# Superposition formulae for the geometric Bäcklund transformations of the
hyperbolic and elliptic sine-Gordon and sinh-Gordon equations

Kelmer, Filipe1, Tenenblat, Keti2

###### Abstract

We provide superposition formulae for the six cases of Bäcklund
transformations corresponding to space-like and time-like surfaces in the
3-dimensional pseudo-euclidean space. In each case, the surfaces have constant
negative or positive Gaussian curvature and they correspond to solutions of
one of the following equations: the sine-Gordon, the sinh-Gordon, the elliptic
sine-Gordon and the elliptic sinh-Gordon equation. The superposition formulae
provide infinitely many solutions algebraically after the first integration of
the Bäcklund transformation. Such transformations and the corresponding
superposition formulae provide solutions of the same hyperbolic equation,
while they show an unusual property for the elliptic equations. The Bäcklund
transformation alternates solutions of the elliptic sinh-Gordon equation with
those of the elliptic sine-Gordon equation and the superposition formulae
provide solutions of the same elliptic equation. Explicit examples and
illustrations are given and their singularities are briefly discussed.

11footnotetext: Department of Mathematics, Universidade de Brasília, Brazil
e-mail<EMAIL_ADDRESS>Partially supported by CNPq grant 132908/2015-8
and CAPES/Brazil-Finance Code 001.22footnotetext: Department of Mathematics,
Universidade de Brasília, Brazil e-mail<EMAIL_ADDRESS>Partially
supported by CNPq, Brasil, Proc. 311916/2021-0 and CAPES/Brazil-Finance Code
001

Keywords: Superposition formulae, Bäcklund Transformation, sine-Gordon
equation, sinh-Gordon equation, elliptic sine-Gordon, elliptic sinh-Gordon.

2020 Mathematics Subject Classification: 58J72, 35J15, 35L10. 53A35, 53A05.

## Introduction

In [5], we provided a unified geometric proof for the six cases of Bäcklund
type Theorem and Integrability Theorem for surfaces in 3-dimensional pseudo-
euclidean space, $\mathbb{R}^{3}_{s}$, $s=0,1$. These cases include the
classical results ($s=0$) and also those in the Lorentz-Minkowski 3-space
($s=1$), previously obtained by McNertney [6], Palmer [7], Tian [9] and Gu-Hu-
Inoguchi [4].

Considering two surfaces
$M^{2}_{r},\bar{M}^{2}_{\bar{r}}\subset\mathbb{R}^{3}_{s}$ where $r$ and
$\bar{r}$ are the indices, $0\leq r,\bar{r}\leq s\leq 1$, the Bäcklund Theorem
states that a necessary condition for the existence of a Bäcklund-type line
congruence in $\mathbb{R}^{3}_{s}$ between two focal surfaces $M^{2}_{r}$ and
$\bar{M}^{2}_{\bar{r}}$, is that both surfaces have the same constant Gaussian
curvature, which may be positive or negative depending on $r$ and $\bar{r}$.
The Integrability Theorem shows that starting with a space-like or time-like
surface $M\subset\mathbb{R}^{3}_{s}$, there exists a 2-parameter family of
surfaces associated to $M$ by a Bäcklund-type line congruence.

Since each such surface corresponds to a solution of one of the following
differential equations: sine-Gordon equation, sinh-Gordon equation, elliptic
sine-Gordon and elliptic sinh-Gordon, the analytic interpretation of these
results provides Bäcklund transformations for the solutions of the
corresponding equations. More precisely, such a transformation provides two-
parameter new solutions for the sine-Gordon (resp. sinh-Gordon) equation from
a given solution of the sine-Gordon (resp. sinh-Gordon) equation, while the
transformation for the elliptic equations provides two-parameter new solutions
of the elliptic sinh-Gordon equation from a given solution of the elliptic
sine-Gordon equation and vice-versa.

In this paper, we provide the superposition formulae for the Bäcklund
transformations discussed in [5]. The commutativity property of the Bäcklund
transformations for distinct parameters provides a superposition formula which
gives new solutions by algebraic expressions. The importance of a
superposition formula is due to the fact that it provides, algebraically,
infinitely many new solutions of the partial differential equation, after the
first integration required by the Bäcklund transformation. We observe that the
sine-Gordon equation has two Bäcklund transformations (3.2) and (3.5), which
are complementary in the sense that a parameter can be chosen in the intervals
$(-1,1)$ or $(1,\infty)$. Similarly, the same occurs with the Bäcklund
transformation of the sinh-Gordon equation (see (3.8) and (3.11)). The
superposition formulae hold for distinct parameters chosen in the same
interval (see Theorems 3.1-3.4). Our superposition formulae include two
unusual cases involving the elliptic sine-Gordon and elliptic sinh-Gordon
equations (see Theorems 3.5 and 3.6). Namely, starting with a solution of the
elliptic sine-Gordon (resp. elliptic sinh-Gordon) equation, by integrating the
Bäcklund transformation for a parameter in the interval $(1,\infty)$, we get
solutions of the elliptic sinh-Gordon (resp. elliptic sine-Gordon) equation
and the superposition formula, for distinct parameters, provides algebraically
new solutions of the original equation we started with. In the last section of
the paper, we provide some examples of solutions for the differential
equations and we visualize their graphs and some illustrations of the
corresponding surfaces. We also comment on the singularities of the solutions.

This paper is organized as follows. In Section 1, we recall from [5], the six
cases of Bäcklund-type Theorem (Theorem 1.1) and the Integrability Theorem
(Theorem 1.2) for surfaces in the 3-dimensional pseudo-euclidean space
$\mathbb{R}^{3}_{s}$. Moreover, by considering the partial differential
equations whose solutions correspond locally to space-like or time-like
surfaces with non-zero constant Gaussian curvature (Theorem 1.3), we provide
the analytic interpretation of the geometric results in terms of solutions of
these partial differential equations (Theorem 1.4). In Section 2, we prove the
Superposition formulae for the sine-Gordon and the sinh-Gordon equations in
Theorem 2.1. The superposition formula for the elliptic sinh-Gordon equation
is given in Theorem 2.2 and for the elliptic sine-Gordon equation in Theorem
2.3. In Section 3, aiming the applications of the last section and also in
order to help the reader to easily access the theory in each case, we give the
explicit Bäcklund transformation and the corresponding superposition formula,
for each one of the six cases considered in the previous sections. In Section
4, we apply the superposition formulae to provide explicit examples and
illustrate the theory.

##  1 Preliminaries

We denote by $M_{r}^{2}\hookrightarrow\mathbb{R}_{s}^{3}$ a semi-Riemannian
surface of index $r$ isometrically immersed in $\mathbb{R}^{3}_{s}$ where the
metric in coordinates $(x_{1},x_{2},x_{3})$ is given by
$dx_{1}^{2}+dx_{2}^{2}+(-1)^{s}dx_{3}^{2}$. The indices $r,\,s\,\in\\{0,1\\}$,
with $0\leq r\leq s\leq 1$. We consider
$X:U\subset\mathbb{R}^{2}\rightarrow\mathbb{R}_{s}^{3}$ to be a local
parametrization of $M_{r}^{2}\subset\mathbb{R}_{s}^{3}$, where $U$ is an open
subset of $\mathbb{R}^{2}$.

Given a vector $v\in\mathbb{R}^{3}_{s}$, we say that $v$ is a  unit vector if
the absolute value $|\langle v,v\rangle|=1$. We say that $v$ is space-like if
$\langle v,v\rangle>0$ or $v=0$, time-like if $\langle v,v\rangle<0$ and
light-like if $\langle v,v\rangle=0$ and $v\neq 0$. We say that $M_{r}^{2}$ is
a space-like (resp. time-like) surface if $r=0$ (resp. $r=1$).

In [5], we introduced the following definition of a Bäcklund-type line
congruence in $\mathbb{R}^{3}_{s}$ for space-like or time-like surfaces in
$\mathbb{R}^{3}_{s}$. Let
$M^{2}_{r},\bar{M}^{2}_{\bar{r}}\hookrightarrow\mathbb{R}^{3}_{s}$ be two
semi-Riemannian surfaces that are isometrically immersed in
$\mathbb{R}_{s}^{3}$ with $0\leq r,\bar{r}\leq s\leq 1$. Let
$\psi:M_{r}\rightarrow\bar{M}_{\bar{r}}$ be a diffeomorphism such that for
each $p\in M$, $\bar{p}=\psi(p)\neq p$ and let $v(p)\in\mathbb{R}^{3}_{s}$ be
the vector from $p$ to $\bar{p}$. We say that $\psi$ is a Bäcklund-type Line
Congruence (BLC) if there exist constants $\lambda>0$ and $\Lambda\geq 0$,
such that the following three conditions hold: 1) The line in
$\mathbb{R}^{3}_{s}$ determined by $v(p)$ is tangent to $M$ and $\bar{M}$ at
$p$ and $\bar{p}$ respectively. 2) For all $p\in M$, $|v(p)|=\sqrt{|\langle
v(p),v(p)\rangle|}=\lambda$. 3)The normals $N(p)$ and $\bar{N}(\bar{p})$ are
not collinear and $\langle N(p),\bar{N}(\bar{p})\rangle=\Lambda$. The constant
$\lambda$ is called the distance of the line congruence. We say that the BLC
in $\mathbb{R}^{3}_{s}$ is space-like (resp. time-like) if for all $p\in M$
the vector $v(p)$ is space-like (resp. time-like), i.e.,
$\epsilon:=\langle\frac{v(p)}{\lambda},\frac{v(p)}{\lambda}\rangle=1$ (resp.
$-1$). One can determine an angle between a pair of independent vectors in
$\mathbb{R}^{3}_{s}$ (see [5]).

By definition, six cases may occur, according to the values of $s,r,\bar{r}$
and $\epsilon$. We introduce the following notation that will be used
throughout this paper. Consider $\xi\in\left\\{1,-1\right\\}$ and denote

$C_{\xi}(\phi)=\left\\{\begin{array}[]{ll}\cos(\phi)&\mbox{if }\xi=1,\\\
\cosh(\phi)&\mbox{if
}\xi=-1,\end{array}\right.\;\mbox{and}\;\;S_{\xi}(\phi)=\left\\{\begin{array}[]{ll}\sin(\phi)&\mbox{if
}\xi=1,\\\ \sinh(\phi)&\mbox{if }\xi=-1,\end{array}\right.$ (1.1)

where $\phi\in(0,\pi)$ if $\xi=1$ and $\phi\in[0,+\infty)$ if $\xi=-1$. Then
$C_{\xi}^{2}(\phi)+\xi S_{\xi}^{2}(\phi)=1$.

In [5], we considered all possible cases of BLCs in $\mathbb{R}^{3}_{s}$ and
provided a unified proof of a Bäcklund type Theorem for surfaces in a semi-
Riemmanian space $\mathbb{R}_{s}^{3}$, which gives a necessary condition, on
the Gaussian curvature for the surfaces. More precisely:

######  Theorem 1.1.

([5] Bäcklund-type Theorem) Let
$M^{2}_{r},\bar{M}^{2}_{\bar{r}}\hookrightarrow\mathbb{R}^{3}_{s}$, with
$0\leq r,\bar{r}\leq s\leq 1$, be pseudo-Riemannian surfaces of
$\mathbb{R}^{3}_{s}$. Suppose that $M$ and $\bar{M}$ are related by a Bäcklund
type line congruence in $\mathbb{R}^{3}_{s}$. Then both surfaces have the same
constant Gaussian curvature, $K=\bar{K}$, given by

$K=\left\\{\begin{array}[]{ll}(-1)^{s+1}\frac{S^{2}_{\epsilon(-1)^{s}}(\phi)}{\lambda^{2}},&\mbox{if
}r=\bar{r},\\\ -\frac{C^{2}_{-1}(\phi)}{\lambda^{2}},&\mbox{if
}r\neq\bar{r},\end{array}\right.$ (1.2)

where $\lambda$ is the distance of the line congruence, $\phi$ is the angle
between the normals and $\epsilon=1$ if the line congruence is space-like or
$\epsilon=-1$ if the congruence is time-like. Whenever $\epsilon=-1$, then
$s=r=\bar{r}=1$.

In Table 1, we recall from [5], the Bäcklund-type Theorem for all possible
types of BLC.

Bäcklund-type Theorem in $\mathbb{R}^{3}_{s}$
---
| $\mathbb{R}^{3}_{s}$ | BLC | 1st surface | 2nd surface | Gaussian curvature
1 |  | Euclidian | Euclidian | Euclidian |  $K=\bar{K}=-\frac{\sin^{2}{\phi}}{\lambda^{2}}$, $\phi\in(0,\pi)$
| $s=0$ | $\epsilon=1$ | $r=0$ | $\bar{r}=0$ |
2 |  | space-like | space-like | space-like |  $K=\bar{K}=\frac{\sinh^{2}{\phi}}{\lambda^{2}}$, $\phi\in(0,+\infty)$
| $s=1$ | $\epsilon=1$ | $r=0$ | $\bar{r}=0$ |
3 |  | space-like | time-like | time-like |  $K=\bar{K}=\frac{\sinh^{2}{\phi}}{\lambda^{2}}$, $\phi\in(0,+\infty)$
| $s=1$ | $\epsilon=1$ | $r=1$ | $\bar{r}=1$ |
4 |  | time-like | time-like | time-like |  $K=\bar{K}=\frac{\sin^{2}{\phi}}{\lambda^{2}}$, $\phi\in(0,\pi)$
| $s=1$ | $\epsilon=-1$ | $r=1$ | $\bar{r}=1$ |
5 |  | space-like | space-like | time-like |  $K=\bar{K}=-\frac{\cosh^{2}{\phi}}{\lambda^{2}}$, $\phi\in[0,+\infty)$
| $s=1$ | $\epsilon=1$ | $r=0$ | $\bar{r}=1$ |
6 |  | space-like | time-like | space-like |  $K=\bar{K}=-\frac{\cosh^{2}{\phi}}{\lambda^{2}}$, $\phi\in[0,+\infty)$
| $s=1$ | $\epsilon=1$ | $r=1$ | $\bar{r}=0$ |
Table 1: Parameters of the congruence and the curvature for each kind of BLC.

Moreover, we also proved the Integrability Theorem, which shows that by
starting with a given surface $M_{r}$ in $\mathbb{R}^{3}_{s}$ of nonzero
constant Gaussian curvature, one can construct new surfaces
$\bar{M}_{\bar{r}}$ in $\mathbb{R}^{3}_{s}$, which are locally related to $M$
by a BLC in $\mathbb{R}^{3}_{s}$.

From now on, without loss of generality, we can consider
$M^{2}_{r}\subset\mathbb{R}^{3}_{s}$ to be a surface with normalized constant
Gaussian curvature $K=\delta\in\\{-1,1\\}$. Therefore, the parameters $\phi$
and $\lambda$ are related by the expression (1.2), where $K=\delta$.

