Solutions to the Sine-Gordon Equation: From Darboux Transformations to Wronskian Representations of the k-Negaton-l-Positon-n-Soliton Solutions

Abstract

With a specific Darboux transformation, we construct solutions to the sine-Gordon equation. We use both the simple Darboux transformation as well as the multiple Darboux transformation, which enables the obtainment of compact solutions of this equation. We give a complete description of the method and the corresponding proofs. We explicitly construct some solutions for the first orders. Using particular generating functions, we give Wronskian representations of the solutions to the sine-Gordon equation. In this case, we give different solutions to this equation. We deduce generalized Wronskian representations of the solutions to the sine-Gordon equation. As an application, we give the general expression of the k-negaton-l-positon-n-soliton solutions of the sine-Gordon equation and we construct some explicit examples of these solutions as well as m complexitons.

1. Introduction

We consider the sine-Gordon equation (SG) which can be written as

$${\mathsf{\partial }}_{xt}u(x,t)=4sinu(x,t).$$

(1)

This equation appeared for the first time in 1870 in differential geometry, where it is shown, in [1], that the angle v between the two sets of asymptotic lines on a surface with constant negative Gaussian curvature obeys the partial differential equation

$$\begin{array}{c}\hfill {\mathsf{\partial }}_p^{2}v-{\mathsf{\partial }}_q^{2}v=sinv,\end{array}$$

(2)

where the coordinate lines on the surface are the asymptotic lines. We recover this equation in other works like [2]. This last equation is slightly different from (1). But it is clear that from Equation (2), the changes of variable given by

$$x={\displaystyle \frac{p+q}{4}} \mathrm{and} y={\displaystyle \frac{p-q}{4}}$$

gives Equation (1).

This equation appears also in various areas of physics like superconductivity, dislocation in crystals, waves in ferromagnetic materials, laser pulses in two-phase media, condensed matter physics, chemical reaction kinetics, and high-energy physics.

This equation gained renewed interest in the 1930s in the context of the discrete analogue of the model in the theory of crystalline dislocations. Following this interest, a mechanical analogue was created through the realization of a set of coupled torsion pendulum. The field of applications has continued to expand, thanks to the emergence of Josephson junction networks and their fluxons, which were studied predominantly in the 1990s and early 2000s. The applications of the model have continued to develop, for example in celestial mechanics or in the evolution of the electromagnetic field on neural microtubules.

This equation is a completely integrable nonlinear equation. Different methods were used to study this equation. For example, we can quote the inverse scattering transform in [3,4,5], the Hirota method in [6], the algebro geometric approach in [7], the use of elliptic functions [8], and the Darboux transformations in [9,10]; however, the method used for the equation was given without direct proofs.

The multi solitons of (SG) equation were constructed by means of the Bäcklund transformation [8]. The complex positon solutions of the sine-Gordon equation were introduced in [11] by means of the Darboux transformation [12,13]; the interactions of multi positons and positon solitons were studied in detail.

The following works can also be cited in reference to this: in [14], some solutions were given by using the arctan transformation, while in [15], other types of solutions have been given by using a specific expansion. A general approach has been proposed in [16] for deriving a large number of nonlinear equations, including the sine-Gordon equation.

In the following, we write the (SG) equation as a compatibility condition of a system of two equations; we use a double Darboux transformation [12,13] defined by the following formulas:

$$\begin{array}{c}\hfill \begin{array}{c}{\mathrm{\Psi }}_1\left(\psi \right)(x,t)=\lambda \phi (x,t)-{\displaystyle \frac{{\lambda }_1{\phi }_1(x,t)}{{\psi }_1(x,t)}}\psi (x,t)\end{array},\end{array}$$

(3)

$$\begin{array}{c}\hfill \begin{array}{c}{\mathrm{\Phi }}_1\left(\phi \right)(x,t)=\lambda \psi (x,t)-{\displaystyle \frac{{\lambda }_1{\psi }_1(x,t)}{{\phi }_1(x,t)}}\phi (x,t)\end{array},\end{array}$$

(4)

with $\phi$, ${\phi }_1$, $\psi$, and ${\psi }_1$ being independent solutions of the system of compatibility of the (SG) equation

$${\psi }_x=v\psi +\lambda \phi ,{\mathsf{\partial }}_t\psi =v{\lambda }^{-1}w\phi ,$$

(5)

$${\phi }_x=\lambda \psi -v\phi ,{\phi }_t=v{\lambda }^{-1}{w}^{-1}\psi .$$

(6)

associated with two different spectral parameters $\lambda$ and ${\lambda }_1$.

With this Darboux transformation, and some choice of generating functions, we construct explicit solutions of this equation for the first orders.

Choosing other generating functions for the Darboux transformations, we obtain solutions of the (SG) equation in terms of Wronskians and construct some explicit solutions to this equation.

We deduce generalized Wronskian representations of the solutions containing derivatives depending on the parameters of the generating functions. We also construct other explicit solutions of the sine-Gordon equation and give some examples of negatons, multi-negatons, positons, multi-positons, and multi-negaton positons.

We will use the following notations. For the initial Darboux transformation, the functions $v_1$, $w_1$, and $u_m$ are defined by the following:

$$\begin{array}{c}\hfill \begin{array}{c}{v}_1=v+{\mathsf{\partial }}_{x}ln\left({\displaystyle \frac{{\phi }_1}{{\psi }_1}}\right),\hfill \end{array}\end{array}$$

(7)

$$\begin{array}{c}\hfill \begin{array}{c}{w}_1=w\times {\left({\displaystyle \frac{{\phi }_1}{{\psi }_1}}\right)}^2,\hfill \end{array}\end{array}$$

(8)

for the ${\phi }_1$ and ${\psi }_1$ solutions of the system of compatibility conditions associated with ${\lambda }_1$

$$\begin{array}{c}{u}_1=u-2iln{\displaystyle \frac{{\phi }_1}{{\psi }_1}},\end{array}$$

(9)

for u solution of the (SG) equation.

