A^{2}]\,.$ (109)

Solution III

It is easy to show that

$u(x,t)=A{\rm dn}(\xi,m)\,,~{}~{}v(x,t)=B\sqrt{m}{\rm
cn}(\xi,m)\,,~{}~{}\xi=\beta(x-ct)\,,~{}~{}0<m<1\,,$ (110)

is an exact solution to the coupled Eqs. (100) and (101) provided Eqs. (103)
and (105) are satisfied and if further

$c=(5-4m)\beta^{2}-6[g_{12}B^{2}+(1-m)g_{11}A^{2}]\,.$ (111)

Note that if $m=1$, then $v(x,t)=u(x,t)$ and that solution has been discussed
in Appendix B.

Solution IV

It is straightforward to show that

$u(x,t)=\frac{A}{{\rm dn}(\xi,m)}\,,~{}~{}v(x,t)=B\sqrt{m}\frac{{\rm
sn}(\xi,m)}{{\rm dn}(\xi,m)}\,,~{}~{}\xi=\beta(x-ct)\,,$ (112)

is an exact solution to the coupled Eqs. (100) and (101) provided

$g_{11}A^{2}-g_{12}B^{2}=g_{21}A^{2}-g_{22}B^{2}=(1-m)\beta^{2}\,,$ (113)
$c=(2-m)\beta^{2}+6g_{12}B^{2}\,,$ (114)
$2(g_{22}-g_{12})B^{2}=(1-m)\beta^{2}\,.$ (115)

Solution V

It is easy to show that

$u(x,t)=\frac{A}{{\rm dn}(\xi,m)}\,,~{}~{}v(x,t)=\frac{B\sqrt{m}{\rm
cn}(\xi,m)}{{\rm dn}(\xi,m)}\,,~{}~{}\xi=\beta(x-ct)\,,$ (116)

is an exact solution to the coupled Eqs. (100) and (101) provided Eq. (114) is
satisfied and further

$g_{11}A^{2}-(1-m)g_{12}B^{2}=g_{21}A^{2}-(1-m)g_{22}B^{2}=(1-m)\beta^{2}\,,$
(117) $2(g_{22}-g_{12})B^{2}=-\beta^{2}\,.$ (118)

Solution VI

It is straightforward to show that

$u(x,t)=\frac{A\sqrt{m}{\rm sn}(\xi,m)}{{\rm
dn}(\xi,m)}\,,~{}~{}v(x,t)=\frac{B\sqrt{m}{\rm cn}(\xi,m)}{{\rm
dn}(\xi,m)}\,,~{}~{}\xi=\beta(x-ct)\,,$ (119)

is an exact solution to the coupled Eqs. (100) and (101) provided

$g_{11}A^{2}+(1-m)g_{12}B^{2}=g_{21}A^{2}+(1-m)g_{22}B^{2}=-(1-m)\beta^{2}\,,$
(120) $c=(5-4m)\beta^{2}+6[g_{11}A^{2}+g_{12}B^{2}]\,,$ (121)
$2(g_{11}-g_{21})A^{2}-2(g_{12}-g_{22})B^{2}=m\beta^{2}\,.$ (122)

Solution VII

The coupled nonlocal MKdV Eqs. (100) and (101) admit the solution

$u(x,t)=1-\frac{A}{B+\cosh^{2}(\xi)}\,,~{}~{}v(x,t)=\frac{\alpha
A}{B+\cosh^{2}(\xi)}\,,~{}~{}~{}A,B>0\,,$ (123)

provided

$\displaystyle
c=4\beta^{2}+6g_{11}\,,~{}~{}g_{21}=g_{11}<0\,,~{}~{}g_{22}=g_{12}\,,~{}~{}Ag_{11}=-(2B+1)\beta^{2}\,,$
$\displaystyle(g_{11}+\alpha^{2}g_{12})A^{2}=-4B(B+1)\beta^{2}\,.$ (124)

On comparing it with the identity (7), we can re-express solution (123) as

$u(x,t)=1-\frac{\beta}{\sqrt{|g_{11}+\alpha^{2}g_{22}|}}[\tanh(\xi+\Delta)-\tanh(\xi-\Delta)]\,,$
(125)

and

$v(x,t)=\frac{\alpha\beta}{\sqrt{|g_{11}+\alpha^{2}g_{22}|}}[\tanh(\xi+\Delta)-\tanh(\xi-\Delta)]\,,$
(126)

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

Solution VIII

Yet another hyperbolic superposed solution to the coupled Eqs. (100) and (101)
is

$u(x,t)=1-\frac{A}{B+\cosh^{2}(\xi)}\,,~{}~{}v(x,t)=-b-\frac{\alpha
A}{B+\cosh^{2}(\xi)}\,,~{}~{}A,B,b>0$ (127)

provided

$\displaystyle c-4\beta^{2}=g_{12}+b^{2}g_{22}=g_{11}+b^{2}g_{12}\,,$
$\displaystyle A(g_{11}-g_{12}\alpha b)==A(g_{21}-g_{22}\alpha
b)=-(2B+1)\beta^{2}\,,$
$\displaystyle(g_{11}+\alpha^{2}g_{12})A^{2}==(g_{21}+\alpha^{2}g_{22})=-4B(B+1)\beta^{2}\,.$
(128)