######  Theorem 1.2.

([5] Integrability Theorem) Let
$M_{r}^{2}(\delta)\hookrightarrow\mathbb{R}^{3}_{s}$ be a surface with
constant Gaussian curvature $K=\delta=\pm 1$, where $r,s\in\\{0,1\\}$,

$0\leq r\leq s=1\quad\mbox{if }\quad\delta=1,\quad\mbox{and }\quad 0\leq r\leq
s\leq 1\quad\mbox{if }\quad\delta=-1.$ (1.3)

Let $p_{0}\in M_{r}^{2}$ be a non-umbilic point. Given a unit tangent vector
${v}_{0}\in T_{p_{0}}M^{2}_{r}$ which is not a principal direction, with
$\langle{v}_{0},{v}_{0}\rangle=\epsilon$ and a constant $\phi$, such that

$\epsilon=\pm 1\quad\mbox{if }\quad\delta=1,\quad\mbox{and
}\quad\epsilon=1\quad\mbox{if }\quad\delta=-1,$ (1.4)
$\phi\in(0,\pi)\quad\mbox{if }\quad\epsilon(-1)^{s}=1,\quad\mbox{and
}\quad\phi\in\left[0,+\infty\right)\quad\mbox{if }\quad\epsilon(-1)^{s}=-1.$

Then, there exists a surface $\bar{M}^{2}_{\bar{r}}\subset\mathbb{R}^{3}_{s}$,
with index $\bar{r}$ satisfying $(-1)^{\bar{r}}=\delta(-1)^{s+r+1}$, which is
related by a BLC to some open neighborhood of $p_{0}\in M_{r}^{2}$, such that
the line of the congruence at $p_{0}$ is in the direction of ${v}_{0}$ and the
distance $\lambda$ of the congruence and the inner product $\Lambda$ between
the normals at corresponding points are given by

$\lambda=\left\\{\begin{array}[]{ll}S_{-\epsilon}(\phi)&\mbox{if
}\,\delta=1,\\\ S_{1}(\phi)&\mbox{if }\,\delta=-1,s=0,\\\
C_{-1}(\phi)&\mbox{if
}\,\delta=-1,s=1,\end{array}\right.\qquad\Lambda=\left\\{\begin{array}[]{lll}C_{-\epsilon}(\phi)&\mbox{if
}\,\delta=1,\\\ C_{1}(\phi)&\mbox{if }\,\delta=-1,s=0,\\\
S_{-1}(\phi)&\mbox{if }\,\delta=-1,s=1,\end{array}\right.$

where $\phi$ is the angle between the normals. In particular, $\lambda$ and
$\Lambda$ are related by the relation

$\delta\left[(-1)^{s+1}\Lambda^{2}-\lambda^{2}\epsilon\right]=1.$ (1.5)

######  Remark 1.1.

The proof of the Integrability Theorem shows that given a surface
$M_{r}^{2}(\delta)\subset\mathbb{R}^{3}_{s}$, of constant Gaussian curvature
$K=\delta=\pm 1$, where $r,s$ and $\delta$ satisfy (1.3), there exists a two-
parameter family of surfaces $\bar{M}$ which is related locally to $M$ by BLC
in $\mathbb{R}^{3}_{s}$. The two parameters are $\phi$ (i.e. the constants
$\lambda$ and $\Lambda$ related by (1.5)) and the one that corresponds to
choosing the unit tangent vector ${v}_{0}$, where $\langle
v_{0},v_{0}\rangle=\epsilon=\pm 1$. Then, Theorem 1.1 implies that each
surface $\bar{M}$ has Gaussian curvature $\bar{K}=\delta$. Moreover, the proof
shows that $\bar{M}$ is obtained by integrating an integrable system. This
procedure is called a Bäcklund Transformation. Observe that when $\delta=1$
and $r=1$ then one can choose the unit vector $v_{0}$ to be spacelike or
timelike i.e., $\epsilon=\pm 1$. Otherwise, if $\delta\neq 1$ or $r\neq 1$,
then $v_{0}$ is always spacelike i.e., $\epsilon=1$ (see Table 1).

By considering an appropriate local coordinate system on the surface, one can
show [5] that locally a space-like or time-like surface in a semi-Riemannian
space, with non zero constant Gaussian curvature and no umbilic points,
corresponds to a solution of a partial differential equation.

######  Theorem 1.3.

[5] Let $M_{r}^{2}(\delta)\subset\mathbb{R}^{3}_{s}$ be a surface with
constant Gaussian curvature $\delta=\pm 1$, $0\leq r\leq s\leq 1$, such that
(1.3) holds. Suppose that $M^{2}_{r}$ has no umbilic points. Then, there
exists a local parametrization $X(x_{1},x_{2})$ of $M^{2}_{r}$ such that for
$i,j\in\\{1,2\\}$,
$\langle\frac{X_{x_{i}}}{|X_{x_{j}}|},\frac{X_{x_{i}}}{|X_{x_{j}}|}\rangle=(-1)^{(i-1)r}\delta_{ij}\epsilon$,
$\epsilon=\pm 1$ and a differentiable function $\alpha(x_{1},x_{2})$ that is a
solution of the differential equation

$\alpha_{x_{1}x_{1}}+\epsilon\delta(-1)^{s}\alpha_{x_{2}x_{2}}=-\epsilon\delta
S_{l}(\alpha),\quad\mbox{where}\quad l=(-1)^{r+s+1}\delta.$ (1.6)

Moreover, in these coordinates, the first and second fundamental forms of
$M^{2}_{r}$ are given, up to orientation $\tau=\pm 1$, respectively, by

$\begin{split}&I=\epsilon
C_{l}^{2}(\frac{\alpha}{2})dx_{1}^{2}+(-1)^{r}\epsilon
S_{l}^{2}(\frac{\alpha}{2})dx^{2}_{2},\\\ &II=\tau
S_{l}(\frac{\alpha}{2})C_{l}(\frac{\alpha}{2})\,\left[\epsilon
dx_{1}^{2}-(-1)^{r}\epsilon ldx^{2}_{2}\right].\end{split}$ (1.7)

Conversely, for fixed integers $\delta=\pm 1$, $r,s\in\\{0,1\\}$, such that
(1.3) is satisfied, $\epsilon=\pm 1$ and $l=(-1)^{r+s+1}\delta$, let
$\alpha(x_{1},x_{2})$ be a non zero solution of (1.6). Then, there exists a
surface $M^{2}_{r}\subset\mathbb{R}^{3}_{s}$ with Gaussian curvature
$K=\delta$, whose fundamental forms are given by (1.7).

We observe that when $\epsilon\delta(-1)^{s}=-1$, then (1.6) reduces to the
sine-Gordon equation if $l=1$, or to the sinh-Gordon equation if $l=-1$, while
whenever $\epsilon\delta(-1)^{s}=1$ then (1.6) reduces to the elliptic sine-
Gordon equation if $l=1$ or to the elliptic sinh-Gordon equation if $l=-1$,

The analytic version of the Integrability Theorem, Theorem 1.2, was given in
[5], by considering the differential equations and their corresponding
Bäcklund transformations. Namely

######  Theorem 1.4.

[5] Consider fixed constants $\delta=\pm 1$, $\epsilon=\pm 1$,
$r,s\in\\{0,1\\}$, $r\leq s$, satisfying (1.3) and (1.4). Let
$\alpha(x_{1},x_{2})$ be a solution of the differential equation (1.6) i.e.

$\alpha_{x_{1}x_{1}}+\epsilon\delta(-1)^{s}\alpha_{x_{2}x_{2}}=-\epsilon\delta
S_{l}(\alpha),$ (1.8)

where $l=(-1)^{r+s+1}\delta$ and $S_{l}$ is given by (1.1). Then the following
system for $\alpha^{\prime}(x_{1},x_{2})$ is integrable

$\left\\{\begin{array}[]{l}\alpha^{\prime}_{x_{1}}-(-1)^{s}\delta\alpha_{x_{2}}=\frac{2}{\lambda}S_{(-1)^{r}}\left(\frac{\alpha^{\prime}}{2}\right)C_{l}\left(\frac{\alpha}{2}\right)-\frac{2(-1)^{s}\tau\Lambda}{\lambda}C_{(-1)^{r}}\left(\frac{\alpha^{\prime}}{2}\right)S_{l}\left(\frac{\alpha}{2}\right)\\\
\alpha^{\prime}_{x_{2}}+\alpha_{x_{1}}=-\frac{2}{\lambda}C_{(-1)^{r}}\left(\frac{\alpha^{\prime}}{2}\right)S_{l}\left(\frac{\alpha}{2}\right)-\frac{2\delta\tau\Lambda}{\lambda}S_{(-1)^{r}}\left(\frac{\alpha^{\prime}}{2}\right)C_{l}\left(\frac{\alpha}{2}\right)\\\
\end{array}\right.,$ (1.9)

where the constants $\lambda$ and $\Lambda$ satisfy (1.5) and $\tau=\pm 1$.
Moreover, $\alpha^{\prime}$ is a solution of

$\alpha^{\prime}_{x_{1}x_{1}}+\epsilon\delta(-1)^{s}\alpha^{\prime}_{x_{2}x_{2}}=-\epsilon\delta
S_{(-1)^{r}}(\alpha^{\prime}).$ (1.10)

######  Remark 1.2.

We point out that whenever $l=(-1)^{r}$ (i.e. $\delta=(-1)^{s+1}$), then (1.8)
and (1.10) coincide and (1.9) is a two parameter (self-)Bäcklund
transformation, where one of the parameters is determined by $\lambda$ and
$\Lambda$ satisfying (1.5) and the other one is the initial condition of
$\alpha^{\prime}$ at a point. It provides a procedure of obtaining new
solutions of (1.8) from a given one, by integrating (1.9). When
$l\neq(-1)^{r}$, i.e., $l=(-1)^{r+1}$, then Theorem 1.4 shows that starting
with a solution of (1.8) and integrating the two-parameter Bäcklund
transformation (1.9) it provides solutions of a different equation given by
(1.10).

The analytic Bäcklund transformation (1.9) was obtained from a geometric
Bäcklund transformation between two surfaces. However, the transformation
(1.9) holds for every initial solution $\alpha$ of (1.8), including the
trivial one (see Example 4.1), which does not correspond to any surface in
$\mathbb{R}^{3}_{1}$. However, $\alpha^{\prime}$ does correspond to a surface
locally (see Section 4).

Although in [5] we gave a unified proof for Theorem 1.4 in all cases, the
superposition formulae for the hyperbolic equations and for the elliptic
equations are quite different and require separate proofs (see Theorems 2.1,
2.2 and 2.3). Therefore, in view of Remark 1.2, Theorem 1.4 can be restated in
the following two corollaries, considering $l=(-1)^{r}$, (i.e.
$\delta=(-1)^{s+1}$) and $l=(-1)^{r+1}$ respectively. The first one will be
useful to obtain the superposition formulae for the sine-Gordon and the sinh-
Gordon equations, while the second one will be used for the superposition
formulae of the elliptic sine-Gordon and elliptic sinh-Gordon equation.

######  Corollary 1.4.1.

Consider fixed constants $\delta=\pm 1$, $\epsilon=\pm 1$, $r,s\in\\{0,1\\}$,
$r\leq s$, satisfying (1.3), (1.4) and $\delta=(-1)^{s+1}$. Let
$\alpha(x_{1},x_{2})$ be a solution of the differential equation

$\alpha_{x_{1}x_{1}}-\alpha_{x_{2}x_{2}}=-\epsilon\delta
S_{(-1)^{r}}(\alpha).$ (1.11)

Then the following system for $\alpha^{\prime}(x_{1},x_{2})$ is integrable

$\left\\{\begin{array}[]{l}\alpha^{\prime}_{x_{1}}+\alpha_{x_{2}}=\frac{2}{\lambda}S_{(-1)^{r}}\left(\frac{\alpha^{\prime}}{2}\right)C_{(-1)^{r}}\left(\frac{\alpha}{2}\right)-\frac{2(-1)^{s}\tau\Lambda}{\lambda}C_{(-1)^{r}}\left(\frac{\alpha^{\prime}}{2}\right)S_{(-1)^{r}}\left(\frac{\alpha}{2}\right),\\\
\alpha^{\prime}_{x_{2}}+\alpha_{x_{1}}=-\frac{2}{\lambda}C_{(-1)^{r}}\left(\frac{\alpha^{\prime}}{2}\right)S_{(-1)^{r}}\left(\frac{\alpha}{2}\right)-\frac{2\delta\tau\Lambda}{\lambda}S_{(-1)^{r}}\left(\frac{\alpha^{\prime}}{2}\right)C_{(-1)^{r}}\left(\frac{\alpha}{2}\right),\\\
\end{array}\right.$ (1.12)

where $\tau=\pm 1$, $\lambda=S_{-\epsilon\delta}(\phi)$,
$\Lambda=C_{-\epsilon\delta}(\phi)$, for a constant $\phi$. Moreoever,
$\alpha^{\prime}$ is a solution of (1.11).

For $\alpha$ and $\alpha^{\prime}$ as in Corollary 1.4.1, we will say that
$\alpha^{\prime}$ is associated to $\alpha$ by the $\phi$-Bäcklund
transformation (1.12). The Bäcklund transformation above includes the
classical case ($s=0$) and the case obtained in [6] when $s=1$, $\epsilon=-1$,
$r=\delta=1$ and $\tau=-1$. Moreover, it shows that for the sine-Gordon
equation (resp. the sinh-Gordon equation) the transformation (1.12) can be
used in two complementary ways with respect to the parameters, namely
$\Lambda\in(-1,1)$ or $\Lambda\in(1,+\infty)$, as one can see in Cases 1 and 2
(resp. Cases 3 and 4) of Section 3.

In order to state the second corollary, considering $l=(-1)^{r+1}$ in Theorem
1.4, we observe that since $l=(-1)^{r+s+1}\delta$, it follows that
$\delta=(-1)^{s}$ and hence (1.3) implies that $s=1$. Therefore $\delta=-1$
and hence it follows from (1.4) that $\epsilon=1$.

######  Corollary 1.4.2.

Consider a fixed constant $r\in\\{0,1\\}$. Let $\alpha(x_{1},x_{2})$ be a
solution of the differential equation

$\alpha_{x_{1}x_{1}}+\alpha_{x_{2}x_{2}}=S_{(-1)^{r+1}}(\alpha).$ (1.13)

Then the following system for $\alpha^{\prime}(x_{1},x_{2})$ is integrable

$\left\\{\begin{array}[]{l}\alpha^{\prime}_{x_{1}}-\alpha_{x_{2}}=\frac{2}{\lambda}S_{(-1)^{r}}\left(\frac{\alpha^{\prime}}{2}\right)C_{(-1)^{r+1}}\left(\frac{\alpha}{2}\right)+\frac{2\tau\Lambda}{\lambda}C_{(-1)^{r}}\left(\frac{\alpha^{\prime}}{2}\right)S_{(-1)^{r+1}}\left(\frac{\alpha}{2}\right)\\\
\alpha^{\prime}_{x_{2}}+\alpha_{x_{1}}=-\frac{2}{\lambda}C_{(-1)^{r}}\left(\frac{\alpha^{\prime}}{2}\right)S_{(-1)^{r+1}}\left(\frac{\alpha}{2}\right)+\frac{2\tau\Lambda}{\lambda}S_{(-1)^{r}}\left(\frac{\alpha^{\prime}}{2}\right)C_{(-1)^{r+1}}\left(\frac{\alpha}{2}\right)\\\
\end{array}\right.,$ (1.14)

where $\tau=\pm 1$, $\lambda=\cosh\phi$, $\Lambda=\sinh\phi$,
$\phi\in[0,\infty)$. Moreoever, $\alpha^{\prime}$ is a solution of

$\alpha^{\prime}_{x_{1}x_{1}}+\alpha^{\prime}_{x_{2}x_{2}}=S_{(-1)^{r}}(\alpha^{\prime}).$
(1.15)

For $\alpha$ and $\alpha^{\prime}$ as in Corollary 1.4.2, we will say that
$\alpha^{\prime}$ is associated to $\alpha$ by the $\phi$-Bäcklund
transformation (1.14). We observe that when $r=0$ this transformation, with
$\tau=-1$, was obtained by Tian [9].

##  2 Superposition formulae

In this section, we provide the superposition formulae for the partial
differential equations considered in the previous section. Such a formula
provides algebraically infinitely many new solutions of the partial
differential equations, after the first integration required by the
corresponding Bäcklund transformation.

Our first result provides the superposition formula for the Bäcklund
transformations of the differential equations given in Corollary 1.4.1. Then,
we prove two theorems giving the superposition formulae for the equations
given in Corollary 1.4.2. These two results are unusual and they show that
starting with a solution of the elliptic sinh-Gordon equation and applying
twice the Bäcklund transformation, with distinct parameters, one gets
solutions of the same equation although the intermediate step gives solutions
of the elliptic sine-Gordon equation. A similar property occurs starting with
a solution of the elliptic sine-Gordon equation.