For the Darboux transformation of order m:

We consider the following determinants $A_m$ and $B_m$ defined by

$$\begin{array}{c}\hfill \begin{array}{c}\hfill A_m=A_m(f_1,f_2,\dots ,f_m)=det{\left({\lambda }_k^{j-1}{f}_{k,j}\right)}_{\{j,k=1,\dots ,m\}},\end{array} \mathrm{with}\end{array}$$

(10)

$$\begin{array}{c}\hfill f_{2j-1,k,}={\psi }_k,f_{2j,k}={\phi }_k,\end{array}$$

(11)

$f_k$ being the m dimensional column vector $\left(\begin{array}{c}{f}_{1,k}\\ ⋮\\ f_{m,k}\end{array}\right),$

$$\begin{array}{c}\hfill \begin{array}{c}{B}_m=B_m(g_1,g_2,\dots ,g_m)=det{\left({\lambda }_k^{j-1}{g}_{j,k}\right)}_{\{j,k=1,\dots ,m\}},\hfill \end{array} \mathrm{with}\end{array}$$

(12)

$$\begin{array}{c}\hfill \begin{array}{c}{g}_{2j-1,k}={\phi }_k,g_{2j,k}={\psi }_k,\hfill \end{array}\end{array}$$

(13)

• $g_k$ being the m dimensional column vector $\left(\begin{array}{c}{g}_{1,k}\\ ⋮\\ g_{m,k}\end{array}\right),$
• for the ${\phi }_k$, ${\psi }_k$, $\phi$, and $\psi$ solutions of the system of compatibility conditions associated, respectively, with ${\lambda }_k$ and $\lambda$.

We take the convention that $B_0=A_0=1$.

We denote F and G as the following m dimensional column vectors

$$F=\left(\begin{array}{c}{F}_1\\ ⋮\\ F_m\end{array}\right),G=\left(\begin{array}{c}{G}_1\\ ⋮\\ G_m\end{array}\right),$$

with $G_{2j-1}=\phi ,G_{2j}=\psi$, $F_{2j-1,k}=\psi ,F_{2j}=\phi$.

When there is no ambiguity, we mention only $A_m$ or $B_m$ in place of $A_m(f_1,\dots ,f_m)$ or $B_m(g_1,\dots ,g_m)$ in order to make the text more readable.

The functions ${\mathrm{\Psi }}_m$, ${\mathrm{\Phi }}_m$, $v_m$, $w_m$ and $u_m$ are defined by

$$\begin{array}{c}\hfill \begin{array}{c}{\mathrm{\Phi }}_m={\displaystyle \frac{B_{m+1}}{B_m}},{\mathrm{\Psi }}_m={\displaystyle \frac{A_{m+1}}{A_m}},\end{array}\end{array}$$

(14)

$$\begin{array}{c}\hfill \begin{array}{c}{w}_m=w\times {\left({\displaystyle \frac{B_m}{A_m}}\right)}^2,v_m=v+{\mathsf{\partial }}_{x}ln\left({\displaystyle \frac{B_m}{A_m}}\right),\hfill \end{array}\end{array}$$

(15)

$$\begin{array}{c}\hfill \begin{array}{c}{u}_m=u-2iln{\displaystyle \frac{B_m}{A_m}}.\hfill \end{array}\end{array}$$

(16)

2. Compatibility Condition

The (SG) equation

$${\mathsf{\partial }}_{xt}u(x,t)=4sinu(x,t),$$

(17)

can be seen as a compatibility condition of (18) and (19):

$${\mathsf{\partial }}_x\psi =v\psi +\lambda \phi ,{\mathsf{\partial }}_t\psi ={\lambda }^{-1}w\phi ,$$

(18)

$${\mathsf{\partial }}_x\phi =\lambda \psi -v\phi ,{\mathsf{\partial }}_t\phi ={\lambda }^{-1}{w}^{-1}\psi .$$

(19)

Proposition 1.

The compatibility condition of the Equations (18) and (19) is equivalent to the Equation (17).

Proof. 

1. We consider $v={\displaystyle \frac{i{\mathsf{\partial }}_{x}u}{2}}$ and $w=e^{iu}$.

By deriving v, we get ${\mathsf{\partial }}_{t}v={\displaystyle \frac{i{\mathsf{\partial }}_{xt}u}{2}}=2isinu=w-w^{-1}$.

By deriving w, we get ${\mathsf{\partial }}_{x}w=i{\mathsf{\partial }}_x\left(u\right)e^{iu}=2vw$,

which also acts reciprocally, so the (SG) Equation (17) is equivalent to ${\mathsf{\partial }}_{t}v=w-w^{-1}$ and ${\mathsf{\partial }}_{x}w=2vw$.