On comparing it with the identity (7), we can re-express solution (127) as

$u(x,t)=1-\frac{\beta}{\sqrt{|g_{11}+\alpha^{2}g_{22}|}}[\tanh(\xi+\Delta)-\tanh(\xi-\Delta)]\,,$
(129)

and

$v(x,t)=-b-\frac{\alpha\beta}{\sqrt{|g_{11}+\alpha^{2}g_{22}|}}[\tanh(\xi+\Delta)-\tanh(\xi-\Delta)]\,,$
(130)

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

##  4 Conclusion and Open Problems

In this paper we have partially extended the notion of superposition to the
coupled nonlocal equations. In particular, we showed that the Ablowitz-
Musslimani variant of the coupled nonlocal NLS equations as well as the
coupled nonlocal mKdV equations admit solutions which can be re-expressed as
the superposition of two hyperbolic as well as two periodic kink and pulse
solutions. In particular, in Sec. II, for the Ablowitz-Musslimani NLS case we
obtained seven hyperbolic superposed solutions as well as fifteen periodic
superposed solutions. On the other hand, for the nonlocal mKdV case, in Sec.
III we obtained two superposed hyperbolic solutions. Besides, in case the two
coupled fields are proportional to each other then in the Ablowitz-Musslimani
case we obtained four superposed periodic solutions while in the mKdV case we
obtained four superposed periodic and one superposed hyperbolic superposed
solution. In addition, for completeness we have also presented some other
solutions admitted by these models.

This paper raises several important questions which are still not understood.
Someof these questions are

  1. 1.

Unlike the coupled nonlocal Ablowitz-Musslimani case, we have not been able to
obtain superposed periodic solutions in the nonlocal mKdV case when the two
fields are not proportional to each other. It would be worthwhile looking for
such solutions.

  2. 2.

For the nonlocal mKdV case, in the hyperbolic case we have only been able to
obtain solutions which can be re-expressed as a superposition of a kink and an
antikink solution but not as a superposition of two kink or two pulse
solutions. It is clearly of interest to obtain such solutions.

  3. 3.

It is clearly of interest to discover similar superposed solutions in the case
of other nonlocal equations such as the nonlocal KdV equation, the nonlocal
Hirota equation, etc. As a first step in that direction one needs to construct
models with coupled nonlocal KdV and coupled nonlocal Hirota equations.

  4. 4.

Finally, what is the interpretation of such superposed solutions? Do they
correspond to bound state or merely an excitation of two kink or two pulse
solutions?

Hopefully, one can find answer to some of the questions raised above.

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.

Appendix A: Solutions of a Coupled Ablowitz-Musslimani nonlocal NLS Model when
$v(x,t)\propto u(x,t)$

In case $v(x,t)=\alpha u(x,t)$ where $\alpha$ is a number, in that case the
two Eqs. (1) and (2) are consistent with each other provided

$g_{11}+\alpha^{2}g_{12}=g_{21}+\alpha^{2}g_{22}\,,$ (131)

and we only need to solve the single equation

$iu_{x,t}+u_{xx}(x)+gu^{2}(x,t)u^{*}(-x,t)=0\,,~{}~{}g=g_{11}+\alpha^{2}g_{12}\,,$
(132)

which is effectively the uncoupled nonlocal Ablowitz-Musslimani variant of the
NLS for which we have [2] already obtained a large number of solutions.
However, it turns out that we have missed several solutions which we now
present one by one.

As mentioned in Sec. II, one major difference between the local and the
nonlocal case is that, unlike the local case, the solutions of the nonlocal
mKdV Eq. (132) are not invariant with respect to shift in $x$.

Solution I

In [2] we had shown that $e^{i\omega t}{\rm dn}(\beta x,m)$ as well as
$e^{i\omega t}{\rm dn}[\beta(x+K[m]),m]$ are the exact solutions of the
nonlocal Eq. (132). Remarkably, it turns out that not only $e^{i\omega t}{\rm
dn}[\beta x,m]$ and $e^{i\omega t}{\rm dn}[\beta(x+K[m]),m]$ but even their
superposition is a solution of the nonlocal NLS Eq. (132). In particular, it
is straightforward to check that

$u(x,t)=e^{i\omega t}[A{\rm dn}(\beta x,m)+\frac{B\sqrt{1-m}}{{\rm dn}(\beta
x,m)}]\,,$ (133)

is also an exact solution of the nonlocal Eq. (132) provided

$g>0\,,~{}~{}gA^{2}=\beta^{2}\,,~{}~{}B=\pm A\,,~{}~{}\omega=[2-m\pm
6\sqrt{1-m}]\beta^{2}\,,$ (134)

where $g$ is as given by Eq. (132) while the $\pm$ sign in $B=\pm A$ and in
$\omega$ are correlated.

Solution II: Peregrine Soliton

It is easy to check that the celebrated Peregrine soliton solution [24, 25] of
the local NLS is also a solution of the nonlocal Eq. (132), i.e. in
particular, it is straightforward to check that

$u(x,t)=\frac{1}{\sqrt{2g}}\bigg{[}1-\frac{4(1+2it)}{(1+2x^{2}+4t^{2})}\bigg{]}e^{it}\,,~{}~{}g>0\,,$
(135)

is an exact solution of Eq. (132).

Solution III: Akhmedive-Eleonskii-Kulagin Breather Solution

It is easy to check that even the celebrated Akhmediev-Eleonskii-Kulagin
breather solution [25, 26] of the local NLS, i.e.