######  Theorem 2.1.

Consider fixed constants $\delta=\pm 1$, $\epsilon=\pm 1$, $r,s\in\\{0,1\\}$,
$r\leq s$, satisfying (1.3) and (1.4), and $\delta=(-1)^{s+1}$. Let
$\alpha(x_{1},x_{2})$ be a solution of (1.11) and let $\alpha^{\prime}$ (resp.
$\alpha^{\prime\prime}$) be a solution of (1.11) associated to $\alpha$ by the
$\phi_{1}$(resp $\phi_{2}$)-Bäcklund transformation (1.12). If
$\phi_{1}\neq\phi_{2}$, then there exists a unique solution $\alpha^{*}$ of
(1.11) such that $\alpha^{*}$ is associated to $\alpha^{\prime}$ (resp.
$\alpha^{\prime\prime}$) by the $\phi_{2}$(resp.$\phi_{1}$)-Bäcklund
transformation (1.12). Moreover, $\alpha^{*}$ is given algebraically by

$T_{(-1)^{r}}\left(\frac{\alpha^{*}-\alpha}{4}\right)=\delta\tau\,\frac{S_{-\delta\epsilon}(\frac{\phi_{2}+\phi_{1}}{2})}{S_{-\delta\epsilon}(\frac{\phi_{2}-\phi_{1}}{2})}\,T_{(-1)^{r}}\left(\frac{\alpha^{\prime}-\alpha^{\prime\prime}}{4}\right),$
(2.1)

where $T_{(-1)^{r}}$ is the function $\tan$ (resp. $\tanh$) when $r=0$ (resp.
$r=1$).

###### Proof.

Let $\alpha(x_{1},x_{2})$ be a solution of (1.11). Let $\alpha^{\prime}$
(resp. $\alpha^{\prime\prime}$) be a solution of (1.11) associated to $\alpha$
by the Bäcklund transformation (1.12) for parameters
$\\{\lambda_{1}=S_{-\epsilon\delta}(\phi_{1}),\;\Lambda_{1}=C_{-\epsilon\delta}(\phi_{1})\\}$
(resp.
$\\{\lambda_{2}=S_{-\epsilon\delta}(\phi_{2}),\;\Lambda_{2}=C_{-\epsilon\delta}(\phi_{2})\\}$),
i.e., $\alpha^{\prime}$ and $\alpha^{\prime\prime}$ satisfy

$\left\\{\begin{array}[]{l}\alpha^{\prime}_{x_{1}}+\alpha_{x_{2}}=\frac{2}{\lambda_{1}}S_{l}\left(\frac{\alpha^{\prime}}{2}\right)C_{l}\left(\frac{\alpha}{2}\right)-\frac{2(-1)^{s}\tau\Lambda_{1}}{\lambda_{1}}C_{l}\left(\frac{\alpha^{\prime}}{2}\right)S_{l}\left(\frac{\alpha}{2}\right),\\\
\alpha^{\prime}_{x_{2}}+\alpha_{x_{1}}=-\frac{2}{\lambda_{1}}C_{l}\left(\frac{\alpha^{\prime}}{2}\right)S_{l}\left(\frac{\alpha}{2}\right)-\frac{2\delta\tau\Lambda_{1}}{\lambda_{1}}S_{l}\left(\frac{\alpha^{\prime}}{2}\right)C_{l}\left(\frac{\alpha}{2}\right),\\\
\end{array}\right.$ (2.2)
$\left\\{\begin{array}[]{l}\alpha^{\prime\prime}_{x_{1}}+\alpha_{x_{2}}=\frac{2}{\lambda_{2}}S_{l}\left(\frac{\alpha^{\prime\prime}}{2}\right)C_{l}\left(\frac{\alpha}{2}\right)-\frac{2(-1)^{s}\tau\Lambda_{2}}{\lambda_{2}}C_{l}\left(\frac{\alpha^{\prime\prime}}{2}\right)S_{l}\left(\frac{\alpha}{2}\right),\\\
\alpha^{\prime\prime}_{x_{2}}+\alpha_{x_{1}}=-\frac{2}{\lambda_{2}}C_{l}\left(\frac{\alpha^{\prime\prime}}{2}\right)S_{l}\left(\frac{\alpha}{2}\right)-\frac{2\delta\tau\Lambda_{2}}{\lambda_{2}}S_{l}\left(\frac{\alpha^{\prime\prime}}{2}\right)C_{l}\left(\frac{\alpha}{2}\right),\\\
\end{array}\right.$ (2.3)

where $\,l=(-1)^{r}$.

Assume there exists $\alpha^{*}$ associated to $\alpha^{\prime}$ (resp.
$\alpha^{\prime\prime}$) by the Bäcklund transformation with parameters
$\\{\lambda_{2},\;\Lambda_{2}\\}$ (resp. $\\{\lambda_{1},\;\Lambda_{1}\\}$).
then $\alpha^{*}$ should satisfy the following systems of equations

$\left\\{\begin{array}[]{l}\alpha^{*}_{x_{1}}+\alpha^{\prime}_{x_{2}}=\frac{2}{\lambda_{2}}S_{l}\left(\frac{\alpha^{*}}{2}\right)C_{l}\left(\frac{\alpha^{\prime}}{2}\right)-\frac{2(-1)^{s}\tau\Lambda_{2}}{\lambda_{2}}C_{l}\left(\frac{\alpha^{*}}{2}\right)S_{l}\left(\frac{\alpha^{\prime}}{2}\right),\\\
\alpha^{*}_{x_{2}}+\alpha^{\prime}_{x_{1}}=-\frac{2}{\lambda_{2}}C_{l}\left(\frac{\alpha^{*}}{2}\right)S_{l}\left(\frac{\alpha^{\prime}}{2}\right)-\frac{2\delta\tau\Lambda_{2}}{\lambda_{2}}S_{l}\left(\frac{\alpha^{*}}{2}\right)C_{l}\left(\frac{\alpha^{\prime}}{2}\right),\\\
\end{array}\right.$ (2.4)

$\left\\{\begin{array}[]{l}\alpha^{*}_{x_{1}}+\alpha^{\prime\prime}_{x_{2}}=\frac{2}{\lambda_{1}}S_{l}\left(\frac{\alpha^{*}}{2}\right)C_{l}\left(\frac{\alpha^{\prime\prime}}{2}\right)-\frac{2(-1)^{s}\tau\Lambda_{1}}{\lambda_{1}}C_{l}\left(\frac{\alpha^{*}}{2}\right)S_{l}\left(\frac{\alpha^{\prime\prime}}{2}\right),\\\
\alpha^{*}_{x_{2}}+\alpha^{\prime\prime}_{x_{1}}=-\frac{2}{\lambda_{1}}C_{l}\left(\frac{\alpha^{*}}{2}\right)S_{l}\left(\frac{\alpha^{\prime\prime}}{2}\right)-\frac{2\delta\tau\Lambda_{1}}{\lambda_{1}}S_{l}\left(\frac{\alpha^{*}}{2}\right)C_{l}\left(\frac{\alpha^{\prime\prime}}{2}\right).\\\
\end{array}\right.$ (2.5)

Subtracting the first (resp. second) equation of (2.3) from the first (resp.
second) equation of (2.2), we obtain

$\begin{array}[]{ll}\alpha^{\prime}_{x_{1}}-\alpha^{\prime\prime}_{x_{1}}=&\frac{2}{\lambda_{1}}S_{l}\left(\frac{\alpha^{\prime}}{2}\right)C_{l}\left(\frac{\alpha}{2}\right)-\frac{2(-1)^{s}\tau\Lambda_{1}}{\lambda_{1}}C_{l}\left(\frac{\alpha^{\prime}}{2}\right)S_{l}\left(\frac{\alpha}{2}\right)\\\
&-\frac{2}{\lambda_{2}}S_{l}\left(\frac{\alpha^{\prime\prime}}{2}\right)C_{l}\left(\frac{\alpha}{2}\right)+\frac{2(-1)^{s}\tau\Lambda_{2}}{\lambda_{2}}C_{l}\left(\frac{\alpha^{\prime\prime}}{2}\right)S_{l}\left(\frac{\alpha}{2}\right),\end{array}$
(2.6)

$\begin{array}[]{ll}\alpha^{\prime}_{x_{2}}-\alpha^{\prime\prime}_{x_{2}}=&-\frac{2}{\lambda_{1}}C_{l}\left(\frac{\alpha^{\prime}}{2}\right)S_{l}\left(\frac{\alpha}{2}\right)-\frac{2\delta\tau\Lambda_{1}}{\lambda_{1}}S_{l}\left(\frac{\alpha^{\prime}}{2}\right)C_{l}\left(\frac{\alpha}{2}\right)\\\
&+\frac{2}{\lambda_{2}}C_{l}\left(\frac{\alpha^{\prime\prime}}{2}\right)S_{l}\left(\frac{\alpha}{2}\right)+\frac{2\delta\tau\Lambda_{2}}{\lambda_{2}}S_{l}\left(\frac{\alpha^{\prime\prime}}{2}\right)C_{l}\left(\frac{\alpha}{2}\right).\end{array}$
(2.7)

Similarly, subtracting the first (resp. second) equation of (2.5) from the
first (resp. second) equation of (2.4), we obtain

$\begin{array}[]{ll}\alpha^{\prime}_{x_{2}}-\alpha^{\prime\prime}_{x_{2}}=&\frac{2}{\lambda_{2}}S_{l}\left(\frac{\alpha^{*}}{2}\right)C_{l}\left(\frac{\alpha^{\prime}}{2}\right)+\frac{2\delta\tau\Lambda_{2}}{\lambda_{2}}C_{l}\left(\frac{\alpha^{*}}{2}\right)S_{l}\left(\frac{\alpha^{\prime}}{2}\right)\\\
&-\frac{2}{\lambda_{1}}S_{l}\left(\frac{\alpha^{*}}{2}\right)C_{l}\left(\frac{\alpha^{\prime\prime}}{2}\right)-\frac{2\delta\tau\Lambda_{1}}{\lambda_{1}}C_{l}\left(\frac{\alpha^{*}}{2}\right)S_{l}\left(\frac{\alpha^{\prime\prime}}{2}\right),\end{array}$
(2.8)
$\begin{array}[]{ll}\alpha^{\prime}_{x_{1}}-\alpha^{\prime\prime}_{x_{1}}=&-\frac{2}{\lambda_{2}}C_{l}\left(\frac{\alpha^{*}}{2}\right)S_{l}\left(\frac{\alpha^{\prime}}{2}\right)-\frac{2\delta\tau\Lambda_{2}}{\lambda_{2}}S_{l}\left(\frac{\alpha^{*}}{2}\right)C_{l}\left(\frac{\alpha^{\prime}}{2}\right)\\\
&+\frac{2}{\lambda_{1}}C_{l}\left(\frac{\alpha^{*}}{2}\right)S_{l}\left(\frac{\alpha^{\prime\prime}}{2}\right)+\frac{2\delta\tau\Lambda_{1}}{\lambda_{1}}S_{l}\left(\frac{\alpha^{*}}{2}\right)C_{l}\left(\frac{\alpha^{\prime\prime}}{2}\right).\\\
\end{array}$ (2.9)

Now comparing equations (2.6) with (2.9) and multiplying all terms by
$\lambda_{1}\lambda_{2}$ we get

$\begin{array}[]{l}\lambda_{1}\left[C_{l}(\frac{\alpha^{*}}{2})S_{l}(\frac{\alpha^{\prime}}{2})-S_{l}(\frac{\alpha^{\prime\prime}}{2})C_{l}(\frac{\alpha}{2})\right]+\lambda_{1}\Lambda_{2}\delta\tau\left[S_{l}(\frac{\alpha^{*}}{2})C_{l}(\frac{\alpha^{\prime}}{2})-C_{l}(\frac{\alpha^{\prime\prime}}{2})S_{l}(\frac{\alpha}{2})\right]+\\\
\lambda_{2}\left[S_{l}(\frac{\alpha^{\prime}}{2})C_{l}(\frac{\alpha}{2})-C_{l}(\frac{\alpha^{*}}{2})S_{l}(\frac{\alpha^{\prime\prime}}{2})\right]+\lambda_{2}\Lambda_{1}\delta\tau\left[C_{l}(\frac{\alpha^{\prime}}{2})S_{l}(\frac{\alpha}{2})-S_{l}(\frac{\alpha^{*}}{2})C_{l}(\frac{\alpha^{\prime\prime}}{2})\right]=0.\end{array}$
(2.10)

where we have used the fact that $\delta=(-1)^{s+1}$.

Using trigonometric identities, the above equation reduces to

$\displaystyle\quad\left[S_{l}(\frac{\alpha^{*}+\alpha^{\prime}}{2})-S_{l}(\frac{\alpha^{\prime\prime}+\alpha}{2})\right](\lambda_{1}+\delta\tau\lambda_{1}\Lambda_{2})+$
$\displaystyle+\left[S_{l}(\frac{\alpha^{*}-\alpha^{\prime}}{2})-S_{l}(\frac{\alpha+\alpha^{\prime\prime}}{2})\right](-\lambda_{1}+\delta\tau\lambda_{1}\Lambda_{2})+$
$\displaystyle+\left[S_{l}(\frac{\alpha+\alpha^{\prime}}{2})-S_{l}(\frac{\alpha^{*}+\alpha^{\prime}}{2})\right](\lambda_{2}+\delta\tau\lambda_{2}\Lambda_{1})+$
$\displaystyle+\left[S_{l}(\frac{\alpha-\alpha^{\prime}}{2})-S_{l}(\frac{\alpha^{*}-\alpha^{\prime\prime}}{2})\right](-\lambda_{2}+\delta\tau\lambda_{2}\Lambda_{1})=0,$

which can be rewritten as

$\displaystyle\quad
C_{l}(\frac{\alpha^{*}+\alpha+\alpha^{\prime}+\alpha^{\prime\prime}}{4})\;S_{l}(\frac{\alpha^{*}-\alpha+\alpha^{\prime}-\alpha^{\prime\prime}}{4})\,(\lambda_{1}+\delta\tau\lambda_{1}\Lambda_{2})+$
$\displaystyle+C_{l}(\frac{\alpha^{*}+\alpha-(\alpha^{\prime}+\alpha^{\prime\prime})}{4})\;S_{l}(\frac{\alpha^{*}-\alpha-(\alpha^{\prime}-\alpha^{\prime\prime})}{4})\,(-\lambda_{1}+\delta\tau\lambda_{1}\Lambda_{2})+$
$\displaystyle+C_{l}(\frac{\alpha^{*}+\alpha+\alpha^{\prime}+\alpha^{\prime\prime}}{4})\;S_{l}(\frac{-(\alpha^{*}-\alpha)+\alpha^{\prime}-\alpha^{\prime\prime}}{4})\,(\lambda_{2}+\delta\tau\lambda_{2}\Lambda_{1})+$
$\displaystyle+C_{l}(\frac{\alpha^{*}+\alpha-(\alpha^{\prime}+\alpha^{\prime\prime})}{4})\;S_{l}(\frac{-(\alpha^{*}-\alpha)-(\alpha^{\prime}-\alpha^{\prime\prime})}{4})\,(-\lambda_{2}+\delta\tau\lambda_{2}\Lambda_{1})=0.$

By applying again trigonometric identities to each term $C_{l}$ and $S_{l}$
above we obtain the equation in terms of $S_{l}$ and $C_{l}$ of the
expressions

$\frac{\alpha^{*}-\alpha}{4},\qquad\frac{\alpha^{\prime}-\alpha^{\prime\prime}}{4},\qquad\frac{\alpha^{*}+\alpha}{4},\qquad\frac{\alpha^{\prime}+\alpha^{\prime\prime}}{4}.$

After cancellations it reduces to

$\displaystyle\delta\tau
C_{l}(\frac{\alpha^{*}+\alpha}{4})C_{l}(\frac{\alpha^{\prime}+\alpha^{\prime\prime}}{4})S_{l}(\frac{\alpha^{*}-\alpha}{4})C_{l}(\frac{\alpha^{\prime}-\alpha^{\prime\prime}}{4})(\lambda_{1}\Lambda_{2}-\lambda_{2}\Lambda_{1})-$
$\displaystyle-l\delta\tau
S_{l}(\frac{\alpha^{*}+\alpha}{4})S_{l}(\frac{\alpha^{\prime}+\alpha^{\prime\prime}}{4})C_{l}(\frac{\alpha^{*}-\alpha}{4})S_{l}(\frac{\alpha^{\prime}-\alpha^{\prime\prime}}{4})(\lambda_{1}\Lambda_{2}+\lambda_{2}\Lambda_{1})+$
$\displaystyle+C_{l}(\frac{\alpha^{*}+\alpha}{4})C_{l}(\frac{\alpha^{\prime}+\alpha^{\prime\prime}}{4})C_{l}(\frac{\alpha^{*}-\alpha}{4})S_{l}(\frac{\alpha^{\prime}-\alpha^{\prime\prime}}{4})(\lambda_{1}+\lambda_{2})-$
$\displaystyle-
lS_{l}(\frac{\alpha^{*}+\alpha}{4})S_{l}(\frac{\alpha^{\prime}+\alpha^{\prime\prime}}{4})S_{l}(\frac{\alpha^{*}-\alpha}{4})C_{l}(\frac{\alpha^{\prime}-\alpha^{\prime\prime}}{4})(\lambda_{1}-\lambda_{2})=0.$