2. a. By deriving the Equations (18) and (19), we get the following relation

$$\begin{array}{c}\hfill \begin{array}{c}{\mathsf{\partial }}_{xt}\psi ={\mathsf{\partial }}_t\left(v\right)\psi +v{\lambda }^{-1}w\phi +\lambda {\lambda }^{-1}{w}^{-1}\psi ={\lambda }^{-1}2vw\phi +{\lambda }^{-1}w(\lambda \psi -v\phi )\end{array}\end{array}$$

(20)

which can be reduced to ${\mathsf{\partial }}_{t}v=w-w^{-1}$.

b. By deriving the Equation (19) we get the following relation

$$\begin{array}{c}\hfill \begin{array}{c}{\mathsf{\partial }}_{xt}\phi =\lambda \left({\lambda }^{-1}w\phi \right)-{\mathsf{\partial }}_t\left(v\right)\phi -v\left({\lambda }^{-1}{w}^{-1}\psi \right)\\ =-{\lambda }^{-1}{\mathsf{\partial }}_x\left(w\right)w^{-2}\psi +{\lambda }^{-1}{w}^{-1}(v\psi +\lambda \phi )\end{array}\end{array}$$

(21)

which can be reduced to ${\mathsf{\partial }}_{x}w=2vw$.

This can act conversely, which gives the equivalence of the proposition. □

We use the previous system, to construct solutions by means of the Darboux transformation, which we define in the following.

3. Covariance Property and Darboux Transformations

We consider arbitrary solutions $\psi$ and $\phi$ of the system (18), (19) associated with $\lambda$ and particular solutions ${\phi }_1$ and ${\psi }_1$ of the system (18), (19) corresponding to ${\lambda }_1$. We define the Darboux transformations ${\mathrm{\Phi }}_1$ and ${\mathrm{\Psi }}_1$ by the formulas

$$\begin{array}{c}{\mathrm{\Phi }}_1=\lambda \psi -{\lambda }_1{\psi }_1{\phi }_1^{-1}\phi ,{\mathrm{\Psi }}_1=\lambda \phi -{\lambda }_1{\phi }_1{\psi }_1^{-1}\psi .\end{array}$$

Theorem 1.

The functions ${\mathrm{\Psi }}_1$ and ${\mathrm{\Phi }}_1$ defined by

$$\begin{array}{c}{\mathrm{\Phi }}_1=\lambda \psi -{\lambda }_1{\psi }_1{\phi }_1^{-1}\phi ,{\mathrm{\Psi }}_1=\lambda \phi -{\lambda }_1{\phi }_1{\psi }_1^{-1}\psi .\end{array}$$

(22)

represent general solutions of the following equations

$${\mathsf{\partial }}_x{\mathrm{\Psi }}_1=v_1{\mathrm{\Psi }}_1+\lambda {\mathrm{\Phi }}_1,{\mathsf{\partial }}_t{\mathrm{\Psi }}_1={\lambda }^{-1}{w}_1{\mathrm{\Phi }}_1,$$

(23)

$${\mathsf{\partial }}_x{\mathrm{\Phi }}_1=\lambda {\mathrm{\Psi }}_1-v_1{\mathrm{\Phi }}_1,{\mathsf{\partial }}_t{\mathrm{\Phi }}_1={\lambda }^{-1}{w}_1^{-1}{\mathrm{\Psi }}_1.$$

(24)

with

$$\begin{array}{c}{v}_1=v+{\mathsf{\partial }}_{x}ln\left({\displaystyle \frac{{\phi }_1}{{\psi }_1}}\right),\hfill \end{array}$$

(25)

$$\begin{array}{c}{w}_1=w\times {\left({\displaystyle \frac{{\phi }_1}{{\psi }_1}}\right)}^2.\hfill \end{array}$$

(26)

For u solution of the (SG) equation, the function

$$\begin{array}{c}{u}_1=u-2iln{\displaystyle \frac{{\phi }_1}{{\psi }_1}}\hfill \end{array}$$

(27)

is another solution to the (SG) equation

$${\mathsf{\partial }}_{xt}{u}_1(x,t)=4sin{u}_1(x,t).$$

Remark 1.

In [9,10], the expressions of $v_1$ and $w_1$ in the case of the simple Darboux transformation as well as the expressions of $v_m$ and $w_m$ for the multiple Darboux transformation (as the will see in the next section) were formulated, but no proof of these results were given.

Moreover, in [9], for this type of equation, the results were given with Darboux transformations that were different than these proposed in the present theorem. This is the reason why we present the proofs for the simple and multiple Darboux transformations in the following.

Proof. 

1. The substitution of ${\mathrm{\Psi }}_1$ and ${\mathrm{\Phi }}_1$ by their expressions (22) in the equation

$${\mathsf{\partial }}_x{\mathrm{\Psi }}_1-v_1{\mathrm{\Psi }}_1-\lambda {\mathrm{\Phi }}_1=0$$

• replacing ${\mathsf{\partial }}_x{\psi }_1$ by $v\psi +\lambda \phi$, and ${\mathsf{\partial }}_x\phi$ by $-v\phi +\lambda \psi$
• gives the following relation

  $$\begin{array}{c}\hfill -v_1\left(\lambda \phi -{\displaystyle \frac{{\lambda }_1{\phi }_1\psi }{{\psi }_1}}\right)-v\left(\lambda \phi -{\displaystyle \frac{{\lambda }_1{\phi }_1\psi }{{\psi }_1}}\right)+\lambda {\lambda }_1{\displaystyle \frac{{\psi }_1\phi }{{\phi }_1}}\\ \hfill -{\lambda }_1^2\psi -\lambda {\lambda }_1{\displaystyle \frac{{\phi }_1\phi }{{\psi }_1}}+{\lambda }_1^2{\displaystyle \frac{{\phi }_1^2\psi }{{\psi }_1^2}}=0.\end{array}$$

  (28)