$u(x,t)=\sqrt{\frac{a^{2}}{2g}}e^{ia^{2}t}\bigg{[}\frac{b^{2}\cosh(\theta)+ib\sqrt{2-b^{2}}}{\sqrt{2}\cosh(\theta)-\sqrt{2-b^{2}}\cos(abx)}\bigg{]}\,,~{}~{}g>0\,,$
(136)

where $\theta=a^{2}b\sqrt{2-b^{2}}t$, is also a solution of the nonlocal Eq.
(132).

Solution IV: Kuznetsov-Ma Soliton

Remarkably, even the celebrated Kuznetsov-Ma soliton solution [25, 27, 28] of
the local NLS is also a solution of the nonlocal Eq. (132). In particular, it
is easy to check that

$u(x,t)=\frac{a}{\sqrt{2g}}e^{ia^{2}t}\bigg{[}1+\frac{2m(m\cos(\theta)+in\sin(\theta)}{n\cosh(\sqrt{2}max)+\cos(\theta)}\bigg{]}\,,~{}~{}g>0\,,$
(137)

where $n^{2}=1+m^{2}\,,~{}~{}\theta=2mna^{2}t$, is an exact solution of the
nonlocal Eq. (132).

We now show that unlike the local NLS or the Yang version of the nonlocal NLS
[29], the Ablowitz-Musslimani variant of the nonlocal NLS admits complex PT-
invariant periodic and hyperbolic superposed solutions.

Solution V

It is straightforward to show that the nonlocal Eq. (132) admits the complex
PT-invariant periodic solution

$u(x,t)=e^{i\omega t}[A{\rm dn}(\beta x,m)+iB\sqrt{m}{\rm sn}(\beta x,m)]\,,$
(138)

provided

$B=\pm
A\,,~{}~{}A=\frac{\beta}{2g}\,,~{}~{}\omega=-\frac{(2m-1)}{2}\beta^{2}\,,~{}~{}g>0\,.$
(139)

Solution VI

Another complex PT-invariant periodic solution of the nonlocal Eq. (132) is

$u(x,t)=e^{i\omega t}[A\sqrt{m}{\rm cn}(\beta x,m)+iB\sqrt{m}{\rm sn}(\beta
x,m)]\,,$ (140)

provided

$B=\pm
A\,,~{}~{}g>0\,,~{}~{}A=\frac{\beta}{2g}\,,~{}~{}\omega=-\frac{(2-m)}{2}\beta^{2}\,.$
(141)

Solution VII

In the limit $m=1$, both the solutions V and VI go over to the complex PT-
invariant hyperbolic solution

$u(x,t)=e^{i\omega t}[A{\rm sech}(\beta x)+iB\tanh(\beta x)]\,,$ (142)

provided

$B=\pm
A\,,~{}~{}g>0\,,~{}~{}A=\frac{\beta}{2g}\,,~{}~{}\omega=-\frac{\beta^{2}}{2}\,.$
(143)

Solution VIII

Remarkably, the nonlocal Eq. (132) also admits the complex PT-invariant
periodic solution

$u(x,t)=e^{i\omega t}[A\sqrt{m}{\rm sn}(\beta x,m)+iB{\rm dn}(\beta x,m)]\,,$
(144)

provided the same relations as given in Eq. (139) are satisfied.

Solution IX

Another complex PT-invariant periodic solution of the nonlocal Eq. (1) is

$u(x,t)=e^{i\omega t}[A\sqrt{m}{\rm sn}(\beta x,m)+iB\sqrt{m}{\rm cn}(\beta
x,m)]\,,$ (145)

provided the same relations as in Eq. (141) are satisfied.

Solution X

In the limit $m=1$, both the solutions VIII and IX go over to the complex PT-
invariant hyperbolic solution

$u(x,t)=e^{i\omega t}[A\tanh(\beta x)+iB{\rm sech}(\beta x)]\,,$ (146)

provided the same relations as in Eq. (143) are satisfied. Thus the Ablowitz-
Musslimani variant of the nonlocal NLS is rather unusual in the sense that the
same model (i.e. with $g>0$) admits not only the kink and the pulse solutions
but also the complex PT-invariant pulse and kink solutions with the PT-
eigenvalue +$1$ as well as $-1$.

We now show that Eq. (132) also satisfies four novel periodic solutions which
can be re-expressed as the superposition of either the two periodic kink or
the two periodic pulse solutions ${\rm sn}(x,m)$ or ${\rm dn}(x,m)$,
respectively.

Solution XI

It is easy to check that the nonlocal NLS Eq. (132) admits the periodic
solution

$u(x,t)=e^{i\omega t}[\frac{A{\rm dn}(\beta x,m){\rm cn}(\beta x,m)}{1+B{\rm
cn}^{2}(\beta x,m)}]\,,~{}~{}B>0\,,$ (147)

provided $g<0$ and further

$\displaystyle 0<m<1\,,~{}~{}B=+\frac{\sqrt{m}}{1-\sqrt{m}}\,,$
$\displaystyle\omega=-[1+m+6\sqrt{m}]\beta^{2}<0\,,~{}~{}g<0\,,~{}~{}|g|A^{2}=\frac{4\sqrt{m}\beta^{2}}{(1-\sqrt{m})^{2}}\,.$
(148)

Note that this solution is not valid for $m=1$, i.e. the nonlocal NLS Eq. (1)
does not admit a corresponding hyperbolic solution.