Dividing by

$C_{l}(\frac{\alpha^{*}+\alpha}{4})C_{l}(\frac{\alpha^{\prime}+\alpha^{\prime\prime}}{4})C_{l}(\frac{\alpha^{*}-\alpha}{4})C_{l}(\frac{\alpha^{\prime}-\alpha^{\prime\prime}}{4}),$

and introducing the notation $T_{l}=\displaystyle\frac{S_{l}}{C_{l}}$, the
equation above reduces to

$\displaystyle\delta\tau
T_{l}(\frac{\alpha^{*}-\alpha}{4})(\lambda_{1}\Lambda_{2}-\lambda_{2}\Lambda_{1})-$
$\displaystyle-l\delta\tau
T_{l}(\frac{\alpha^{*}+\alpha}{4})T_{l}(\frac{\alpha^{\prime}+\alpha^{\prime\prime}}{4})T_{l}(\frac{\alpha^{\prime}-\alpha^{\prime\prime}}{4})(\lambda_{1}\Lambda_{2}+\lambda_{2}\Lambda_{1})+$
$\displaystyle+T_{l}(\frac{\alpha^{\prime}-\alpha^{\prime\prime}}{4})(\lambda_{1}+\lambda_{2})-$
$\displaystyle-
lT_{l}(\frac{\alpha^{*}+\alpha}{4})T_{l}(\frac{\alpha^{\prime}+\alpha^{\prime\prime}}{4})T_{l}(\frac{\alpha^{*}-\alpha}{4})(\lambda_{1}-\lambda_{2})=0.$
(2.11)

Since

$\Lambda_{i}=C_{-\delta\epsilon}(\phi_{i}),\qquad\lambda_{i}=S_{-\delta\epsilon}(\phi_{i}),\qquad
i=1,2.$

we have that

$\lambda_{1}\Lambda_{2}\pm\lambda_{2}\Lambda_{1}=2S_{-\delta\epsilon}(\frac{\phi_{1}\pm\phi_{2}}{2})C_{-\delta\epsilon}(\frac{\phi_{1}\pm\phi_{2}}{2}),$
$\lambda_{1}\pm\lambda_{2}=2S_{-\delta\epsilon}(\frac{\phi_{1}\pm\phi_{2}}{2})C_{-\delta\epsilon}(\frac{\phi_{1}\mp\phi_{2}}{2}).$

Substituting these expressions into (2), we conclude that the following
product vanishes

$\displaystyle\left[T_{l}(\frac{\alpha^{*}-\alpha}{4})S_{-\delta\epsilon}(\frac{\phi_{2}-\phi_{1}}{2})-\delta\tau
T_{l}(\frac{\alpha^{\prime}-\alpha^{\prime\prime}}{4})S_{-\delta\epsilon}(\frac{\phi_{2}+\phi_{1}}{2})\right]\cdot$
$\displaystyle\cdot\left[-\delta\tau
C_{-\delta\epsilon}(\frac{\phi_{1}-\phi_{2}}{2})+lT_{l}(\frac{\alpha^{*}+\alpha}{4})T_{l}(\frac{\alpha^{\prime}+\alpha^{\prime\prime}}{4})C_{-\delta\epsilon}(\frac{\phi_{1}+\phi_{2}}{2})\right]=0.$

Similar computations, comparing equations (2.7) and (2.8) show that

$\displaystyle\left[T_{l}(\frac{\alpha^{*}-\alpha}{4})S_{-\delta\epsilon}(\frac{\phi_{2}-\phi_{1}}{2})-\delta\tau
T_{l}(\frac{\alpha^{\prime}-\alpha^{\prime\prime}}{4})S_{-\delta\epsilon}(\frac{\phi_{2}+\phi_{1}}{2})\right]\cdot$
$\displaystyle\cdot\left[C_{-\delta\epsilon}(\frac{\phi_{1}+\phi_{2}}{2})-\delta\tau
lT_{l}(\frac{\alpha^{*}+\alpha}{4})T_{l}(\frac{\alpha^{\prime}+\alpha^{\prime\prime}}{4})C_{-\delta\epsilon}(\frac{\phi_{1}-\phi_{2}}{2})\right]=0.$

From the last two equations we conclude that

$T_{l}(\frac{\alpha^{*}-\alpha}{4})S_{-\delta\epsilon}(\frac{\phi_{2}-\phi_{1}}{2})-\delta\tau
T_{l}(\frac{\alpha^{\prime}-\alpha^{\prime\prime}}{4})S_{-\delta\epsilon}(\frac{\phi_{2}+\phi_{1}}{2})=0,$

which is exactly the superposition formula given by (2.1).

It remains to prove that $\alpha^{*}$ defined by (2.1) satisfies (1.11). We
are assuming that $\alpha^{\prime}$ and $\alpha^{\prime\prime}$ satisfy (2.2)
and (2.3), respectively and $\alpha$ satisfies (1.11). Theorem 1.4 implies
that $\alpha^{\prime}$ and $\alpha^{\prime\prime}$ also satisfy (1.11). We
introduce the following notation

$Z:=\frac{S_{-\delta\epsilon}(\frac{\phi_{2}+\phi_{1}}{2})}{S_{-\delta\epsilon}(\frac{\phi_{2}-\phi_{1}}{2})},\qquad
D:=\alpha^{\prime\prime}-\alpha^{\prime},\qquad\Sigma:=\alpha^{\prime\prime}+\alpha^{\prime},\qquad
G:=-\delta\tau\,Z\,T_{l}\left(\frac{D}{4}\right).$

It follows from (2.1) that

$\alpha^{*}_{x_{i}x_{i}}=\alpha_{x_{i}x_{i}}+\frac{4}{1+l\,G^{2}}\left(\frac{-2l\,G}{1+l\,G^{2}}(G_{x_{i}})^{2}+G_{x_{i}x_{i}}\right),\qquad
i=1,2.$ (2.12)

Therefore proving that $\alpha^{*}$ satisfies (1.11) is equivalent to proving

$\frac{4}{1+l\,G^{2}}\left(\frac{-2l\,G}{1+l\,G^{2}}[(G_{x_{1}})^{2}-(G_{x_{2}})^{2}]+G_{x_{1}x_{1}}-G_{x_{2}x_{2}}\right)+\epsilon\delta\left(S_{l}(\alpha^{*})-S_{l}(\alpha)\right)=0,$
(2.13)

where we used the fact that $\alpha$ is a solution of (1.11).

Observe that

$S_{l}(\alpha^{*})-S_{l}(\alpha)=2S_{l}(\frac{\alpha^{*}-\alpha}{2})\,C_{l}(\frac{\alpha^{*}+\alpha}{2})=\frac{4G}{1+l\,G^{2}}C_{l}(\frac{\alpha^{*}+\alpha}{2}),$

and it follows from (2.1) and a long computation that

$C_{l}(\frac{\alpha^{*}+\alpha}{2})=\frac{1}{1+l\,G^{2}}\left[(1-l\,G^{2})C_{l}(\alpha)-2l\,GS_{l}(\alpha)\right].$

Therefore, (2.13) reduces to

$-2l[(G_{x_{1}})^{2}-(G_{x_{2}})^{2}]+\frac{(1+l\,G^{2})}{G}(G_{x_{1}x_{1}}-G_{x_{2}x_{2}})+\epsilon\delta\left[(1-l\,G^{2})C_{l}(\alpha)-2l\,GS_{l}(\alpha)\right]=0.$
(2.14)

Using the expression of $G$, we have

$G_{x_{i}}=\frac{G\,D_{x_{i}}}{2S_{l}(\frac{D}{2})}\qquad\quad
G_{x_{i}x_{i}}=\frac{G}{2S_{l}(\frac{D}{2})}\left(D_{x_{i}x_{i}}+\frac{l}{2}T_{l}\left(\frac{D}{4}\right)(D_{x_{i}})^{2}\right),\;\;i=1,2.$

Moreover,

$\frac{l}{4S_{l}(\frac{D}{2})}\left(\frac{-2G^{2}}{S_{l}(\frac{D}{2})}+(1+l\,G^{2})T_{l}(\frac{D}{4})\right)=\frac{-l(Z^{2}-1)}{8(C_{l}(\frac{D}{4}))^{2}}.$

Substituting these expressions into (2.14), we conclude that we need to prove
that

$\displaystyle\frac{-l(Z^{2}-1)}{8C_{l}^{2}(\frac{D}{4})}\,[(D_{x_{1}})^{2}-(D_{x_{2}})^{2}]+\frac{1+l\,G^{2}}{2S_{l}(\frac{D}{2})}(D_{x_{1}x_{1}}-D_{x_{2}x_{2}})+$
$\displaystyle+\epsilon\delta\left[(1-l\,G^{2})C_{l}(\alpha)-2l\,GS_{l}(\alpha)\right]=0.$
(2.15)

Since $\alpha^{\prime}$ and $\alpha^{\prime\prime}$ satisfy (1.11), it follows
that

$D_{x_{1}x_{1}}-D_{x_{2}x_{2}}=-2\epsilon\delta
S_{l}\left(\frac{D}{2}\right)C_{l}\left(\frac{\Sigma}{2}\right),$ (2.16)

where $\Sigma=\alpha^{\prime\prime}+\alpha^{\prime}$. Moreover, since
$\alpha^{\prime}$ and $\alpha^{\prime\prime}$ satisfy (2.2) and (2.3)
respectively, a long but straightforward computation shows that

$\displaystyle(D_{x_{1}})^{2}-(D_{x_{2}})^{2}=\frac{-4}{\lambda_{1}^{2}\lambda_{2}^{2}}\left[-\left(C_{l}^{2}(\frac{\alpha}{2})S_{l}^{2}(\frac{\alpha^{\prime}}{2})-C_{l}^{2}(\frac{\alpha^{\prime}}{2})S_{l}^{2}(\frac{\alpha}{2})\right)(1-\Lambda_{1}^{2})\lambda_{2}^{2}\right.$
$\displaystyle-\left(C_{l}^{2}(\frac{\alpha}{2})S_{l}^{2}(\frac{\alpha^{\prime\prime}}{2})-C_{l}^{2}(\frac{\alpha^{\prime\prime}}{2})S_{l}^{2}(\frac{\alpha}{2})\right)(1-\Lambda_{2}^{2})\lambda_{1}^{2}+$
$\displaystyle+2\left(C_{l}^{2}(\frac{\alpha}{2})S_{l}(\frac{\alpha^{\prime}}{2})S_{l}(\frac{\alpha^{\prime\prime}}{2})-C_{l}(\frac{\alpha^{\prime}}{2})C_{l}(\frac{\alpha^{\prime\prime}}{2})S_{l}^{2}(\frac{\alpha}{2})\right)(1-\Lambda_{1}\Lambda_{2})\lambda_{1}\lambda_{2}$
$\displaystyle\left.-S_{l}(\alpha)S_{l}(\frac{D}{2})(\Lambda_{2}-\Lambda_{1})\delta\tau\lambda_{1}\lambda_{2}\right].$

Therefore, it follows from the fact that

$1-\Lambda_{i}^{2}=-\epsilon\delta\lambda_{i}^{2},\qquad
1-\Lambda_{1}\Lambda_{2}=\frac{-\epsilon\delta(Z^{2}+1)}{Z^{2}-1}\lambda_{1}\lambda_{2},\qquad\Lambda_{2}-\Lambda_{1}=\frac{2\epsilon\delta
Z}{Z^{2}-1}\lambda_{1}\lambda_{2},$

that

$\displaystyle(D_{x_{1}})^{2}-(D_{x_{2}})^{2}=\frac{-16\epsilon\delta}{Z^{2}-1}\left[\left(C_{l}^{2}(\frac{\alpha}{2})C_{l}^{2}(\frac{\Sigma}{4})-S_{l}^{2}(\frac{\alpha}{2})S_{l}^{2}(\frac{\Sigma}{4})\right)S_{l}^{2}(\frac{D}{4})Z^{2}\right.$
$\displaystyle-\left(C_{l}^{2}(\frac{\alpha}{2})S_{l}^{2}(\frac{\Sigma}{4})-S_{l}^{2}(\frac{\alpha}{2})C_{l}^{2}(\frac{\Sigma}{4})\right)C_{l}^{2}(\frac{D}{4})$
$\displaystyle\left.-\frac{1}{2}S_{l}(\alpha)S_{l}(\frac{D}{2})\delta\tau
Z\right],$

where we used the fact that
$S_{l}^{2}(\frac{D}{4})Z^{2}=G^{2}C_{l}^{2}(\frac{D}{4})$.

Substituting this expression and (2.16) into (2), we obtain a second degree
polynomial in $G$, whose coefficients vanish identically. ∎

Theorem 2.1 shows that the composition of Bäcklund transformations, with
distinct parameters, is commutative. Moreover, starting with a solution of the
sine-Gordon equation (resp. sinh-Gordon equation) on gets infinitely many
solutions of the same equation, by an algebraic expression, after the fist
integration of the Bäcklund transformation, as one can see in the following
diagram.

$\begin{array}[]{ccccccc}&&\alpha^{\prime}&\stackrel{{\scriptstyle\phi_{2}}}{{\longrightarrow}}&\bar{\alpha}&&\longrightarrow\\\
&\stackrel{{\scriptstyle\phi_{1}}}{{\nearrow}}&&\stackrel{{\scriptstyle\phi_{1}}}{{\nearrow}}&&\stackrel{{\scriptstyle\phi_{3}}}{{\searrow}}\\\
\alpha&\stackrel{{\scriptstyle\phi_{2}}}{{\longrightarrow}}&\alpha^{\prime\prime}&&&&\hat{\alpha}\\\
&\stackrel{{\scriptstyle\phi_{3}}}{{\searrow}}&&\stackrel{{\scriptstyle\phi_{3}}}{{\searrow}}&&\stackrel{{\scriptstyle\phi_{1}}}{{\nearrow}}\\\
&&\alpha^{\prime\prime\prime}&\stackrel{{\scriptstyle\phi_{2}}}{{\longrightarrow}}&\tilde{\alpha}&&\longrightarrow\end{array}$
(2.17)

Our next two theorems provide the superposition formulae for the elliptic
sinh-Gordon and the elliptic sine-Gordon equations. The Bäcklund
transformations for these equations, given in Corollary 1.4.2, provide a
solution of the elliptic sine-Gordon equation starting from a solution of the
elliptic sinh-Gordon equation when $r=0$ and vice-versa when $r=1$. However,
the superposition formula gives a new solution of the equation we started
with, before applying any Bäcklund transformation.