So $v_1$ can be expressed as

$$\begin{array}{c}\hfill v_1=-v+{\left(\lambda \phi -{\displaystyle \frac{{\lambda }_1{\phi }_1\psi }{{\psi }_1}}\right)}^{-1}\\ \hfill \left({\displaystyle \frac{{\lambda }_1{\psi }_1}{{\phi }_1}}\left(\lambda \phi -{\displaystyle \frac{{\lambda }_1{\phi }_1\psi }{{\psi }_1}}\right)-{\displaystyle \frac{{\lambda }_1{\phi }_1}{{\psi }_1}}\left(\lambda \phi -{\displaystyle \frac{{\lambda }_1{\phi }_1\psi }{{\psi }_1}}\right)\right).\end{array}$$

(29)

It can be rewritten as

$$\begin{array}{c}\hfill v_1=-v+{\displaystyle \frac{{\lambda }_1{\psi }_1}{{\phi }_1}}-{\displaystyle \frac{{\lambda }_1{\phi }_1}{{\psi }_1}}.\end{array}$$

(30)

Using the relations ${\mathsf{\partial }}_x{\psi }_1=v{\psi }_1+{\lambda }_1{\phi }_1$ and ${\mathsf{\partial }}_x{\phi }_1=-v{\phi }_1+{\lambda }_1{\psi }_1$, it is clear that ${\displaystyle \frac{-\lambda {\phi }_1}{{\psi }_1}}=-{\displaystyle \frac{{\mathsf{\partial }}_x{\psi }_1}{{\psi }_1}}+v$ and ${\displaystyle \frac{{\lambda }_1{\psi }_1}{{\phi }_1}}={\displaystyle \frac{{\mathsf{\partial }}_x{\phi }_1}{{\phi }_1}}+v$ and so we get

$$\begin{array}{c}\hfill v_1=-v+{\displaystyle \frac{{\mathsf{\partial }}_x{\phi }_1}{{\phi }_1}}+v-{\displaystyle \frac{{\mathsf{\partial }}_x{\psi }_1}{{\psi }_1}}+v,\end{array}$$

(31)

or

$$\begin{array}{c}\hfill v_1=v+{\mathsf{\partial }}_{x}ln\left({\displaystyle \frac{{\phi }_1}{{\psi }_1}}\right),\end{array}$$

(32)

and proves the expression (25) for $v_1$.

2. To prove the second relation (26) for $w_1$, it is sufficient to simplify the expression

$$\begin{array}{c}\hfill {\mathsf{\partial }}_t{\mathrm{\Psi }}_1-{\lambda }^{-1}{w}_1{\mathrm{\Phi }}_1=0.\end{array}$$

(33)

Replacing ${\mathrm{\Psi }}_1$ and ${\mathrm{\Phi }}_1$ by their expressions (22) in the previous expression, we get

$$\lambda {\mathsf{\partial }}_t\phi -{\displaystyle \frac{{\lambda }_1\psi {\mathsf{\partial }}_t{\phi }_1}{{\psi }_1}}-{\displaystyle \frac{{\lambda }_1{\phi }_1{\mathsf{\partial }}_t\psi }{{\psi }_1}}+{\displaystyle \frac{{\lambda }_1{\phi }_1\psi {\mathsf{\partial }}_t{\psi }_1}{{\psi }_1^2}}-w_1\left(\psi -{\displaystyle \frac{{\lambda }_1{\psi }_1\phi }{\lambda {\phi }_1}}\right)=0.$$

It can be rewritten as

$$w\left({\displaystyle \frac{\psi {\phi }_1^2}{{\psi }_1^2}}-{\displaystyle \frac{{\lambda }_1{\phi }_1\phi }{\lambda {\psi }_1^2}}\right)-w_1\left(\psi -{\displaystyle \frac{{\lambda }_1{\psi }_1\phi }{\lambda {\phi }_1}}\right)=0$$

or

$$w{\left({\displaystyle \frac{{\phi }_1}{{\psi }_1}}\right)}^2\left(\psi -{\displaystyle \frac{{\lambda }_1{\psi }_1\phi }{\lambda {\phi }_1}}\right)=w_1\left(\psi -{\displaystyle \frac{{\lambda }_1{\psi }_1\phi }{\lambda {\phi }_1}}\right)$$

which proves the relation (26) for $w_1$

$$w_1=w{\left({\displaystyle \frac{{\phi }_1}{{\psi }_1}}\right)}^2.$$

3. By construction $u=-iln\left(w\right)$ and $u_1=-iln\left(w_1\right)$ is a solution to the (SG) equation.

Thus, it is obvious that

$$u_1=-iln\left(w_1\right)=-iln\left(w\right)-iln\left({\left({\displaystyle \frac{{\phi }_1}{{\psi }_1}}\right)}^2\right)=u-2iln\left({\displaystyle \frac{{\phi }_1}{{\psi }_1}}\right)$$

is a solution of the (SG) equation.