On comparing Eqs. (147) and the identity (7) and using Eq. (4), one can re-
express the periodic solution (147) as the superposition of a periodic kink
and a periodic antikink solution, i.e.

$u(x,t)=e^{i\omega t}\sqrt{\frac{m}{2|g|}}\beta[{\rm sn}(\beta
x+\Delta,m)-{\rm sn}(\beta x-\Delta,m)]\,.$ (149)

Here $\Delta$ is defined by ${\rm sn}(\sqrt{m}\Delta,1/m)=\pm m^{1/4}$, where
use has been made of the identity [30]

$\sqrt{m}{\rm sn}(y,m)={\rm sn}(\sqrt{m}y,1/m)\,.$ (150)

Solution XII

Remarkably, Eq. (1) also admits another periodic solution

$u(x,t)=e^{i\omega t}\frac{A{\rm sn}(\beta x,m)}{1+B{\rm cn}^{2}(\beta
x,m)}\,,~{}~{}B>0\,,$ (151)

provided

$\displaystyle 0<m<1\,,~{}~{}B=\frac{\sqrt{m}}{1-\sqrt{m}}\,,~{}~{}g<0\,,$
(152)
$\displaystyle\omega=[6\sqrt{m}-(1+m)]\beta^{2}\,,~{}~{}|g|A^{2}=4\sqrt{m}\beta^{2}\,.$

Note that this solution too is not valid in the hyperbolic limit of $m=1$. On
comparing the solution (151) and the novel identity (6) and using Eq. (152),
the periodic solution XII given by Eq. (151) can be re-expressed as the
superposition of the two periodic kink solutions

$u(x,t)=ie^{i\omega t}\sqrt{\frac{m}{|g|}}\beta\bigg{[}{\rm sn}(\beta
x+\Delta,m)+{\rm sn}(\beta x-\Delta,m)\bigg{]}\,.$ (153)

Here $\Delta$ is defined by ${\rm sn}(\sqrt{m}\Delta,1/m)=\pm m^{1/4}$, where
use has been made of the identity (150).

It is worth noting that for both the solutions XI and XII, not only the value
of $B$ is the same but even $g<0$ for both the solutions.

Solution XIII

It is easy to check that the nonlocal NLS Eq. (132) admits another periodic
solution

$u(x,t)=e^{i\omega t}\frac{A{\rm sn}(\beta x,m){\rm cn}(\beta x,m)}{1+B{\rm
cn}^{2}(\beta x,m)}\,,$ (154)

provided

$\displaystyle 0<m<1\,,~{}~{}B=\frac{1-\sqrt{1-m}}{\sqrt{1-m}}\,,~{}~{}g<0\,,$
$\displaystyle\omega=(2-m-6\sqrt{1-m})\beta^{2}\,,~{}~{}A^{2}=\frac{2(1-\sqrt{1-m})^{2}\beta^{2}}{\sqrt{1-m}}\,.$
(155)

Note that whereas this solution is valid if $g<0$, the same solution in the
Yang’s nonlocal case, as well as in the local NLS case is valid only if $g>0$.

On comparing the solution (154) with the identity (2) and using Eq. (4) we
find that the solution as given by Eq. (154) can be re-expressed as a
superposition of the two periodic pulse solutions, i.e.

$u(x,t)=e^{i\omega t}\beta\sqrt{\frac{1}{2|g|}}\big{(}{\rm
dn}[\beta(x)-K(m)/2,m]-{\rm dn}[\beta(x)+K(m)/2,m]\big{)}\,.$ (156)

Solution XIV

Remarkably, the nonlocal NLS Eq. (132) also admits another periodic solution

$u(x,t)=e^{i\omega t}\frac{A{\rm dn}(\beta x,m)}{1+B{\rm cn}^{2}(\beta
x,m)}\,,$ (157)

provided

$\displaystyle 0<m<1\,,~{}~{}B=\frac{1-\sqrt{1-m}}{\sqrt{1-m}}\,,~{}~{}g>0\,,$
$\displaystyle\omega=[2-m+6\sqrt{1-m}]\beta^{2}\,,~{}~{}A^{2}=\frac{4}{\sqrt{1-m}}\beta^{2}\,.$
(158)

Note that this solutions is also not valid for $m=1$. Thus for this solution
$g>0$, $\omega>0$.

On comparing the solution (157) and the identity (10) and using Eq. (4), the
periodic solution (157) can be re-expressed as a superposition of the two
periodic pulse solutions, i.e.

$u(x,t)=e^{i\omega t}\frac{\beta}{\sqrt{g}}[{\rm dn}(\beta x+K(m)/2,m)+{\rm
dn}(\beta x-K(m)/2,m)]\,,$ (159)

where $\Delta=\pm K(m)/2$

It is worth noting that for both the superposed periodic pulse solutions XIII
and XIV, while the value of $B$ is the same but the value of $g$ is opposite
for the two solutions.

Note that out of the 14 solutions, the solutions XI to XIV are only valid if
$m\neq 1$. Further, while the solutions XI, XII and XIII are valid if $g<0$,
the remaining eleven solutions are valid if $g>0$.