######  Theorem 2.2.

Let $\alpha(x_{1},x_{2})$ be a solution of

$\alpha_{x_{1}x_{1}}+\alpha_{x_{2}x_{2}}=\sinh\alpha.$ (2.18)

Let $\alpha^{\prime}$ and $\alpha^{\prime\prime}$ be solutions of the elliptic
sine-Gordon equation associated to $\alpha$ by $\phi_{1}$(resp.
$\phi_{2}$)-Bäcklund transformation (1.14) where $r=0$ and $\tau=\pm 1$. If
$\phi_{1}\neq\phi_{2}$, then there exists a unique solution $\alpha^{*}$ of
(2.18) which is associated to $\alpha^{\prime}$ and $\alpha^{\prime\prime}$ by
$\phi_{2}$(resp. $\phi_{1}$)-Bäcklund transformation (1.14), with $r=0$.
Moreover, $\alpha^{*}$ is given algebraically by

$\tanh\frac{\alpha^{*}}{2}=\frac{P-Q\,\tanh(\displaystyle{\frac{\alpha}{2}})}{Q-P\,\tanh(\displaystyle{\frac{\alpha}{2}})},$
(2.19)

where

$P=\tau(\sinh\phi_{2}-\sinh\phi_{1})\sin(\frac{\alpha^{\prime\prime}-\alpha^{\prime}}{2}),$
$Q=(1+\sinh\phi_{1}\sinh\phi_{2})\cos(\frac{\alpha^{\prime\prime}-\alpha^{\prime}}{2})-\cosh\phi_{1}\cosh\phi_{2}.$

###### Proof.

Let $\alpha$ be a solution of the elliptic sinh-Gordon equation i.e., (1.13)
with $r=0$. Let $\alpha^{\prime}$ and $\alpha^{\prime\prime}$ be solutions of
the elliptic sine-Gordon equation i.e., (1.15) with $r=0$, associated to
$\alpha$ by $\phi_{1}$(resp. $\phi_{2}$)-Bäcklund transformation (1.14) with
$r=0$, $\phi_{1}\neq\phi_{2}$. Then

$\left\\{\begin{array}[]{l}\alpha^{\prime}_{x_{1}}-\alpha_{x_{2}}=\frac{2}{\cosh\phi_{1}}\sin(\frac{\alpha^{\prime}}{2})\cosh(\frac{\alpha}{2})+\frac{2\tau\sinh\phi_{1}}{\cosh\phi_{1}}\cos(\frac{\alpha^{\prime}}{2})\sinh(\frac{\alpha}{2}),\\\
\alpha^{\prime}_{x_{2}}+\alpha_{x_{1}}=-\frac{2}{\cosh\phi_{1}}\cos(\frac{\alpha^{\prime}}{2})\sinh(\frac{\alpha}{2})+\frac{2\tau\sinh\phi_{1}}{\cosh\phi_{1}}\sin(\frac{\alpha^{\prime}}{2})\cosh(\frac{\alpha}{2}),\\\
\end{array}\right.$ (2.20)

$\left\\{\begin{array}[]{l}\alpha^{\prime\prime}_{x_{1}}-\alpha_{x_{2}}=\frac{2}{\cosh\phi_{2}}\sin(\frac{\alpha^{\prime\prime}}{2})\cosh(\frac{\alpha}{2})+\frac{2\tau\sinh\phi_{2}}{\cosh\phi_{2}}\cos(\frac{\alpha^{\prime\prime}}{2})\sinh(\frac{\alpha}{2}),\\\
\alpha^{\prime\prime}_{x_{2}}+\alpha_{x_{1}}=-\frac{2}{\cosh\phi_{2}}\cos(\frac{\alpha^{\prime\prime}}{2})\sinh(\frac{\alpha}{2})+\frac{2\tau\sinh\phi_{2}}{\cosh\phi_{2}}\sin(\frac{\alpha^{\prime\prime}}{2})\cosh(\frac{\alpha}{2}).\\\
\end{array}\right.$ (2.21)

Suppose $\alpha^{*}$ exists then it should satisfy the following systems of
equations

$\left\\{\begin{array}[]{l}\alpha^{*}_{x_{1}}-\alpha^{\prime}_{x_{2}}=\frac{2}{\cosh\phi_{2}}\sinh(\frac{\alpha^{*}}{2})\cos(\frac{\alpha^{\prime}}{2})+\frac{2\tau\sinh\phi_{2}}{\cosh\phi_{2}}\cosh(\frac{\alpha^{*}}{2})\sin(\frac{\alpha^{\prime}}{2}),\\\
\alpha^{*}_{x_{2}}+\alpha^{\prime}_{x_{1}}=-\frac{2}{\cosh\phi_{2}}\cosh(\frac{\alpha^{*}}{2})\sin(\frac{\alpha^{\prime}}{2})+\frac{2\tau\sinh\phi_{2}}{\cosh\phi_{2}}\sinh(\frac{\alpha^{*}}{2})\cos(\frac{\alpha^{\prime}}{2}),\\\
\end{array}\right.$ (2.22)
$\left\\{\begin{array}[]{l}\alpha^{*}_{x_{1}}-\alpha^{\prime\prime}_{x_{2}}=\frac{2}{\cosh\phi_{1}}\sinh(\frac{\alpha^{*}}{2})\cos(\frac{\alpha^{\prime\prime}}{2})+\frac{2\tau\sinh\phi_{1}}{\cosh\phi_{1}}\cosh(\frac{\alpha^{*}}{2})\sin(\frac{\alpha^{\prime\prime}}{2}),\\\
\alpha^{*}_{x_{2}}+\alpha^{\prime\prime}_{x_{1}}=-\frac{2}{\cosh\phi_{1}}\cosh(\frac{\alpha^{*}}{2})\sin(\frac{\alpha^{\prime\prime}}{2})+\frac{2\tau\sinh\phi_{1}}{\cosh\phi_{1}}\sinh(\frac{\alpha^{*}}{2})\cos(\frac{\alpha^{\prime\prime}}{2}).\\\
\end{array}\right.$ (2.23)

Subtracting the first (resp. second) equation of (2.21) from the first (resp.
second) equation of (2.20) we get the system of equations

$\begin{array}[]{l}\alpha^{\prime}_{x_{1}}-\alpha^{\prime\prime}_{x_{1}}=\frac{2}{\cosh\phi_{1}}\sin(\frac{\alpha^{\prime}}{2})\cosh(\frac{\alpha}{2})+\frac{2\tau\sinh\phi_{1}}{\cosh\phi_{1}}\cos(\frac{\alpha^{\prime}}{2})\sinh(\frac{\alpha}{2})\\\
\qquad\qquad-\frac{2}{\cosh\phi_{2}}\sin(\frac{\alpha^{\prime\prime}}{2})\cosh(\frac{\alpha}{2})-\frac{2\tau\sinh\phi_{2}}{\cosh\phi_{2}}\cos(\frac{\alpha^{\prime\prime}}{2})\sinh(\frac{\alpha}{2}),\\\
\alpha^{\prime}_{x_{2}}-\alpha^{\prime\prime}_{x_{2}}=-\frac{2}{\cosh\phi_{1}}\cos(\frac{\alpha^{\prime}}{2})\sinh(\frac{\alpha}{2})+\frac{2\tau\sinh\phi_{1}}{\cosh\phi_{1}}\sin(\frac{\alpha^{\prime}}{2})\cosh(\frac{\alpha}{2})\\\
\qquad\qquad+\frac{2}{\cosh\phi_{2}}\cos(\frac{\alpha^{\prime\prime}}{2})\sinh(\frac{\alpha}{2})-\frac{2\tau\sinh\phi_{2}}{\cosh\phi_{2}}\sin(\frac{\alpha^{\prime\prime}}{2})\cosh(\frac{\alpha}{2}).\end{array}$
(2.24)

Similarly, subtracting the first (resp. second) equation of (2.23) from the
first (resp. second) equation of (2.22) we get

$\begin{array}[]{l}\alpha^{\prime}_{x_{2}}-\alpha^{\prime\prime}_{x_{2}}=-\frac{2}{\cosh\phi_{2}}\sinh(\frac{\alpha^{*}}{2})\cos(\frac{\alpha^{\prime}}{2})-\frac{2\tau\sinh\phi_{2}}{\cosh\phi_{2}}\cosh(\frac{\alpha^{*}}{2})\sin(\frac{\alpha^{\prime}}{2})\\\
\qquad\qquad+\frac{2}{\cosh\phi_{1}}\sinh(\frac{\alpha^{*}}{2})\cos(\frac{\alpha^{\prime\prime}}{2})+\frac{2\tau\sinh\phi_{1}}{\cosh\phi_{1}}\cosh(\frac{\alpha^{*}}{2})\sin(\frac{\alpha^{\prime\prime}}{2}),\\\
\alpha^{\prime}_{x_{1}}-\alpha^{\prime\prime}_{x_{1}}=-\frac{2}{\cosh\phi_{2}}\cosh(\frac{\alpha^{*}}{2})\sin(\frac{\alpha^{\prime}}{2})+\frac{2\tau\sinh\phi_{2}}{\cosh\phi_{2}}\sinh(\frac{\alpha^{*}}{2})\cos(\frac{\alpha^{\prime}}{2})\\\
\qquad\qquad+\frac{2}{\cosh\phi_{1}}\cosh(\frac{\alpha^{*}}{2})\sin(\frac{\alpha^{\prime\prime}}{2})-\frac{2\tau\sinh\phi_{1}}{\cosh\phi_{1}}\sinh(\frac{\alpha^{*}}{2})\cos(\frac{\alpha^{\prime\prime}}{2}).\end{array}$
(2.25)

Comparing equation (2.24) with (2.25), we obtain two algebraic expressions
that must be satisfied by $\cosh(\alpha^{*}/2)$ and $\sinh(\alpha^{*}/2)$. By
solving this system of algebraic equations, we conclude that $\alpha^{*}/2$ is
given by (2.19).

Now we need to prove that $\alpha^{*}$ satisfies the elliptic sinh-Gordon
equation (2.18). Recall that $\alpha^{\prime}$ and $\alpha^{\prime\prime}$ are
solutions of the elliptic sine-Gordon equation. We introduce the following
notation

$\begin{array}[]{lcl}A=\tau(\sinh\phi_{2}-\sinh\phi_{1}),&B=\cosh\phi_{1}\cosh\phi_{2},&L=1+\sinh\phi_{1}\sinh\phi_{2},\\\
&D=\alpha^{\prime\prime}-\alpha^{\prime}.&\end{array}$ (2.26)

Then the functions $P$ and $Q$ are rewritten as

$P=A\sin\left(\frac{D}{2}\right),\qquad
Q=L\cos\left(\frac{D}{2}\right)-B\quad\mbox{and}\quad B^{2}-L^{2}=A^{2}.$
(2.27)

We denote the right hand side of (2.19) by

$H=\frac{G-\tanh(\frac{\alpha}{2})}{1-G\tanh(\frac{\alpha}{2})},\quad\mbox{where}\quad
G=\frac{P}{Q}.$ (2.28)

It follows that proving that $\alpha^{*}$ satisfies (2.18) is equivalent to
proving that

$\frac{2H}{1-H^{2}}|\nabla H|^{2}+\triangle H-H=0.$ (2.29)

Since $\alpha$ satisfies (2.18) it follows that

$\displaystyle\triangle\tanh(\frac{\alpha}{2})=\frac{1}{2\cosh^{2}(\frac{\alpha}{2})}\left(\sinh\alpha-\tanh(\frac{\alpha}{2})|\nabla\alpha|^{2}\right),$
$\displaystyle|\nabla\tanh(\frac{\alpha}{2})|^{2}=\frac{1}{4\cosh^{4}(\frac{\alpha}{2})}|\nabla\alpha|^{2}.$

Therefore, proving (2.29) reduces to showing that $G$ satisfies

$\frac{2G}{1-G^{2}}|\nabla G|^{2}+\triangle G-G(\cosh\alpha-G\sinh\alpha)=0.$
(2.30)

Long but straightforward computations show that

$\displaystyle\triangle
G=\frac{A}{2Q^{2}}\left(L-B\cos(\frac{D}{2})\right){\triangle
D}-\frac{A}{4Q^{3}}(BQ+2A^{2})\sin(\frac{D}{2})\,|\nabla D|^{2},$
$\displaystyle|\nabla
G|^{2}=\frac{A^{2}}{4Q^{4}}\left(L-B\cos(\frac{D}{2})\right)^{2}\,|\nabla
D|^{2},$ $\displaystyle
1-G^{2}=\frac{1}{Q^{2}}\left(L-B\cos(\frac{D}{2})\right)^{2}.$

Moreover, since $D=\alpha^{\prime\prime}-\alpha^{\prime}$ where
$\alpha^{\prime}$ and $\alpha^{\prime\prime}$ are solutions of the elliptic
sine-Gordon equation, it follows that

$\displaystyle\triangle
D=\sin\alpha^{\prime\prime}-\sin\alpha^{\prime}=2\sin\frac{D}{2}\cos(\frac{\alpha^{\prime\prime}+\alpha^{\prime}}{2}).$

Hence proving (2.30) reduces to proving

$-B|\nabla
D|^{2}+\frac{2}{\sin(\frac{D}{2})}\left(L-B\cos(\frac{D}{2})\right)\triangle
D-4Q\cosh\alpha+4A\sin(\frac{D}{2})\sinh\alpha=0.$ (2.31)

Now $\alpha^{\prime}$ and $\alpha^{\prime\prime}$ satisfy (2.20) and (2.21)
respectively. These systems of equations provide

$\displaystyle|\nabla D|^{2}$ $\displaystyle=$
$\displaystyle\frac{4}{B}\left[\cosh\alpha\left(B-L\cos(\frac{D}{2})\right)+\cos(\frac{\alpha^{\prime\prime}+\alpha^{\prime}}{2})\left(L-B\cos(\frac{D}{2})\right)+\right.$
$\displaystyle\qquad+\left.A\sin(\frac{D}{2})\sinh\alpha\right].$

This last expression proves that (2.31) is identically satisfied, which
concludes the proof that $\alpha^{*}$ satisfies the elliptic sinh-Gordon
equation ∎

The next result provides the superposition formula for the Bäcklund
transformation (1.14), when $r=1$, which gives solutions for the elliptic
sine-Gordon equation.

######  Theorem 2.3.

Let $\alpha(x_{1},x_{2})$ be a solution of

$\alpha_{x_{1}x_{1}}+\alpha_{x_{2}x_{2}}=\sin\alpha.$ (2.32)

Let $\alpha^{\prime}$ and $\alpha^{\prime\prime}$ be solutions of the elliptic
sinh-Gordon equation associated to $\alpha$ by a $\phi_{1}$(resp.
$\phi_{2}$)-Bäcklund transformation (1.14), where $r=1$ and $\tau=\pm 1$. If
$\phi_{1}\neq\phi_{2}$, then there exists a unique solution $\alpha^{*}$ of
(2.32) which is associated to $\alpha^{\prime}$ and $\alpha^{\prime\prime}$ by
the $\phi_{2}$(resp. $\phi_{1}$)-Bäcklund transformation (1.14), with $r=1$.
Moreover, $\alpha^{*}$ is given algebraically by

$\tan\frac{\alpha^{*}}{2}=\frac{\tilde{P}-\tilde{Q}\,\displaystyle{\tan(\frac{\alpha}{2})}}{\tilde{Q}+\tilde{P}\,\displaystyle{\tan(\frac{\alpha}{2})}},$
(2.33)

where

$\tilde{P}=-\tau(\sinh\phi_{1}-\sinh\phi_{2})\,\sinh(\frac{\alpha^{\prime\prime}-\alpha^{\prime}}{2})$
(2.34)

and

$\tilde{Q}=(1+\sinh\phi_{1}\sinh\phi_{2})\cosh\frac{\alpha^{\prime\prime}-\alpha^{\prime}}{2}-\cosh\phi_{1}\cosh\phi_{2}.$
(2.35)

###### Proof.

The proof is similar to the previous one. Let $\alpha$ be a solution of the
elliptic sine-Gordon equation i.e., (1.13) with $r=1$. Let $\alpha^{\prime}$
and $\alpha^{\prime\prime}$ be solutions of the elliptic sinh-Gordon (i.e.
(1.15) with $r=1$) associated to $\alpha$ by $\phi_{1}$(resp.
$\phi_{2}$)-Bäcklund transformation (1.14) with $r=1$, where
$\phi_{1}\neq\phi_{2}$, then

$\left\\{\begin{array}[]{l}\alpha^{\prime}_{x_{1}}-\alpha_{x_{2}}=\frac{2}{\cosh\phi_{1}}\sinh(\frac{\alpha^{\prime}}{2})\cos(\frac{\alpha}{2})+\frac{2\tau\sinh\phi_{1}}{\cosh\phi_{1}}\cosh(\frac{\alpha^{\prime}}{2})\sin(\frac{\alpha}{2}),\\\
\alpha^{\prime}_{x_{2}}+\alpha_{x_{1}}=-\frac{2}{\cosh\phi_{1}}\cosh(\frac{\alpha^{\prime}}{2})\sin(\frac{\alpha}{2})+\frac{2\tau\sinh\phi_{1}}{\cosh\phi_{1}}\sinh(\frac{\alpha^{\prime}}{2})\cos(\frac{\alpha}{2}),\\\
\end{array}\right.$ (2.36)