This completes the proof of the theorem. □

4. Multiple Darboux Transformation

We consider the following determinants $A_m$ and $B_m$ defined by

$$\begin{array}{c}\hfill \begin{array}{c}{A}_m=A_m(f_1,f_2,\dots ,f_m)=det{\left({\lambda }_k^{j-1}{f}_{k,j}\right)}_{\{j,k=1,\dots ,m\}},\hfill \end{array} \mathrm{with}\end{array}$$

(34)

$$\begin{array}{c}\hfill \begin{array}{c}{f}_{2j-1,k,}={\psi }_k,f_{2j,k}={\phi }_k.\hfill \end{array}\end{array}$$

(35)

$f_k$ being the m dimensional column vector $\left(\begin{array}{c}{f}_{1,k}\\ ⋮\\ f_{m,k}\end{array}\right)$

$$\begin{array}{c}\hfill \begin{array}{c}{B}_m=B_m(g_1,g_2,\dots ,g_m)=det{\left({\lambda }_k^{j-1}{g}_{j,k}\right)}_{\{j,k=1,\dots ,m\}},\hfill \end{array} \mathrm{with}\end{array}$$

(36)

$$\begin{array}{c}\hfill \begin{array}{c}{g}_{2j-1,k}={\phi }_k,g_{2j,k}={\psi }_k,\hfill \end{array}\end{array}$$

(37)

$g_k$ being the m dimensional column vector $\left(\begin{array}{c}{g}_{1,k}\\ ⋮\\ g_{m,k}\end{array}\right)$

Remark 2.

We choose the following convention $B_0=A_0=1$.

We denote F and G as the following m dimensional column vectors

$$F=\left(\begin{array}{c}{F}_1\\ ⋮\\ F_m\end{array}\right),G=\left(\begin{array}{c}{G}_1\\ ⋮\\ G_m\end{array}\right),$$

with $G_{2j-1}=\phi ,G_{2j}=\psi$, $F_{2j-1,k}=\psi ,F_{2j}=\phi$.

Remark 3.

When there is no ambiguity, we mention only $A_m$ or $B_m$ at the place of $A_m(f_1,\dots ,f_m)$ or $B_m(g_1,\dots ,g_m)$ in order to lighten the text.

Then we have the following statement:

Theorem 2.

The functions ${\mathrm{\Psi }}_m$ and ${\mathrm{\Phi }}_m$ are defined by

$$\begin{array}{c}\hfill \begin{array}{c}{\mathrm{\Phi }}_m={\displaystyle \frac{B_{m+1}}{B_m}},{\mathrm{\Psi }}_m={\displaystyle \frac{A_{m+1}}{A_m}}.\end{array}\end{array}$$

(38)

which represent the general solutions of the following equations

$${\mathsf{\partial }}_x{\mathrm{\Psi }}_m=v_m{\mathrm{\Psi }}_m+\lambda {\mathrm{\Phi }}_m,{\mathsf{\partial }}_t{\mathrm{\Psi }}_m={\lambda }^{-1}{w}_m{\mathrm{\Phi }}_m,$$

(39)

$${\mathsf{\partial }}_x{\mathrm{\Phi }}_m=\lambda {\mathrm{\Psi }}_m-v_m{\mathrm{\Phi }}_m,{\mathsf{\partial }}_t{\mathrm{\Phi }}_m={\lambda }^{-1}{w}_m^{-1}{\mathrm{\Psi }}_m.$$

(40)

with

$$\begin{array}{c}\hfill \begin{array}{c}{w}_m=w\times {\left({\displaystyle \frac{B_m}{A_m}}\right)}^2,v_m=v+{\mathsf{\partial }}_{x}ln\left({\displaystyle \frac{B_m}{A_m}}\right).\hfill \end{array}\end{array}$$

(41)

The function

$$\begin{array}{c}\hfill \begin{array}{c}{u}_m=u-2iln{\displaystyle \frac{B_m}{A_m}}\hfill \end{array}\end{array}$$

(42)

is a solution to the (SG) equation

$${\mathsf{\partial }}_{xt}{u}_m(x,t)=4sin{u}_m(x,t).$$

Remark 4.

To shorten the text, if there is no ambiguity, we mention only ${\mathrm{\Psi }}_m(f_i,g_i,F,G)$ or ${\mathrm{\Phi }}_m(f_i,g_i,F,G)$ in place of ${\mathrm{\Psi }}_m(f_1,\dots ,f_m,g_1,\dots ,g_m,F,G)$ or ${\mathrm{\Phi }}_m(f_1,\dots ,f_m,g_1,\dots ,g_m,F,G)$.

We give another formulation for ${\mathrm{\Psi }}_m\left(f\right)$ and ${\mathrm{\Phi }}_m\left(f\right)$. Let $f_1,\dots ,f_m,F,g_1,\dots ,g_m,G$ be independent solutions of (18), (19), respectively, associated with ${\lambda }_1,\dots ,{\lambda }_m,\lambda$.

Then, we have the following expression for ${\mathrm{\Psi }}_m$ and ${\mathrm{\Phi }}_m$.

Lemma 1.

$$\begin{array}{c}\hfill {\mathrm{\Psi }}_m(f_1,\dots ,f_m,g_1,\dots ,g_m,F,G)=\lambda {\mathrm{\Phi }}_{m-1}(g_1,\dots ,g_{m-1},G)\\ \hfill -{\lambda }_m{\displaystyle \frac{{\mathrm{\Phi }}_{m-1}(g_1,\dots ,g_m)}{{\mathrm{\Psi }}_{m-1}(f_1,\dots ,f_m)}}{\mathrm{\Psi }}_{m-1}(f_1,\dots ,f_{m-1},F)\end{array}$$

(43)

$$\begin{array}{c}\hfill {\mathrm{\Phi }}_m(f_1,\dots ,f_m,g_1,\dots ,g_m,F,G)=\lambda {\mathrm{\Psi }}_{m-1}(f_1,\dots ,f_{m-1},F)\\ \hfill -{\lambda }_m{\displaystyle \frac{{\mathrm{\Psi }}_{m-1}(f_1,\dots ,f_m)}{{\mathrm{\Phi }}_{m-1}(g_1,\dots ,g_m)}}{\mathrm{\Phi }}_{m-1}(g_1,\dots ,g_{m-1},G)\end{array}$$

(44)

Then ${\mathrm{\Psi }}_m$, ${\mathrm{\Phi }}_m$ can be written as

$$\begin{array}{c}\hfill {\mathrm{\Psi }}_m={\displaystyle \frac{A_{m+1}(f_1,\dots ,f_m,F)}{A_m(f_1,\dots ,f_m)}}.\end{array}$$

(45)

$$\begin{array}{c}\hfill {\mathrm{\Phi }}_m={\displaystyle \frac{B_{m+1}(g_1,\dots ,g_m,G)}{B_m(g_1,\dots ,g_m)}}.\end{array}$$

(46)

Proof of the Lemma.