Appendix B: Solutions of a Coupled Nonlocal mKdV Model When $v(x,t)\propto
u(x,t)$

In case $v(x,t)=\alpha u(x,t)$ where $\alpha$ is a number, then the two
coupled Eqs. (100) and (101) are consistent with each other provided

$g_{11}+\alpha^{2}g_{12}=g_{21}+\alpha^{2}g_{22}\,.$ (160)

Remarkably, in this case we only need to solve an uncoupled nonlocal mKdV
equation

$u_{t}(x,t)+u_{xxx}+6gu(x,t)u(-x,-t)u_{x}(x,t)=0\,,~{}~{}g=g_{11}+\alpha^{2}g_{12}\,.$
(161)

In a recent paper [13] we have obtained several exact solutions to an
uncoupled nonlocal mKdV equation using which we can immediately write down the
exact solutions to the coupled nonlocal mKdV Eqs. (100) and (101) in case the
two nonlocal fields $v(x,t)$ and $u(x,t)$ are proportional to each other.

Solution I

It is easy to show that one of the exact solution of the nonlocal mKdV Eq.
(161) is

$u(x,t)=A{\rm dn}[\beta(x-ct),m]\,,$ (162)

provided

$g>0\,,~{}~{}gA^{2}=\beta^{2}\,,~{}~{}c=(2-m)\beta^{2}\,.$ (163)

Solution II

Similarly

$u(x,t)=A\sqrt{m}{\rm cn}[\beta(x-ct),m]\,,$ (164)

is an exact solution to Eq. (161) provided

$g>1\,,~{}~{}gA^{2}=\beta^{2}\,,~{}~{}c=(2m-1)\beta^{2}\,.$ (165)

Solution III

Remarkably, even a linear superposition of the solutions I and II is also a
solution of Eq. (161), i.e.

$u(x,t)=A{\rm dn}[\beta(x-ct),m]+B\sqrt{m}{\rm cn}[\beta(x-ct),m]\big{]}\,,$
(166)

provided

$g>0\,,~{}~{}4gA^{2}=\beta^{2}\,,~{}~{}B=\pm
A\,,~{}~{}c=\frac{(1+m)}{2}\beta^{2}\,.$ (167)

Solution IV

In the limit $m=1$, the solutions I, II and III (with $B=A$) go over to the
hyperbolic solution

$u(x,t)=A{\rm sech}[\beta(x-ct)]\,,$ (168)

provided

$g>1\,,~{}~{}gA^{2}=\beta^{2}\,,~{}~{}v=\beta^{2}\,,$ (169)

while solution III with $B=-A$ goes over to the vacuum solution $u=0$.

Solution V

Remarkably, unlike the local mKdV, for the nonlocal case even

$u(x,t)=A\sqrt{m}{\rm sn}[\beta(x-ct),m]\,,$ (170)

is an exact solution to Eq. (161) provided

$g>0\,,~{}~{}gA^{2}=\beta^{2}\,,~{}~{}c=-(1+m)\beta^{2}\,.$ (171)

Solution VI

In the limit $m=1$, the solution V goes over to the hyperbolic solution

$u(x,t)=A\tanh[\beta(x-ct)]\,,$ (172)

provided

$g>0\,,~{}~{}gA^{2}=\beta^{2}\,,~{}~{}c=-2\beta^{2}\,.$ (173)

Thus unlike the local mKdV, the nonlocal attractive mKdV Eq. (161) (i.e. with
$g=+1$) admits both the kink and the pulse solutions.

Unlike the local case, the solutions of the nonlocal mKdV Eq. (161) are not
invariant with respect to shifts in $x$ and $t$. For example, while $A{\rm
dn}[\beta(x-ct+x_{0}),m]$ is an exact solution of the local mKdV no matter
what $x_{0}$ is, it is not an exact solution of the nonlocal Eq. (161).
However, for the special values of $x_{0}$, ${\rm sn}(x,m)$, ${\rm cn}(x,m)$
and ${\rm dn}(x,m)$ are still the solutions of the nonlocal Eq. (161). In
particular, we now show that when $x_{0}=K(m)$, where $K(m)$ is the complete
elliptic integral of the first kind, there are exact solutions of the nonlocal
Eq. (161) in both the focusing ($g>0$) and the defocusing ($g<0$) cases. This
is because of the relations [3]

$\displaystyle{\rm dn}[x+K(m),m]=\frac{\sqrt{1-m}}{{\rm dn}(x,m)}\,,~{}~{}{\rm
sn}[x+K(m),m]=\frac{{\rm cn}(x,m)}{{\rm dn}(x,m)}\,,$ $\displaystyle{\rm
cn}[x+K(m),m]=-\frac{\sqrt{1-m}{\rm sn}(x,m)}{{\rm dn}(x,m)}\,.$ (174)

Solution VII

It is easy to show that

$u(x,t)=\frac{A}{{\rm dn}[\beta(x-ct),m]}\,,$ (175)

is an exact solution to the Eq. (161) provided

$g>0\,,~{}~{}gA^{2}=(1-m)\beta^{2}\,,~{}~{}c=(2-m)\beta^{2}\,.$ (176)

Solution VIII

It is easy to show that

$u(x,t)=\frac{A\sqrt{m}{\rm sn}[\beta(x-ct),m]}{{\rm dn}[\beta(x-ct),m]}\,,$
(177)

is an exact solution to the nonlocal mKdV Eq. (161) provided

$g<0\,,~{}~{}|g|A^{2}=(1-m)\beta^{2}\,,~{}~{}c=(2m-1)\beta^{2}\,.$ (178)