$\left\\{\begin{array}[]{l}\alpha^{\prime\prime}_{x_{1}}-\alpha_{x_{2}}=\frac{2}{\cosh\phi_{2}}\sinh(\frac{\alpha^{\prime\prime}}{2})\cos(\frac{\alpha}{2})+\frac{2\tau\sinh\phi_{2}}{\cosh\phi_{2}}\cosh(\frac{\alpha^{\prime\prime}}{2})\sin(\frac{\alpha}{2}),\\\
\alpha^{\prime\prime}_{x_{2}}+\alpha_{x_{1}}=-\frac{2}{\cosh\phi_{2}}\cosh(\frac{\alpha^{\prime\prime}}{2})\sin(\frac{\alpha}{2})+\frac{2\tau\sinh\phi_{2}}{\cosh\phi_{2}}\sinh(\frac{\alpha^{\prime\prime}}{2})\cos(\frac{\alpha}{2}).\\\
\end{array}\right.$ (2.37)

Suppose $\alpha^{*}$ exists then it should satisfy the following systems of
equations

$\left\\{\begin{array}[]{l}\alpha^{*}_{x_{1}}-\alpha^{\prime}_{x_{2}}=\frac{2}{\cosh\phi_{2}}\sin(\frac{\alpha^{*}}{2})\cosh(\frac{\alpha^{\prime}}{2})+\frac{2\tau\sinh\phi_{2}}{\cosh\phi_{2}}\cos(\frac{\alpha^{*}}{2})\sinh(\frac{\alpha^{\prime}}{2}),\\\
\alpha^{*}_{x_{2}}+\alpha^{\prime}_{x_{1}}=-\frac{2}{\cosh\phi_{2}}\cos(\frac{\alpha^{*}}{2})\sinh(\frac{\alpha^{\prime}}{2})+\frac{2\tau\sinh\phi_{2}}{\cosh\phi_{2}}\sin(\frac{\alpha^{*}}{2})\cosh(\frac{\alpha^{\prime}}{2}),\\\
\end{array}\right.$ (2.38)
$\left\\{\begin{array}[]{l}\alpha^{*}_{x_{1}}-\alpha^{\prime\prime}_{x_{2}}=\frac{2}{\cosh\phi_{1}}\sin(\frac{\alpha^{*}}{2})\cosh(\frac{\alpha^{\prime\prime}}{2})+\frac{2\tau\sinh\phi_{1}}{\cosh\phi_{1}}\cos(\frac{\alpha^{*}}{2})\sinh(\frac{\alpha^{\prime\prime}}{2}),\\\
\alpha^{*}_{x_{2}}+\alpha^{\prime\prime}_{x_{1}}=-\frac{2}{\cosh\phi_{1}}\cos(\frac{\alpha^{*}}{2})\sinh(\frac{\alpha^{\prime\prime}}{2})+\frac{2\tau\sinh\phi_{1}}{\cosh\phi_{1}}\sin(\frac{\alpha^{*}}{2})\cosh(\frac{\alpha^{\prime\prime}}{2}).\\\
\end{array}\right.$ (2.39)

Subtracting the first (resp. second) equation of (2.38) from the first (resp.
second) equation of (2.39) we get the system of equations

$\begin{array}[]{l}\alpha^{\prime}_{x_{1}}-\alpha^{\prime\prime}_{x_{1}}=\frac{2}{\cosh\phi_{1}}\sinh(\frac{\alpha^{\prime}}{2})\cos(\frac{\alpha}{2})+\frac{2\tau\sinh\phi_{1}}{\cosh\phi_{1}}\cosh(\frac{\alpha^{\prime}}{2})\sin(\frac{\alpha}{2})\\\
\qquad\qquad-\frac{2}{\cosh\phi_{2}}\sinh(\frac{\alpha^{\prime\prime}}{2})\cos(\frac{\alpha}{2})-\frac{2\tau\sinh\phi_{2}}{\cosh\phi_{2}}\cosh(\frac{\alpha^{\prime\prime}}{2})\sin(\frac{\alpha}{2}),\\\
\alpha^{\prime}_{x_{2}}-\alpha^{\prime\prime}_{x_{2}}=-\frac{2}{\cosh\phi_{1}}\cosh(\frac{\alpha^{\prime}}{2})\sin(\frac{\alpha}{2})+\frac{2\tau\sinh\phi_{1}}{\cosh\phi_{1}}\sinh(\frac{\alpha^{\prime}}{2})\cos(\frac{\alpha}{2})\\\
\qquad\qquad+\frac{2}{\cosh\phi_{2}}\cosh(\frac{\alpha^{\prime\prime}}{2})\sin(\frac{\alpha}{2})-\frac{2\tau\sinh\phi_{2}}{\cosh\phi_{2}}\sinh(\frac{\alpha^{\prime\prime}}{2})\cos(\frac{\alpha}{2}).\end{array}$
(2.40)

Similarly, subtracting the first (resp. second) equation of (2.39) from the
first (resp. second) equation of (2.38) we get

$\begin{array}[]{l}\alpha^{\prime}_{x_{2}}-\alpha^{\prime\prime}_{x_{2}}=-\frac{2}{\cosh\phi_{2}}\sin(\frac{\alpha^{*}}{2})\cosh(\frac{\alpha^{\prime}}{2})-\frac{2\tau\sinh\phi_{2}}{\cosh\phi_{2}}\cos(\frac{\alpha^{*}}{2})\sinh(\frac{\alpha^{\prime}}{2})\\\
\qquad\qquad+\frac{2}{\cosh\phi_{1}}\sin(\frac{\alpha^{*}}{2})\cosh(\frac{\alpha^{\prime\prime}}{2})+\frac{2\tau\sinh\phi_{1}}{\cosh\phi_{1}}\cos(\frac{\alpha^{*}}{2})\sinh(\frac{\alpha^{\prime\prime}}{2}),\\\
\alpha^{\prime}_{x_{1}}-\alpha^{\prime\prime}_{x_{1}}=-\frac{2}{\cosh\phi_{2}}\cos(\frac{\alpha^{*}}{2})\sinh(\frac{\alpha^{\prime}}{2})+\frac{2\tau\sinh\phi_{2}}{\cosh\phi_{2}}\sin(\frac{\alpha^{*}}{2})\cosh(\frac{\alpha^{\prime}}{2})\\\
\qquad\qquad+\frac{2}{\cosh\phi_{1}}\cos(\frac{\alpha^{*}}{2})\sinh(\frac{\alpha^{\prime\prime}}{2})-\frac{2\tau\sinh\phi_{1}}{\cosh\phi_{1}}\sin(\frac{\alpha^{*}}{2})\cosh(\frac{\alpha^{\prime\prime}}{2}).\end{array}$
(2.41)

Comparing equation (2.40) with (2.41), we obtain two algebraic expressions
that must be satisfied by $\cos(\alpha^{*}/2)$ and $\sin(\alpha^{*}/2)$. By
solving this system of equations, we conclude that $\alpha^{*}/2$ is given by
(2.33), where $\tilde{P}$ and $\tilde{Q}$ are given by (2.34) and (2.35).

We need to prove that $\alpha^{*}$ defined by (2.33) satisfies the elliptic
sine-Gordon equation (2.32). Recall that $\alpha^{\prime}$ and
$\alpha^{\prime\prime}$ are solutions of the elliptic sinh-Gordon equation. We
use the previous notation (2.26) for the constants $A,\,B,\,L$ and the
function $D=\alpha^{\prime\prime}-\alpha^{\prime}$. Then the functions
$\tilde{P}$ and $\tilde{Q}$ are rewritten as

$\tilde{P}=A\sinh(\frac{D}{2}),\qquad\tilde{Q}=L\cosh(\frac{D}{2})-B$ (2.42)

and $B^{2}-L^{2}=A^{2}$. We denote the right hand side of (2.33) by

$\tilde{H}=\frac{\tilde{G}-\tan(\frac{\alpha}{2})}{1+\tilde{G}\tan(\frac{\alpha}{2})},\quad\mbox{where}\quad\tilde{G}=\frac{\tilde{P}}{\tilde{Q}}.$
(2.43)

It follows that proving that $\alpha^{*}$ satisfies (2.32) is equivalent to
proving that

$\frac{2\tilde{H}}{1+\tilde{H}^{2}}|\nabla\tilde{H}|^{2}-\triangle\tilde{H}+\tilde{H}=0.$
(2.44)

Since $\alpha$ satisfies (2.32) it follows that

$\displaystyle\triangle\tan(\frac{\alpha}{2})=\frac{1}{2\cos^{3}(\frac{\alpha}{2})}\left(\sin(\frac{\alpha}{2})|\nabla\alpha|^{2}+\cos(\frac{\alpha}{2})\sin\alpha\right),$
(2.45)
$\displaystyle|\nabla\tan(\frac{\alpha}{2})|^{2}=\frac{1}{4\cos^{4}(\frac{\alpha}{2})}|\nabla\alpha|^{2}.$
(2.46)

Therefore, proving (2.44) reduces to showing that $\tilde{G}$ satisfies

$\triangle\tilde{G}-\frac{2\tilde{G}}{1+\tilde{G}^{2}}|\nabla\tilde{G}|^{2}-\tilde{G}(\cos\alpha+\tilde{G}\sin\alpha)=0.$
(2.47)

A long but traightforward computation shows that

$\displaystyle\triangle\tilde{G}=\frac{A}{2\tilde{Q}^{2}}\left(L-B\cosh(\frac{D}{2})\right){\triangle
D}+\frac{A}{4\tilde{Q}^{3}}(B\tilde{Q}-2A^{2})\sinh(\frac{D}{2})\,|\nabla
D|^{2},$
$\displaystyle|\nabla\tilde{G}|^{2}=\frac{A^{2}}{4\tilde{Q}^{4}}\left(L-B\cosh(\frac{D}{2})\right)^{2}\,|\nabla
D|^{2},$ (2.48) $\displaystyle
1+\tilde{G}^{2}=\frac{1}{\tilde{Q}^{2}}\left(L-B\cosh(\frac{D}{2})\right)^{2}.$

Since $\alpha^{\prime}$ and $\alpha^{\prime\prime}$ are solutions of the
elliptic sinh-Gordon equation, and $D=\alpha^{\prime\prime}-\alpha^{\prime}$
it follows that

$\displaystyle\triangle
D=\sinh\alpha^{\prime\prime}-\sinh\alpha^{\prime}=2\sinh(\frac{D}{2})\cosh(\frac{\alpha^{\prime\prime}+\alpha^{\prime}}{2}).$
(2.49)

Therefore, proving (2.47) reduces to proving

$\frac{B}{4}|\nabla
D|^{2}+\left(L-B\cosh(\frac{D}{2})\right)\cosh(\frac{\alpha^{\prime\prime}+\alpha^{\prime}}{2})-\tilde{P}\sin\alpha-\tilde{Q}\cos\alpha=0.$
(2.50)

Moreover, $\alpha^{\prime}$ and $\alpha^{\prime\prime}$ satisfy (2.36) and
(2.37) respectively, which provide $\nabla\alpha^{\prime}$ and
$\nabla\alpha^{\prime\prime}$ and $|\nabla D|^{2}$. Finally, substituting this
expression into (2.50), we obtain an identity. Therefore, we conclude that
$\alpha^{*}$ satisfies the elliptic sine-Gordon equation. ∎

The composition of Bäcklund transformations for the elliptic sinh-Gordon
equation (ESHG) and the elliptic sine-Gordon equation (ESGE) is illustrated in
the following diagrams. Starting with a solution of one of these equations, we
need the superposition formula (i.e. a composition of two Bäcklund
transformations) to produce new solutions of the same equation.

$\begin{array}[]{ccccccc}{\color[rgb]{1,0,0}ESHG}&&{\color[rgb]{0,0,1}ESG}&&{\color[rgb]{1,0,0}ESHG}&&{\color[rgb]{0,0,1}ESG}\\\
&&\alpha_{1}&\stackrel{{\scriptstyle\phi_{2}}}{{\longrightarrow}}&{\alpha_{12}}&&\longrightarrow\\\
&\stackrel{{\scriptstyle\phi_{1}}}{{\nearrow}}&&\stackrel{{\scriptstyle\phi_{1}}}{{\nearrow}}&&\stackrel{{\scriptstyle\phi_{3}}}{{\searrow}}\\\
\alpha&\stackrel{{\scriptstyle\phi_{2}}}{{\longrightarrow}}&\alpha_{2}&&&&\alpha_{123}\\\
&\stackrel{{\scriptstyle\phi_{3}}}{{\searrow}}&&\stackrel{{\scriptstyle\phi_{3}}}{{\searrow}}&&\stackrel{{\scriptstyle\phi_{1}}}{{\nearrow}}\\\
&&\alpha_{3}&\stackrel{{\scriptstyle\phi_{2}}}{{\longrightarrow}}&{\alpha_{23}}&&\longrightarrow\end{array}\qquad\;\begin{array}[]{ccccccc}{\color[rgb]{0,0,1}ESG}&&{\color[rgb]{1,0,0}ESHG}&&{\color[rgb]{0,0,1}ESG}&&\\!\\!{\color[rgb]{1,0,0}ESHG}\\\
&&\alpha_{1}&\stackrel{{\scriptstyle\phi_{2}}}{{\longrightarrow}}&{\alpha_{12}}&&\longrightarrow\\\
&\stackrel{{\scriptstyle\phi_{1}}}{{\nearrow}}&&\stackrel{{\scriptstyle\phi_{1}}}{{\nearrow}}&&\stackrel{{\scriptstyle\phi_{3}}}{{\searrow}}\\\
\alpha&\stackrel{{\scriptstyle\phi_{2}}}{{\longrightarrow}}&\alpha_{2}&&&&{\alpha_{123}}\\\
&\stackrel{{\scriptstyle\phi_{3}}}{{\searrow}}&&\stackrel{{\scriptstyle\phi_{3}}}{{\searrow}}&&\stackrel{{\scriptstyle\phi_{1}}}{{\nearrow}}\\\
&&\alpha_{3}&\stackrel{{\scriptstyle\phi_{2}}}{{\longrightarrow}}&\alpha_{23}&&\longrightarrow\end{array}$

##  3 Six cases

In this section, aiming the applications of Section 4 and also in order to
help the reader to easily access the theory in each case, we give the explicit
Bäcklund transformation and the corresponding superposition formula, for each
one of the six cases considered in the previous sections.

Case 1. Considering $s=0$, $\epsilon=1$, $r=0$ and $\delta=-1$ then $l=1$ and
(1.5) implies that $\Lambda^{2}+\lambda^{2}=1$ hence, $\Lambda=\cos\phi$ and
$\lambda=\sin\phi$, where $\phi\in(0,\pi)$.

######  Theorem 3.1.

For any solution $\alpha(x_{1},x_{2})$ of

$\alpha_{x_{1}x_{1}}-\alpha_{x_{2}x_{2}}=\sin\alpha,$ (3.1)

the following system of differential equations for
$\alpha^{\prime}(x_{1},x_{2})$, where $\phi\in(0,\pi)$, is integrable

$\left\\{\begin{array}[]{l}\alpha^{\prime}_{x_{1}}+\alpha_{x_{2}}=\frac{2}{\sin\phi}\sin\left(\frac{\alpha^{\prime}}{2}\right)\cos\left(\frac{\alpha}{2}\right)-\frac{2\tau\cos\phi}{\sin\phi}\cos\left(\frac{\alpha^{\prime}}{2}\right)\sin\left(\frac{\alpha}{2}\right),\\\
\alpha^{\prime}_{x_{2}}+\alpha_{x_{1}}=-\frac{2}{\sin\phi}\cos\left(\frac{\alpha^{\prime}}{2}\right)\sin\left(\frac{\alpha}{2}\right)+\frac{2\tau\cos\phi}{\sin\phi}\sin\left(\frac{\alpha^{\prime}}{2}\right)\cos\left(\frac{\alpha}{2}\right).\\\
\end{array}\right.$ (3.2)

Moreover, $\alpha^{\prime}$ is also a solution of (3.1). If $\alpha^{\prime}$
and $\alpha^{\prime\prime}$ are solutions of (3.1) obtained by the Bäcklund
transformation (3.2), for constants $\phi_{1}\neq\phi_{2}$, then a fourth
solution $\alpha^{*}$ of (3.1) is obtained algebraically with the
superposition formula

$\tan(\frac{\alpha^{*}-\alpha}{4})=-\tau\,\frac{\sin(\frac{\phi_{2}+\phi_{1}}{2})}{\sin(\frac{\phi_{2}-\phi_{1}}{2})}\,\tan(\frac{\alpha^{\prime}-\alpha^{\prime\prime}}{4}).$
(3.3)

This theorem gives the classical Bäcklund transformation (3.2) and the
superposition formula (3.3) for the sine-Gordon equation obtained by Bäcklund
[1],[2] and Bianchi [3], when $\tau=-1$. It can also be found for example in
[8] (see pg. 44).