1. In the case $m=1$

$$\begin{array}{c}\hfill \begin{array}{c}{\mathrm{\Phi }}_1=\lambda \psi -{\lambda }_1{\psi }_1{\phi }_1^{-1}\phi ,{\mathrm{\Psi }}_1=\lambda \phi -{\lambda }_1{\phi }_1{\psi }_1^{-1}\psi .\end{array}\end{array}$$

(47)

verify

$$\begin{array}{c}\hfill {\mathrm{\Phi }}_1\left(G\right)=\lambda {\mathrm{\Psi }}_0\left(F\right)-{\lambda }_1{\displaystyle \frac{{\mathrm{\Psi }}_0\left(f_1\right)}{{\mathrm{\Phi }}_0\left(g_1\right)}}{\mathrm{\Phi }}_0\left(G\right),\\ \hfill {\mathrm{\Psi }}_1\left(F\right)=\lambda {\mathrm{\Phi }}_0\left(G\right)-{\lambda }_1{\displaystyle \frac{{\mathrm{\Phi }}_0\left(g_1\right)}{{\mathrm{\Psi }}_0\left(f_1\right)}}{\mathrm{\Psi }}_0\left(F\right),\end{array}$$

(48)

with ${\mathrm{\Psi }}_0\left(F\right)=\psi$, ${\mathrm{\Phi }}_0\left(G\right)=\phi$, ${\mathrm{\Psi }}_0\left(f_1\right)={\psi }_1$, ${\mathrm{\Phi }}_0\left(g_1\right)={\phi }_1$.

These previous relations (47) can be written as

$$\begin{array}{c}\hfill \mathrm{\Psi }(f_1,F)={\displaystyle \frac{A_2(f_1,F)}{A_1\left(f_1\right)}},\mathrm{\Phi }(g_1,G)={\displaystyle \frac{B_2(g_1,G)}{B_1\left(g_1\right)}}.\end{array}$$

(49)

which proves the Lemma in the case $m=1$.

2. From the relations (43), (44)

$$\begin{array}{c}\hfill {\mathrm{\Psi }}_m(f_1,\dots ,f_m,g_1,\dots ,g_m,F,G)=\lambda {\mathrm{\Phi }}_{m-1}(g_1,\dots ,g_{m-1},G)\\ \hfill -{\lambda }_m{\displaystyle \frac{{\mathrm{\Phi }}_{m-1}(g_1,\dots ,g_m)}{{\mathrm{\Psi }}_{m-1}(f_1,\dots ,f_m)}}{\mathrm{\Psi }}_{m-1}(f_1,\dots ,f_{m-1},F)\end{array}$$

$$\begin{array}{c}\hfill {\mathrm{\Phi }}_m(f_1,\dots ,f_m,g_1,\dots ,g_m,F,G)=\lambda {\mathrm{\Psi }}_{m-1}(f_1,\dots ,f_{m-1},F)\\ \hfill -{\lambda }_m{\displaystyle \frac{{\mathrm{\Psi }}_{m-1}(f_1,\dots ,f_m)}{{\mathrm{\Phi }}_{m-1}(g_1,\dots ,g_m)}}{\mathrm{\Phi }}_{m-1}(g_1,\dots ,g_{m-1},G)\end{array}$$

and the recurrence hypothesis

$$\begin{array}{c}{\mathrm{\Phi }}_{m-1}={\displaystyle \frac{B_m}{B_{m-1}}},{\mathrm{\Psi }}_{m-1}={\displaystyle \frac{A_m}{A_{m-1}}},\end{array}$$

it is clear, by the definition of ${\mathrm{\Psi }}_m$ and ${\mathrm{\Phi }}_m$, that ${\mathrm{\Psi }}_m(f_1,\dots ,f_m,g_1,\dots ,g_m,F,G)=0$ and ${\mathrm{\Psi }}_m(f_1,\dots ,f_m,g_1,\dots ,g_m,F,G)=0$ for $F=f_1,\dots ,f_{m-1}$, and $G=g_1,\dots ,g_{m-1}$.

We can evaluate ${\mathrm{\Phi }}_m$ and ${\mathrm{\Psi }}_m$ for $F=f_m$ and $G=g_m$.