Solution IX

It is easy to show that

$u(x,t)=\frac{A\sqrt{m}{\rm cn}[\beta(x-ct),m]}{{\rm dn}[\beta(x-ct),m]}\,,$
(179)

is an exact solution to the Eq. (161) provided

$g<0\,,~{}~{}|g|A^{2}=\beta^{2}\,,~{}~{}c=-(1+m)\beta^{2}\,,~{}~{}m\neq 1\,.$
(180)

Solution X

Remarkably, it turns out that not only ${\rm dn}[\beta(x-ct),m]$ and ${\rm
dn}[\beta(x-ct+K[m]),m]$ but even their superposition is a solution of the
nonlocal mKdV Eq. (161). In particular, it is easy to check that

$u(x,t)=e^{i\omega t}\left(A{\rm dn}[\beta(x-ct),m]+\frac{B\sqrt{1-m}}{{\rm
dn}[\beta(x-ct),m]}\right)\,,$ (181)

is also an exact solution of the nonlocal Eq. (161) provided

$g>0\,,~{}~{}gA^{2}=\beta^{2}\,,~{}~{}B=\pm A\,,~{}~{}c=[2-m\pm
6\sqrt{1-m}]\beta^{2}\,,$ (182)

where the $\pm$ sign in $B=\pm A$ and in $v$ are correlated.

Solution XI: The Bion Solution

The well known breather (also called bion) solution of the attractive local
mKdV Equation

$u_{t}+u_{xxx}+6gu^{2}u_{x}=0\,,$ (183)

is [25]

$u(x,t)=-\frac{2}{\sqrt{g}}\frac{d}{dx}\tan^{-1}\bigg{[}\frac{c\sin(ax+bt+a_{0})}{a\cosh(cx+dt+c_{0})}\bigg{]}\,,$
(184)

provided

$g>0\,,~{}~{}b=a(a^{2}-3c^{2})\,,~{}~{}d=c(3a^{2}-c^{2})\,.$ (185)

Here $a_{0},c_{0}$ are arbitrary constants. It is then clear that the bion
solution as given by Eq. (184) is also the bion solution of the nonlocal mKdV
Eq. (161) provided $a_{0}=c_{0}=0$.

Solution XII: Periodic Generalization of the Bion Solution

It has been shown [31] that the periodic generalization of the bion solution
of the local attractive mKdV Eq. (183) is

$u(x,t)=-\frac{2}{\sqrt{g}}\frac{d}{dx}\tan^{-1}\bigg{[}\alpha{\rm
sn}(ax+bt+a_{0},k){\rm dn}(cx+dt+c_{0},m)\bigg{]}\,,$ (186)

provided

$\displaystyle a^{4}k=c^{4}(1-m)\,,~{}~{}\alpha=\frac{c}{a}\,,$ $\displaystyle
b=a[a^{2}(1+k)-3c^{2}(2-m)]\,,~{}~{}d=c[3a^{2}(1+k)-(2-m)c^{2}]\,.~{}~{}~{}$
(187)

As expected, in the limit $m\rightarrow 1,k\rightarrow 0$, the periodic bion
solution (186) goes over to the bion solution (184) and the relations between
$c$ and $d$ as well as between $a$ and $b$ as given by Eq. (4) go over to the
one given in Eq. (185).

It is then clear that the periodic bion solution as given by Eq. (186) is also
the periodic bion solution of the nonlocal mKdV Eq. (161) provided
$a_{0}=c_{0}=0$.

Solution XIII: Rational Solution

It is easy to check that following the rational solution of the local mKdV Eq.
(183) [25], the solution of the nonlocal mKdV Eq. (161) is

$\sqrt{g}u(x,t)=c-\frac{4c}{4c(x-6c^{2}t)^{2}+1}\,.$ (188)

Solution XIV

It has been shown [32] that there is another periodic solution of the
attractive “local” mKdV Eq. (183) and it is easy to check that it is also the
solution of the nonlocal mKdV Eq. (161) and is given by

$u(x,t)=-\frac{2}{\sqrt{g}}\frac{d}{dx}\tan^{-1}\bigg{[}\alpha\sc(ax+bt,k){\rm
dn}(cx+dt,m)\bigg{]}\,,$ (189)

provided

$\displaystyle
g>0\,,~{}~{}(1-k)a^{4}=(1-m)c^{4}\,,~{}~{}\alpha^{2}=-\frac{c}{a}\,,$
$\displaystyle
b=-a[a^{2}(2-k)+3c^{2}(2-m)]\,,~{}~{}d=-c[3a^{2}(2-k)+(2-m)c^{2}]\,.~{}~{}~{}~{}$
(190)

Solution XV

Following [32] it is easy to see that

$u(x,t)=-\frac{2}{\sqrt{g}}\frac{d}{dx}\tan^{-1}\bigg{[}\alpha{\rm
sn}(ax+bt,k){\rm sn}(cx+dt,m)\bigg{]}\,,$ (191)

is an exact solution of the nonlocal mKdV Eq. (161) provided

$\displaystyle g>0\,,~{}~{}ka^{4}=mc^{4}\,,~{}~{}\alpha^{2}=\sqrt{km}\,,$
$\displaystyle
b=a[a^{2}(1+k)+3c^{2}(1+m)]\,,~{}~{}d=c[3a^{2}(1+k)+(1+m)c^{2}]\,.~{}~{}~{}$
(192)

Solution XVI

Following [32], it is easy to see that another periodic solution of the
nonlocal mKdV Eq. (161) is