Case 2. Considering $s=1$, $\epsilon=1$, $r=0$ and $\delta=1$ then $l=1$ and
(1.5) implies that $\Lambda^{2}-\lambda^{2}=1$, hence $\Lambda=\cosh\phi$ and
$\lambda=\sinh\phi$, where $\phi\in(0,\infty)$.

######  Theorem 3.2.

For any solution $\alpha(x_{1},x_{2})$ of

$\alpha_{x_{1}x_{1}}-\alpha_{x_{2}x_{2}}=-\sin\alpha,$ (3.4)

the following system of differential equations for
$\alpha^{\prime}(x_{1},x_{2})$, $\phi\in(0,\infty)$, is integrable

$\left\\{\begin{array}[]{l}\alpha^{\prime}_{x_{1}}+\alpha_{x_{2}}=\frac{2}{\sinh\phi}\sin\left(\frac{\alpha^{\prime}}{2}\right)\cos\left(\frac{\alpha}{2}\right)+\frac{2\tau\cosh\phi}{\sinh\phi}\cos\left(\frac{\alpha^{\prime}}{2}\right)\sin\left(\frac{\alpha}{2}\right),\\\
\alpha^{\prime}_{x_{2}}+\alpha_{x_{1}}=-\frac{2}{\sinh\phi}\cos\left(\frac{\alpha^{\prime}}{2}\right)\sin\left(\frac{\alpha}{2}\right)-\frac{2\tau\cosh\phi}{\sinh\phi}\sin\left(\frac{\alpha^{\prime}}{2}\right)\cos\left(\frac{\alpha}{2}\right).\\\
\end{array}\right.$ (3.5)

Moreover, $\alpha^{\prime}$ is also a solution of (3.4). If $\alpha^{\prime}$
and $\alpha^{\prime\prime}$ are solutions of (3.4) obtained by the Bäcklund
transformation (3.5), for constants $\phi_{1}\neq\phi_{2}$, then a fourth
solution $\alpha^{*}$ of (3.4) is obtained algebraically with the
superposition formula

$\tan(\frac{\alpha^{*}-\alpha}{4})=\tau\,\frac{\sinh(\frac{\phi_{2}+\phi_{1}}{2})}{\sinh(\frac{\phi_{2}-\phi_{1}}{2})}\,\tan(\frac{\alpha^{\prime}-\alpha^{\prime\prime}}{4}),$
(3.6)

We observe that Cases 1 and 2 deal with the sine-Gordon equation and they are
complementary in the following sense: while in Case 1 the parameter
$\phi\in(0,\pi)$ and hence $\Lambda=\cos\phi\in(-1,1)$, in Case 2, the
parameter $\phi\in(0,\infty)$ and hence $\Lambda=\cosh\phi\in(1,+\infty)$.

Case 3. Considering $s=1$, $\epsilon=1$, $r=1$ and $\delta=1$ then $l=-1$ and
(1.5) implies that $\Lambda^{2}-\lambda^{2}=1$, hence $\Lambda=\cosh\phi$ and
$\lambda=\sinh\phi$, where $\phi\in(0,\infty)$.

######  Theorem 3.3.

For any solution $\alpha(x_{1},x_{2})$ of

$\alpha_{x_{1}x_{1}}-\alpha_{x_{2}x_{2}}=-\sinh\alpha,$ (3.7)

the following system of differential equations for
$\alpha^{\prime}(x_{1},x_{2})$, where $\phi\in(0,\infty)$, is integrable

$\left\\{\begin{array}[]{l}\alpha^{\prime}_{x_{1}}+\alpha_{x_{2}}=\frac{2}{\sinh\phi}\sinh\left(\frac{\alpha^{\prime}}{2}\right)\cosh\left(\frac{\alpha}{2}\right)+\frac{2\tau\cosh\phi}{\sinh\phi}\cosh\left(\frac{\alpha^{\prime}}{2}\right)\sinh\left(\frac{\alpha}{2}\right)\\\
\alpha^{\prime}_{x_{2}}+\alpha_{x_{1}}=-\frac{2}{\sinh\phi}\cosh\left(\frac{\alpha^{\prime}}{2}\right)\sinh\left(\frac{\alpha}{2}\right)-\frac{2\tau\cosh\phi}{\sinh\phi}\sinh\left(\frac{\alpha^{\prime}}{2}\right)\cosh\left(\frac{\alpha}{2}\right)\\\
\end{array}\right.,$ (3.8)

Moreover, $\alpha^{\prime}$ is also a solution of (3.7). If $\alpha^{\prime}$
and $\alpha^{\prime\prime}$ are solutions of (3.7) obtained by the Bäcklund
transformation (3.8), for constants $\phi_{1}\neq\phi_{2}$, then a fourth
solution $\alpha^{*}$ of (3.7) is obtained algebraically with the
superposition formula

$\tanh(\frac{\alpha^{*}-\alpha}{4})=\tau\,\frac{\sinh(\frac{\phi_{2}+\phi_{1}}{2})}{\sinh(\frac{\phi_{2}-\phi_{1}}{2})}\,\tanh(\frac{\alpha^{\prime}-\alpha^{\prime\prime}}{4}),$
(3.9)

Case 4. Considering $s=1$, $\epsilon=-1$, $r=1$ and $\delta=1$ then $l=-1$ and
(1.5) implies that $\Lambda^{2}+\lambda^{2}=1$, hence $\Lambda=\cos\phi$ and
$\lambda=\sin\phi$, where $\phi\in(0,\pi)$.

######  Theorem 3.4.

For any solution $\alpha(x_{1},x_{2})$ of

$\alpha_{x_{1}x_{1}}-\alpha_{x_{2}x_{2}}=\sinh\alpha,$ (3.10)

the following system of differential equations for
$\alpha^{\prime}(x_{1},x_{2})$, where $\phi\in(0,\pi)$, is integrable

$\left\\{\begin{array}[]{l}\alpha^{\prime}_{x_{1}}+\alpha_{x_{2}}=\frac{2}{\sin\phi}\sinh\left(\frac{\alpha^{\prime}}{2}\right)\cosh\left(\frac{\alpha}{2}\right)+\frac{2\tau\cos\phi}{\sin\phi}\cosh\left(\frac{\alpha^{\prime}}{2}\right)\sinh\left(\frac{\alpha}{2}\right),\\\
\alpha^{\prime}_{x_{2}}+\alpha_{x_{1}}=-\frac{2}{\sin\phi}\cosh\left(\frac{\alpha^{\prime}}{2}\right)\sinh\left(\frac{\alpha}{2}\right)-\frac{2\tau\cos\phi}{\sin\phi}\sinh\left(\frac{\alpha^{\prime}}{2}\right)\cosh\left(\frac{\alpha}{2}\right).\\\
\end{array}\right.$ (3.11)

Moreover, $\alpha^{\prime}$ is also a solution of (3.10). If $\alpha^{\prime}$
and $\alpha^{\prime\prime}$ are solutions of (3.10) obtained by the Bäcklund
transformation (3.11), for constants $\phi_{1}\neq\phi_{2}$, then a fourth
solution $\alpha^{*}$ of (3.10) is obtained algebraically with the
superposition formula

$\tanh(\frac{\alpha^{*}-\alpha}{4})=\tau\,\frac{\sin(\frac{\phi_{2}+\phi_{1}}{2})}{\sin(\frac{\phi_{2}-\phi_{1}}{2})}\,\tanh(\frac{\alpha^{\prime}-\alpha^{\prime\prime}}{4}),$
(3.12)

As in the first two cases, we observe that Cases 3 and 4 deal with the sinh-
Gordon equation and they are complementary in the following sense: in Case 3
the parameter $\phi\in(0,\infty)$ and hence $\Lambda\in(1,+\infty)$, while in
Case 4 we have $\phi\in(0,\pi)$ and hence $\Lambda\in(-1,1)$.

Case 5. Considering $s=1$, $\epsilon=1$, $r=0$ and $\delta=-1$ then $l=-1$ and
(1.5) implies that $-\Lambda^{2}+\lambda^{2}=1$, hence $\Lambda=\sinh\phi$ and
$\lambda=\cosh\phi$, where $\phi\in[0,\infty)$.

######  Theorem 3.5.

For any solution $\alpha(x_{1},x_{2})$ of the elliptic sinh-Gordon equation

$\alpha_{x_{1}x_{1}}+\alpha_{x_{2}x_{2}}=\sinh\alpha,$ (3.13)

the following system of differential equations for
$\alpha^{\prime}(x_{1},x_{2})$, where $\phi\in[0,\infty)$, is integrable

$\left\\{\begin{array}[]{l}\alpha^{\prime}_{x_{1}}-\alpha_{x_{2}}=\frac{2}{\cosh\phi}\sin\left(\frac{\alpha^{\prime}}{2}\right)\cosh\left(\frac{\alpha}{2}\right)+\frac{2\tau\sinh\phi}{\cosh\phi}\cos\left(\frac{\alpha^{\prime}}{2}\right)\sinh\left(\frac{\alpha}{2}\right),\\\
\alpha^{\prime}_{x_{2}}+\alpha_{x_{1}}=-\frac{2}{\cosh\phi}\cos\left(\frac{\alpha^{\prime}}{2}\right)\sinh\left(\frac{\alpha}{2}\right)+\frac{2\tau\sinh\phi}{\cosh\phi}\sin\left(\frac{\alpha^{\prime}}{2}\right)\cosh\left(\frac{\alpha}{2}\right).\\\
\end{array}\right.$ (3.14)

Moreover, $\alpha^{\prime}$ is a solution of the elliptic sine-Gordon equation

$\alpha^{\prime}_{x_{1}x_{1}}+\alpha^{\prime}_{x_{2}x_{2}}=\sin\alpha^{\prime}.$
(3.15)

If $\alpha^{\prime}$ and $\alpha^{\prime\prime}$ are solutions of (3.15)
obtained by the Bäcklund transformation (3.14), for constants
$\phi_{1}\neq\phi_{2}$, then a solution $\alpha^{*}$ of (3.13) is obtained
algebraically with the superposition formula

$\tanh\frac{\alpha^{*}}{2}=\frac{P-Q\,\tanh\displaystyle{\frac{\alpha}{2}}}{Q-P\,\tanh\displaystyle{\frac{\alpha}{2}}},$
(3.16)

where

$P=\tau(\sinh\phi_{2}-\sinh\phi_{1})\sin(\frac{\alpha^{\prime\prime}-\alpha^{\prime}}{2}),$
$Q=(1+\sinh\phi_{1}\sinh\phi_{2})\cos(\frac{\alpha^{\prime\prime}-\alpha^{\prime}}{2})-\cosh\phi_{1}\cosh\phi_{2}.$

Case 6. Finally, considering $s=1$, $\epsilon=1$, $r=1$ and $\delta=-1$ then
$l=1$ and (1.5) implies that $-\Lambda^{2}+\lambda^{2}=1$, hence
$\Lambda=\sinh\phi$ and $\lambda=\cosh\phi$, where $\phi\in[0,\infty)$.

######  Theorem 3.6.

For any solution $\alpha(x_{1},x_{2})$ of the elliptic sine-Gordon equation
(3.15), the following system of differential equations for
$\alpha^{\prime}(x_{1},x_{2})$, where $\phi\in[0,\infty)$, is integrable

$\left\\{\begin{array}[]{l}\alpha^{\prime}_{x_{1}}-\alpha_{x_{2}}=\frac{2}{\cosh\phi}\sinh\left(\frac{\alpha^{\prime}}{2}\right)\cos\left(\frac{\alpha}{2}\right)+\frac{2\tau\sinh\phi}{\cosh\phi}\cosh\left(\frac{\alpha^{\prime}}{2}\right)\sin\left(\frac{\alpha}{2}\right),\\\
\alpha^{\prime}_{x_{2}}+\alpha_{x_{1}}=-\frac{2}{\cosh\phi}\cosh\left(\frac{\alpha^{\prime}}{2}\right)\sin\left(\frac{\alpha}{2}\right)+\frac{2\tau\sinh\phi}{\cosh\phi}\sinh\left(\frac{\alpha^{\prime}}{2}\right)\cos\left(\frac{\alpha}{2}\right).\\\
\end{array}\right.$ (3.17)

Moreover, $\alpha^{\prime}$ is a solution of the elliptic sinh-Gordon equation
(3.13). If $\alpha^{\prime}$ and $\alpha^{\prime\prime}$ are solutions of
(3.13) obtained by the Bäcklund transformation (3.17) for constants
$\phi_{1}\neq\phi_{2}$, then a solution $\alpha^{*}$ of the elliptic sine-
Gordon equation (3.15) is obtained algebraically with the superposition
formula

$\tan\frac{\alpha^{*}}{2}=\frac{\tilde{P}-\tilde{Q}\,\displaystyle{\tan\frac{\alpha}{2}}}{\tilde{Q}+\tilde{P}\,\displaystyle{\tan\frac{\alpha}{2}}},$
(3.18)

where

$\tilde{P}=\tau(\sinh\phi_{2}-\sinh\phi_{1})\,\sinh(\frac{\alpha^{\prime\prime}-\alpha^{\prime}}{2})$
$\tilde{Q}=(1+\sinh\phi_{1}\sinh\phi_{2})\cosh\frac{\alpha^{\prime\prime}-\alpha^{\prime}}{2}-\cosh\phi_{1}\cosh\phi_{2}.$

##  4 Examples

In this section, we present some examples and illustrations of the previous
results. Since explicit solutions of the sine-Gordon equation, the
corresponding surfaces and applications of Theorem 3.1, namely the Bäclund
transformation (3.2) and the superposition formula (3.3) are well known (see
for example [8]), we will concentrate in applying the new results. Our first
example provides the most straightforward application of Theorem 1.4.

######  Example 4.1.

Let $\delta=\pm 1$, $\epsilon=\pm 1$, $r,s\in\\{0,1\\}$, $r\leq s$, be fixed
constants satisfying (1.3) and (1.4). Consider $\alpha=0$ a trivial solution
of (1.8). Then the Bäcklund transformation (1.9) reduces to

$\left\\{\begin{array}[]{l}\alpha^{\prime}_{x_{1}}=\frac{2}{\lambda}S_{(-1)^{r}}(\frac{\alpha^{\prime}}{2}),\\\
\alpha^{\prime}_{x_{2}}=-\frac{2\delta\tau\Lambda}{\lambda}S_{(-1)^{r}}(\frac{\alpha^{\prime}}{2}),\\\
\end{array}\right.$

where $\tau=\pm 1$, and the constants $\Lambda,\lambda$ satisfy (1.5). The
solutions of this system are given by

$\alpha^{\prime}(x_{1},x_{2})=\left\\{\begin{array}[]{ll}4\arctan\exp\xi(x_{1},x_{2}),&\quad\mbox{if
}r=0,\\\ 2\ln\tanh\left(-\frac{1}{2}\xi(x_{1},x_{2})\right),&\quad\mbox{if
}r=1,\\\ \end{array}\right.$ (4.1)

where

$\xi(x_{1},x_{2})=\frac{1}{\lambda}x_{1}-\delta\tau\frac{\Lambda}{\lambda}x_{2}+c,$

and $c$ is a constant that depends on a given initial condition. Moreover,
$\alpha^{\prime}$ is a solution of (1.10).