a. Using the recurrence hypothesis

$$\begin{array}{c}{\mathrm{\Psi }}_m{(f_1,\dots ,f_m,g_1,\dots ,g_m,F,G)}_{F=f_m,G=g_m}\hfill \\ ={\lambda }_m{\displaystyle \frac{B_m(g_1,\dots ,g_m)}{B_{m-1}(g_1,\dots ,g_{m-1})}}-{\lambda }_m{\displaystyle \frac{{\displaystyle \frac{B_m(g_1,\dots ,g_m)}{B_{m-1}(g_1,\dots ,g_{m-1})}}}{{\displaystyle \frac{A_m(f_1,\dots ,f_m)}{A_{m-1}(f_1,\dots ,f_{m-1})}}}}{\displaystyle \frac{A_m(f_1,\dots ,f_m)}{A_{m-1}(f_1,\dots ,f_{m-1})}}=0\hfill \end{array}$$

b. We also have

$$\begin{array}{c}{\mathrm{\Phi }}_m{(f_1,\dots ,f_m,g_1,\dots ,g_m,F,G)}_{F=f_m,G=g_m}\hfill \\ ={\lambda }_m{\displaystyle \frac{A_m(f_1,\dots ,f_m)}{A_{m-1}(f_1,\dots ,f_{m-1})}}-{\lambda }_m{\displaystyle \frac{{\displaystyle \frac{A_m(f_1,\dots ,f_m)}{A_{m-1}(f_1,\dots ,f_{m-1})}}}{{\displaystyle \frac{B_m(g_1,\dots ,g_m)}{B_{m-1}(g_1,\dots ,g_{m-1})}}}}{\displaystyle \frac{B_m(g_1,\dots ,g_m)}{B_{m-1}(g_1,\dots ,g_{m-1})}}=0\hfill \end{array}$$

Moreover, $C_m$ and $D_m$ are defined by

$$\begin{array}{c}\hfill C_m={\displaystyle \frac{A_{m+1}}{A_m}},D_m={\displaystyle \frac{B_{m+1}}{B_m}}.\end{array}$$

(50)

verify $C_m=0$ and $D_m=0$ for $F=f_1,\dots ,f_m$ and $G=g_1,\dots ,g_m$ and have the same coefficient in ${\lambda }^m$, respectively, so that ${\mathrm{\Psi }}_m$ and ${\mathrm{\Phi }}_m$.

However, ${\mathrm{\Psi }}_m$ and ${\mathrm{\Phi }}_m$ are defined as equal to 0 for $F=f_1,\dots ,f_m$ and $G=g_1,\dots ,g_m$, respectively. The coefficients of these two polynomials are solutions to a system of m equations with m unknowns, respectively. Thus, they are determined in a unique way for the reason that $A_m\ne 0$ and $B_m\ne 0$.

The terms

$${\displaystyle \frac{A_{m+1}}{A_m}},{\displaystyle \frac{B_{m+1}}{B_m}}$$

are equal to 0 for $f=f_1,\dots ,f_m$ and has the same coefficient in ${\lambda }^m$ that ${\mathrm{\Psi }}_m$ and ${\mathrm{\Phi }}_m$ respectively, which proves the relation

$$\begin{array}{c}\hfill {\mathrm{\Psi }}_m={\displaystyle \frac{A_{m+1}}{A_m}}, \mathrm{and} {\mathrm{\Phi }}_m={\displaystyle \frac{B_{m+1}}{B_m}}.\end{array}$$

(51)

The property at the rank is m and so is the lemma. □

Proof of the Theorem.

The theorem in the case $m=1$ has been already proven in the previous section.

We suppose that the theorem is verified for $m-1$.

1. In this part, to lighten the text and avoid ambiguity, we will denote the following

$$\begin{array}{c}{\mathrm{\Psi }}_m(f_1,\dots ,f_m,g_1,\dots ,g_m,F,G)\mathrm{by}{\mathrm{\Psi }}_m,\hfill \\ {\mathrm{\Phi }}_m(f_1,\dots ,f_m,g_1,\dots ,g_m,F,G)\mathrm{by}{\mathrm{\Phi }}_m,\hfill \\ {\mathrm{\Psi }}_{m-1}(f_1,\dots ,f_{m-1},g_1,\dots ,g_{m-1},F,G)\mathrm{by}{\mathrm{\Psi }}_{m-1}\left(F\right),\hfill \\ {\mathrm{\Phi }}_{m-1}(f_1,\dots ,f_{m-1},g_1,\dots ,g_{m-1},F,G)\mathrm{by}{\mathrm{\Phi }}_{m-1}\left(G\right),\hfill \\ {\mathrm{\Psi }}_{m-1}(f_1,\dots ,f_{m-1},g_1,\dots ,g_{m-1},f_m,g_m)\mathrm{by}{\mathrm{\Psi }}_{m-1}\left(f_m\right),\hfill \\ {\mathrm{\Phi }}_{m-1}(f_1,\dots ,f_{m-1},g_1,\dots ,g_{m-1},f_m,g_m)\mathrm{by}{\mathrm{\Phi }}_{m-1}\left(g_m\right).\hfill \end{array}$$

The substitution of ${\mathrm{\Psi }}_m$ and ${\mathrm{\Phi }}_m$ by their expressions

$$\begin{array}{c}\hfill {\mathrm{\Psi }}_m=\lambda {\mathrm{\Phi }}_{m-1}\left(G\right)-{\lambda }_m{\displaystyle \frac{{\mathrm{\Phi }}_{m-1}\left(g_m\right)}{{\mathrm{\Psi }}_{m-1}\left(f_m\right)}}{\mathrm{\Psi }}_{m-1}\left(F\right)\end{array}$$

(52)

$$\begin{array}{c}\hfill {\mathrm{\Phi }}_m=\lambda {\mathrm{\Psi }}_{m-1}\left(F\right)-{\lambda }_m{\displaystyle \frac{{\mathrm{\Psi }}_m{f}_m}{{\mathrm{\Phi }}_{m-1}\left(g_m\right)}}{\mathrm{\Phi }}_{m-1}\left(G\right)\end{array}$$