$u(x,t)=-\frac{2}{\sqrt{|g|}}\frac{d}{dx}\tan^{-1}\bigg{[}\alpha{\rm
cn}(ax+bt,k){\rm cn}(cx+dt,m)\bigg{]}\,,$ (193)

provided

$\displaystyle
g<0\,,~{}~{}k(1-k)a^{4}=m(1-m)c^{4}\,,~{}~{}\alpha^{2}=\frac{km}{(1-k)(1-m)}\,,$
$\displaystyle
b=a[a^{2}(1-2k)+3c^{2}(1-2m)]\,,~{}~{}d=c[3a^{2}(1-2k)+(1-2m)c^{2}]\,.~{}~{}~{}~{}$
(194)

Recently, we [13] have obtained four novel periodic and one hyperbolic
solutions of local mKdV Eq. (183) and shown that they can be re-expressed as
superposed kink or pulse solutions. We now show that the nonlocal mKdV Eq.
(161) also admits these five superposed solutions.

Solution XVII

Following [13] it is easy to see that

$u(x,t)=\frac{A{\rm dn}(\xi,m){\rm cn}(\xi,m)}{1+B{\rm
cn}^{2}(\xi,m)}\,,~{}~{}B>0\,,\xi=\beta(x-ct)\,,$ (195)

is an exact solution of the nonlocal mKdV Eq. (161) provided

$\displaystyle g<0\,,~{}~{}0<m<1\,,~{}~{}B=\frac{\sqrt{m}}{1-\sqrt{m}}\,,$
$\displaystyle
c=-[1+m+6\sqrt{m}]\beta^{2}<0\,,~{}~{}|g|A^{2}=\frac{4\sqrt{m}\beta^{2}}{(1-\sqrt{m})^{2}}\,.$
(196)

Note that this solution is not valid for $m=1$, i.e. the nonlocal mKdV Eq.
(161) does not admit a corresponding hyperbolic solution. Notice that for this
solution $v<0$.

On using the identity (7) one can then rewrite the periodic pulse solution
(195) as a superposition of a periodic kink and a periodic antikink solution,
i.e.

$u(x,t)=\sqrt{\frac{2m}{|g|}}\beta[{\rm sn}(\xi+\Delta,m)-{\rm
sn}(\xi-\Delta,m)]\,,~{}~{}\xi=\beta(x-ct)\,.$ (197)

Here $\Delta$ is defined by ${\rm sn}(\sqrt{m}\Delta,1/m)=\pm m^{1/4}$, where
use has been made of the identity (150).

Superposed Solution XVIII

Following [13] it is easy to show that the nonlocal mKdV Eq. (161) admits the
periodic kink solution

$u(x,t)=\frac{A{\rm sn}(\xi,m)}{1+B{\rm cn}^{2}(\xi,m)}\,,~{}~{}B>0\,,$ (198)

provided

$\displaystyle g<0\,,~{}~{}0<m<1\,,~{}~{}B=\frac{\sqrt{m}}{1-\sqrt{m}}\,,$
$\displaystyle
c=[6\sqrt{m}-(1+m)]\beta^{2}\,,~{}~{}|g|A^{2}=4\sqrt{m}\beta^{2}\,.$ (199)

Note that this solution does not exist for $m=1$, i.e. the corresponding
hyperbolic solution does not exist.

On using the identity (6), the periodic solution (198) can be re expressed as

$u(x,t)=i\sqrt{\frac{m}{|g|}}\beta[{\rm sn}(\xi+\Delta,m)+{\rm
sn}(\xi-\Delta,m)]\,,~{}~{}\xi=\beta(x-ct)\,.$ (200)

Here $\Delta$ is defined by ${\rm sn}(\sqrt{m}\Delta,1/m)=\pm m^{1/4}$, where
use has been made of the identity (150).

It is worth noting that for both the superposed solutions XVIII and XIX of the
nonlocal repulsive mKdV Eq. (161), the value of $B$ as well as $g$ are the
same.

Solution XIX

Following [13] it is easy to show that another periodic solution to the
nonlocal mKdV Eq. (161) is

$u(x,t)=\frac{A{\rm sn}(\xi,m){\rm cn}(\xi,m)}{1+B{\rm cn}^{2}(\xi,m)}\,,$
(201)

provided

$\displaystyle
0<m<1\,,~{}~{}B=\frac{1-\sqrt{1-m}}{\sqrt{1-m}}\,,~{}~{}c=(2-m-6\sqrt{1-m})\beta^{2}\,,$
$\displaystyle
g<0\,,~{}~{}|g|A^{2}=\frac{4(1-\sqrt{1-m})^{2}\beta^{2}}{\sqrt{1-m}}\,.$ (202)

On using the identity (2), the periodic solution (201) can be re-expressed as
a superposition of the two periodic pulse solutions, i.e.

$u(x,t)=\frac{\beta}{\sqrt{|g|}}\big{(}{\rm dn}[\xi-K(m)/2,m]-{\rm
dn}[\xi+K(m)/2,m]\big{)}\,,~{}~{}\xi=\beta(x-ct)\,.$ (203)

Solution XX

Following [13], yet another periodic solution to the nonlocal mKdV Eq. (161)
is

$u(x,t)=\frac{A{\rm dn}(\xi,m)}{1+B{\rm cn}^{2}(\xi,m)}\,,$ (204)

provided

$\displaystyle 0<m<1\,,~{}~{}B=\frac{1-\sqrt{1-m}}{\sqrt{1-m}}\,,~{}~{}g>0\,,$
$\displaystyle
c=[2-m+6\sqrt{1-m}]\beta^{2}\,,~{}~{}gA^{2}=\frac{4}{\sqrt{1-m}}\beta^{2}\,.$
(205)

Thus for this solution $c>0$.