Although $\alpha=0$ does not correspond to any surface, we observe that
$\alpha^{\prime}$ does correspond to a surface (see for instance (4.3), in
Example 4.2).

In the following examples, we include explicit solutions of the differential
equations and some of the surfaces of constant curvature that correspond to
those solutions. We recall from [5], that under the hypothesis of Theorem 1.2,
if $X(x_{1},x_{2})$ is a parametrized surface in $\mathbb{R}^{3}_{s}$ of
curvature $\delta=\pm 1$, then there exists a two-parameter family of surfaces
$\bar{X}$ related to $X$ by a Bäcklund-type Line Congruence. Moreover, it
follows from Theorem 1.4 that if $\alpha$ is a solution of (1.8), that
corresponds to the surface $X$, then the Bäcklund transformation (1.9) is
integrable for $\alpha^{\prime}$ and the surface $\bar{X}$ that corresponds to
$\alpha^{\prime}$ is given by

$\bar{X}=X+\lambda\left(\frac{C_{(-1)^{r}}(\frac{\alpha^{\prime}}{2})}{C_{l}(\frac{\alpha}{2})}X_{x_{1}}+\frac{S_{(-1)^{r}}(\frac{\alpha^{\prime}}{2})}{S_{l}(\frac{\alpha}{2})}X_{x_{2}}\right).$
(4.2)

######  Example 4.2.

We consider the constants $\delta=1,\,\epsilon=-1,\,r=s=1$, then (1.8) is the
sinh-Gordon equation (SHGE). We will illustrate Theorem 3.4, by starting with
the trivial solution $\alpha=0$. From Example 4.1, the Bäcklund transformation
(3.11) provides new solutions for the SHGE given by (4.1), with $r=1$. Let
$\alpha_{j}=2\ln\tanh(-\xi_{j}(x_{1},x_{2})/2)$, $j=1,2,3$, be a solution of
the SHGE given by $\xi_{j}=1/\sin(\phi_{j})(x_{1}+\cos(\phi_{j})x_{2})+c_{j}$,
for distinct parameters $\phi_{j}\in(0,\pi)$, $\tau=-1$ and arbitrary
constants $c_{j}$. Observe that for each $j$, $\alpha_{j}$ is defined on the
half-plane $\xi_{j}<0$. We will apply the superposition formula to distinct
pairs of $\alpha_{j}$.

Now we choose $\phi_{1}=\pi/2$, $\phi_{2}=\pi/3$ and $\phi_{3}=\pi/8$. By
applying the superposition formula (3.12) to $\alpha$ and $\alpha_{j}$,
$j=1,2$ and the parameters parameters $\phi_{1}$, $\phi_{2}$ (here we are
denoting the solutions $\alpha^{\prime}$, $\alpha^{\prime\prime}$ and
$\alpha^{*}$ by $\alpha_{1}$, $\alpha_{2}$ and $\alpha_{12}$ respectively),
one gets the solution $\alpha_{12}$. Similarly, by applying the superposition
formula to $\alpha$, $\alpha_{2}$ and $\alpha_{3}$ with the parameters
$\phi_{2}$ and $\phi_{3}$, one gets the solution $\alpha_{23}$. These
solutions of the sinh-Gordon equation are given by

$\alpha_{12}=4\mbox{ arctanh
}\left(\frac{\sin(\frac{5\pi}{12})}{\sin(\frac{\pi}{12})}\tanh(\frac{\alpha_{1}-\alpha_{2}}{4})\right),\quad\;\alpha_{23}=4\mbox{
arctanh
}\left(\frac{\sin(\frac{11\pi}{48})}{\sin(\frac{5\pi}{48})}\tanh(\frac{\alpha_{2}-\alpha_{3}}{4})\right),$

where $\tanh((\alpha_{1}-\alpha_{2})/{4})$ and
$\tanh((\alpha_{2}-\alpha_{3})/{4})$ are functions of $(x_{1},x_{2})$
determined by $\exp(\alpha_{j}/2)=\tanh(-\xi_{j}/2)$. The superposition of
$\alpha_{12}$ and $\alpha_{23}$ with the parameters $\phi_{1}$ and $\phi_{3}$,
provides the solution $\alpha_{123}$ (see Diagram (2.17)),

$\alpha_{123}=\alpha_{2}+4\mbox{ arctanh
}\left(\frac{\sin(\frac{5\pi}{16})}{\sin(\frac{3\pi}{16})}\tanh(\frac{\alpha_{12}-\alpha_{23}}{4})\right).$

The graph of these functions, with the constants $c_{j}=0$, can be visualized
in Figures (1(a)) - (1(f)). This procedure provides infinitely many solutions
of the sinh-Gordon equation, whose explicit expressions may get more and
complicated.

The time-like surface $X_{1}$ in $\mathbb{R}^{3}_{1}$, with constant curvature
$K=1$, that corresponds to the function $\alpha_{1}$, is given by

$X_{1}=\left(\frac{\cos x_{2}}{\sinh x_{1}},\,\frac{\sin x_{2}}{\sinh
x_{1}},\,x_{1}-\frac{\cosh x_{1}}{\sinh x_{1}}\right).$ (4.3)

The surface $X_{12}$ corresponding to $\alpha_{12}$ is given by

$X_{12}=X_{1}+\sin(\phi_{2})\left(\frac{\cosh(\frac{\alpha_{12}}{2})}{\sinh(\frac{\alpha_{1}}{2})}X_{1,x_{1}}+\frac{\sinh(\frac{\alpha_{12}}{2})}{\sinh(\frac{\alpha_{1}}{2})}X_{1,x_{2}}\right).$

The solution $\alpha_{123}$ corresponds to a surface $X_{123}$.

$X_{123}=X_{12}+\sin\phi_{3}\left(\frac{\cosh(\frac{\alpha_{123}}{2})}{\sinh(\frac{\alpha_{12}}{2})}X_{12,x_{1}}+\frac{\sinh(\frac{\alpha_{123}}{2})}{\sinh(\frac{\alpha_{12}}{2})}X_{12,x_{2}}\right).$

These surfaces are illustrated in Figures (2(a)) - (2(c)).

(a) $\alpha_{1}$ (b) $\alpha_{2}$

(c) $\alpha_{3}$ (d) $\alpha_{12}$

(e) $\alpha_{23}$ (f) $\alpha_{123}$

Figure 4.1: (a), (b) and (c) are solutions of the sinh-Gordon equation
obtained from the null solution by the Bäcklund transformation with parameters
$\phi_{1}=\pi/2$, $\phi_{2}=\pi/3$ and $\phi_{3}=\pi/8$ respectively. (d) is a
solution obtained from $(a)$ and $(b)$ by the superposition formula. (e) is a
solution obtained from (b) and (c) by the superposition formula and (f) is
obtained from (d) and (e) by the superposition formula.

(a) $X_{1}$ (b) $X_{12}$

(c) $X_{123}$

Figure 4.2: (a)$X_{1}$ is a time-like surface with Gaussian curvature $K=1$
contained in $\mathbb{R}^{3}_{1}$ corresponding to the solution $\alpha_{1}$.
(b) $X_{12}$ is a time-like surface obtained from the surface (a) by a
Bäcklund transformation, with parameter $\phi_{2}=\pi/3$ and it corresponds to
the solution $\alpha_{12}$. (c) $X_{123}$ is the surface obtained from
$X_{12}$ by a Bäcklund transformation, with parameter $\phi_{3}=\pi/8$ and it
corresponds to $\alpha_{123}$.

######  Example 4.3.

We consider the constants $\delta=-1,\,\epsilon=1,\,r=0,\,s=1$, then (1.8) is
the elliptic sinh-Gordon equation (ESHGE). We will illustrate Theorem 3.5, by
starting with the trivial solution $\alpha=0$. The Bäcklund transformation
(3.14) provides solutions for the elliptic sine-Gordon equation (ESGE) given
by (4.1), where $r=0$, $\Lambda=\sinh\phi$, $\lambda=\cosh\phi$ and any
$\phi\in[0,+\infty)$. Therefore, by choosing $\tau=1$ and
$\phi_{1}\neq\phi_{2}$ in (4.1), we have the solutions $\alpha_{1}$ and
$\alpha_{2}$ for the ESGE given by

$\displaystyle\alpha_{j}=4\arctan e^{\xi_{j}},\mbox{ where
\quad}\xi_{j}=\frac{1}{\cosh\phi_{j}}(x_{1}+\sinh\phi_{j}x_{2})+c_{j},\quad
j=1,2,\quad c_{j}\in\mathbb{R}.$

By applying the superposition formula (3.16), we obtain a new solution
$\alpha_{12}$ for the ESHGE,

$\tanh(\frac{\alpha_{12}}{2})=\frac{(\sinh\phi_{2}-\sinh\phi_{1})\sin(\frac{\alpha_{2}-\alpha_{1}}{2})}{(1+\sinh\phi_{1}\sinh\phi_{2})\cos(\frac{\alpha_{2}-\alpha_{1}}{2})-\cosh\phi_{1}\cosh\phi_{2}}.$
(4.4)

We observe that the function $\alpha_{12}$ has two curves of singularities
(when the right hand side of (4.4) tends to $\pm 1$) defined by

$\frac{e^{\xi_{2}}-e^{\xi_{1}}}{1+e^{\xi_{2}+\xi_{1}}}\pm\frac{\sinh\phi_{2}-\sinh\phi_{1}}{1+\cosh(\phi_{2}+\phi_{1})}=0.$
(4.5)

In particular, by choosing $\phi_{1}=0$, $\phi_{2}=\ln[(1+\sqrt{5})/2]$ and
the constants $c_{j}=0$, we have $\xi_{1}=x_{1}$ and
$\xi_{2}=(2x_{1}+x_{2})/\sqrt{5}$. Hence, starting with the null solution of
the ESHGE, the superposition formula provides a new explicit solution of the
same equation, namely

$\alpha_{12}=2\mbox{ arctanh
}\left(\frac{\tan(\frac{\alpha_{2}-\alpha_{1}}{4})}{(1-\frac{\sqrt{5}}{2})-(1+\frac{\sqrt{5}}{2})(\tan(\frac{\alpha_{2}-\alpha_{1}}{4}))^{2}}\right),$
(4.6)

where

$\tan(\frac{\alpha_{2}-\alpha_{1}}{4})=\frac{e^{(2x_{1}+x_{2})/\sqrt{5}}-e^{x_{1}}}{1+e^{(2x_{1}+x_{2})/\sqrt{5}+x_{1}}}.$

In Figure 3(c), one can visualize the graph of the solution of the ESHGE given
by $\alpha_{12}$ and also the two curves of singularities.

(a) $\alpha_{1}$ (b) $\alpha_{2}$

(c) $\alpha_{12}$

Figure 4.3: (a) and (b) are solutions of the elliptic sine-Gordon equation,
obtained from the null solution of the elliptic sinh-Gordon equation, by
Bäcklund transformations with parameters $\phi_{1}=0$ and
$\phi_{2}=\ln(1+\sqrt{5}$. (c) $\alpha_{12}$ is a solution of the elliptic
sinh-Gordon equation, with two curves of singularities, obtained by using the
superposition formula.

We observe that the surface associated to $\alpha_{1}$ (the intermediate step
which is a solution of the ESGE) is given by (4.9) in the next example.

######  Example 4.4.

In this example, we illustrate Theorem 3.6. Let $\,\alpha\,$ be the solution
of the elliptic sine-Gordon equation (ESGE) given by
$\tan(\alpha/4)=e^{x_{1}}$. By solving the Bäcklund transformation (3.17) for
the parameters $\phi_{1}=0$, and any $\phi_{2}\in(0,\infty)$, we obtain
$\alpha_{1}$ and $\alpha_{2}$ solutions of the ESHGE, given as follows,

$\displaystyle\tanh(\frac{\alpha_{1}}{4})=(-x_{2}+c_{1})\rm\,sech\,x_{1},$
(4.7)
$\displaystyle\tanh(\frac{\alpha_{2}}{2})=\displaystyle{\frac{\sinh{\phi_{2}(\sinh\xi-\sinh
x_{1})}}{\sinh\xi\sinh x_{1}+1-\cosh\phi_{2}\cosh\xi\cosh x_{1}}},$ (4.8)

where $\xi=(x_{1}+\sinh(\phi_{2})\,x_{2})/\cosh\phi_{2}+c_{2}$ and
$c_{2}\in\mathbb{R}$. The solution $\alpha_{1}$ has two curves of
singularities defined by $x_{2}=\pm\cosh x_{1}+c_{1}$ (see Fig. 4(b)) and the
two curves of singularities of $\alpha_{2}$ are defined by
$\sinh\phi_{2}(\sinh\xi-\sinh x_{1})\pm(\sinh\xi\sinh
x_{1}+1-\cosh\phi_{2}\cosh\xi\cosh x_{1})=0$. The superposition formula
(3.18), defines explicitly a function $\alpha_{12}$ which is a new solution of
the ESGE, given by

$\tan(\frac{\alpha_{12}}{2})=\frac{\sinh x_{1}P_{12}+Q_{12}}{\sinh
x_{1}Q_{12}-P_{12}},$

where $P_{12}=\sinh\phi_{2}\sinh((\alpha_{2}-\alpha_{1})/2)$ and
$Q_{12}=\cosh((\alpha_{2}-\alpha_{1})/2)-\cosh\phi_{2}$ are determined by
(4.7) and (4.8). Observe that the solution $\alpha_{12}$ has many curves of
singularities defined by

$\arctan\left((\sinh x_{1}P_{12}+Q_{12})/(\sinh
x_{1}Q_{12}-P_{12})\right)=\pi/2+k\pi,$

where $k$ is any integer (see Figure 4(d)). In particular, by choosing
$\phi_{2}=\ln(1+\sqrt{2})$ and the constants $c_{1}=c_{2}=0$, one can
visualize the graph of the functions in Figures (4(a)) - (4(d)).

Now we consider the surfaces in $\mathbb{R}^{3}_{1}$ of constant Gaussian
curvature $K=-1$, that correspond to the functions above. The time-like
surface given by

$X(x_{1},x_{2})=(x_{1}-\tanh x_{1},\rm\,sech\,x_{1}\cosh
x_{2},\rm\,sech\,x_{1}\sinh x_{2}),$ (4.9)

corresponds to the solution $\alpha$. The geometric Bäcklund transformation of
the surface $X$ for the parameters $\phi_{1}$ and $\phi_{2}$, provides two
space-like surfaces of constant negative curvature given by

$X_{j}=X+\cosh\phi_{j}\left(-\cosh(\alpha_{j}/2)/\tanh(x_{1})X_{x_{1}}+\sinh(\alpha_{j}/2)\cosh(x_{1})X_{x_{2}}\right),\qquad
j=1,2.$

The fourth surface $X_{12}$ given by

$X_{12}=X_{1}+\cosh\phi_{2}(1-\tanh^{2}(\frac{\alpha_{1}}{4}))\left[\dfrac{\cos(\alpha_{12}/2)}{1+\tanh^{2}({\alpha_{1}}/{4})}X_{1,x_{1}}+\dfrac{\sin(\alpha_{12}/2)}{2\tanh({\alpha_{1}}/{4})}X_{1,x_{2}}\right],$

where $\tanh(\alpha_{1}/{4})$ is given by (4.7), is a time-like surface that
corresponds to $\alpha_{12}$ and it is associated to $X_{1}$ and to $X_{2}$ by
a Bäcklund transformation with $\phi_{2}$ and $\phi_{1}$, respectively.

(a) $\alpha$

(b) $\alpha_{1}$

(c) $\alpha_{2}$

(d) $\alpha_{12}$

Figure 4.4: (a) $\alpha$ is a solution of ESGE. (b) and (c) are solutions of
the ESHGE obtained from $\alpha$ by the Bäcklund transformations with
parameter $\phi_{1}=0$ and $\phi_{2}=\ln(\sqrt{2}+1)$ respectively. (d) is a
solution of the ESGE obtained from (a) by using the superposition formula with
parameters $\phi_{1}$ and $\phi_{2}$.

## References

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