On using the identity (10) the periodic solution (204) can be re-expressed as
superposition of two periodic pulse solutions, i.e.

$u(x,t)=\beta[{\rm dn}(\beta x+K(m)/2,m)+{\rm dn}(\beta x-K(m)/2,m)]\,.$ (206)

Superposed Solution XXI

Following [5, 13] it is easy to check that the nonlocal mKdV Eq. (161) admits
a hyperbolic pulse solution

$\psi(x,t)=1-\frac{A}{B+\cosh^{2}(\xi)}\,,~{}~{}B>0\,,~{}~{}\xi=\beta(x-ct)\,,$
(207)

provided

$g<0\,,~{}~{}\sqrt{|g|}A=2\sqrt{B(B+1)}\beta\,,~{}~{}\beta^{2}=\frac{4(B+1)}{(B+2)^{2}}<1\,,~{}~{}c=4\beta^{2}-6\,.$
(208)

On comparing with the hyperbolic identity (13), the solution (207) can be re-
expressed as the superposition of a (hyperbolic) kink and an antikink
solution, i.e.

$u(x,t)=1-\frac{\beta}{\sqrt{|g|}}[\tanh(\xi+\Delta)-\tanh(\xi-\Delta)]\,,$
(209)

where $\xi=\beta(x-ct)$ while $\sinh(\Delta)=\sqrt{B}$.

## References

  * [1] A. Khare and A. Saxena, Phys. Lett. 377 (2013) 2761.
  * [2] A. Khare and A. Saxena, J. Math. Phys. 55 (2014) 032701\.
  * [3] See for example, M. Abramowitz and I. Stegun, Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables, Dover, NY (1964).
  * [4] A. Khare and A. Saxena, Phys. Lett. A 380 (2016) 856.
  * [5] A.P. Tanakeyav, V.V. Smagin, M.A. Borich and A.S. Zhuravlev, The Physics of Metals and Metallography 107 (2009) 229; ibid. 107 (2009) 245.
  * [6] V.V. Smagin, A.P. Tanakeyav and M.A. Borich, The Physics of Metals and Metallography 108 (2009) 425.
  * [7] R. F. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. D 12 (1975) 2443.
  * [8] D. K. Campbell and A. R. Bishop, Nucl. Phys. B 200 (1982) 297.
  * [9] A. Saxena and A. R. Bishop, Phys. Rev. A 44 (1991) R2251.
  * [10] O. Schnetz, M. Thies and K. Urlichs, Annals Phys. 321 (2006) 2604.
  * [11] A. Khare and A. Saxena, arXiv:2202.06223 (2022).
  * [12] A. Khare and A. Saxena, Int. J. Mod. Phys. B 36 (2022) 2250142; arXiv: 2202.12444 (2022).
  * [13] A. Khare and A. Saxena, arXiv:2207.06359 (2022).
  * [14] M.J. Ablowitz and Z.H. Musslimani, Phys. Rev. Lett. 110 (2013) 064105.
  * [15] J. Yang, Phys. Rev. E 98 (2018) 042202.
  * [16] M.J. Ablowitz and Z.H. Musslimani, Nonlinearity 29 (2016) 915.
  * [17] F. He, E. Fan and J. Xu, arXiv:1804.10863 (2018).
  * [18] J. Cen, F. Correa and A. Fring, J. Math. Phys. 60 (2019) 081508.
  * [19] Y. Xia, R. Yao and X. Xin, Chinese Phys. B 31 (2021) 020401\.
  * [20] S. V. Manakov, Sov. Phys. JETP 38 (1973) 248; V. E. Zhakharov and S. V. Manakov, Sov. Phys. JETP 42 (1976) 842.
  * [21] A.V. Mikhailov, Physica D 3 (1981) 73.
  * [22] V.E. Zakharov and E.I. Schulman, Physica D 4 (1982) 270.
  * [23] V.S. Gerdjikov, in Proceedings of the sixth International Conference on Geometry, Integrability and Quantization, Varna, Bulgaria, June 3-10, 2004, edited by I.M. Mladenov and A.C. Hirshfeld (Softex, Sofia 2005) pp. 1-48.
  * [24] D.H. Peregrine, J. Aust. Math. Soc. Ser. B, Appl. Math. 25 (1983) 16.
  * [25] See for example, P.G. Drazin and R.S. Johnson, Solitons: An Introduction (Cambridge Univ. Press, 1989).
  * [26] N.N. Akhmediev, V.M. Eleonskii and N.E. Kulagin, JETP 62 (1985) 894.
  * [27] E.A. Kuznetsov, Sovt. Phys. Doklady 22 (1977) 507\.
  * [28] Y.C. Ma, Stud. Appl. Math. 60 (1979) 43.
  * [29] J. Yang, Phys. Rev. E 98 (2018) 042202.
  * [30] P.F. Byrd and M.D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (2nd ed.), Springer-Verlag, Berlin (1971).
  * [31] P.G. Kevrikidis, A. Khare and A. Saxena, Phys. Rev. E 68 (2003) 047701.
  * [32] P.G. Kevrikidis, A. Khare and A. Saxena and G. Hearing, J. Phys. A 37 (2004) 10959.