(53)

in the equation

$${\mathsf{\partial }}_x{\mathrm{\Psi }}_m\left(F\right)-v_m{\mathrm{\Psi }}_m-\left(F\right)-\lambda {\mathrm{\Phi }}_m\left(G\right)=0$$

replacing

$$\begin{array}{c}{\mathsf{\partial }}_x{\mathrm{\Psi }}_{m-1}\left(F\right)\mathrm{by}{v}_{m-1}{\mathrm{\Psi }}_{m-1}\left(F\right)+\lambda {\mathrm{\Phi }}_{m-1}\left(G\right),\hfill \\ {\mathsf{\partial }}_x{\mathrm{\Phi }}_{m-1}\left(G\right)\mathrm{by}-v_{m-1}{\mathrm{\Phi }}_{m-1}\left(G\right)+\lambda {\mathrm{\Psi }}_{m-1}\left(F\right),\hfill \\ {\mathsf{\partial }}_x{\mathrm{\Psi }}_{m-1}\left(g_m\right)\mathrm{by}{v}_{m-1}{\mathrm{\Psi }}_{m-1}\left(g_m\right)+\lambda {\mathrm{\Phi }}_{m-1}\left(g_m\right),\hfill \\ {\mathsf{\partial }}_x{\mathrm{\Phi }}_{m-1}\left(g_m\right)\mathrm{by}-v_{m-1}{\mathrm{\Phi }}_{m-1}\left(f_m\right)+\lambda {\mathrm{\Psi }}_{m-1}\left(f_m\right),\hfill \end{array}$$

gives the following relation

$$\begin{array}{c}-v_m\left(\lambda {\mathrm{\Phi }}_{m-1}\left(G\right)-{\displaystyle \frac{{\lambda }_m{\mathrm{\Phi }}_{m-1}\left(g_m\right){\mathrm{\Psi }}_{m-1}\left(F\right)}{{\mathrm{\Psi }}_{m-1}\left(f_m\right)}}\right)\hfill \\ -v_{m-1}\left(\lambda {\mathrm{\Phi }}_{m-1}\left(G\right)-{\displaystyle \frac{{\lambda }_m{\mathrm{\Phi }}_{m-1}\left(g_m\right){\mathrm{\Psi }}_{m-1}\left(F\right)}{{\mathrm{\Psi }}_{m-1}\left(f_m\right)}}\right)\hfill \\ -\lambda {\lambda }_m{\displaystyle \frac{{\mathrm{\Phi }}_{m-1}\left(G\right){\mathrm{\Psi }}_{m-1}\left(f_m\right)}{{\mathrm{\Phi }}_{m-1}\left(g_m\right)}}-{\lambda }_m^2{\mathrm{\Psi }}_{m-1}\left(F\right)\hfill \\ -\lambda {\lambda }_m{\displaystyle \frac{{\mathrm{\Phi }}_{m-1}\left(g_m\right){\mathrm{\Phi }}_{m-1}\left(G\right)}{{\mathrm{\Psi }}_{m-1}\left(f_m\right)}}+{\lambda }_m^2{\displaystyle \frac{{\mathrm{\Phi }}_{m-1}^2\left(g_m\right){\mathrm{\Psi }}_{m-1}\left(F\right)}{{\mathrm{\Psi }}_{m-1}^2\left(f_m\right)}}=0.\hfill \end{array}$$

So $v_1$ can be expressed as

$$\begin{array}{c}{v}_m=-v_{m-1}+{\left(\lambda {\mathrm{\Phi }}_{m-1}\left(G\right)-{\displaystyle \frac{{\lambda }_m{\mathrm{\Phi }}_{m-1}\left(g_m\right){\mathrm{\Psi }}_{m-1}\left(F\right)}{{\mathrm{\Psi }}_{m-1}\left(f_m\right)}}\right)}^{-1}\times \hfill \\ \left[{\displaystyle \frac{{\lambda }_m{\mathrm{\Psi }}_{m-1}\left(f_m\right)}{{\mathrm{\Phi }}_{m-1}\left(g_m\right)}}\left(\lambda {\mathrm{\Phi }}_{m-1}\left(G\right)-{\displaystyle \frac{{\lambda }_m{\mathrm{\Phi }}_{m-1}\left(g_m\right){\mathrm{\Psi }}_{m-1}\left(F\right)}{{\mathrm{\Psi }}_{m-1}\left(f_m\right)}}\right)\right.\hfill \\ \left.-{\displaystyle \frac{{\lambda }_m{\mathrm{\Phi }}_{m-1}\left(g_m\right)}{{\mathrm{\Psi }}_{m-1}\left(f_m\right)}}\left(\lambda {\mathrm{\Phi }}_{m-1}\left(G\right)-{\displaystyle \frac{{\lambda }_m{\mathrm{\Phi }}_{m-1}\left(g_m\right){\mathrm{\Psi }}_{m-1}\left(F\right)}{{\mathrm{\Psi }}_{m-1}\left(f_m\right)}}\right)\right].\hfill \end{array}$$

It can be rewritten as

$$\begin{array}{c}\hfill v_m=-v_{m-1}+{\displaystyle \frac{{\lambda }_m{\mathrm{\Psi }}_{m-1}\left(f_m\right)}{{\mathrm{\Phi }}_{m-1}\left(g_m\right)}}-{\displaystyle \frac{{\lambda }_m{\mathrm{\Phi }}_{m-1}\left(g_m\right)}{{\mathrm{\Psi }}_{m-1}\left(f_m\right)}},\end{array}$$

(54)

or

$$\begin{array}{c}\hfill v_m=-v_{m-1}+{\displaystyle \frac{{\mathsf{\partial }}_x{\mathrm{\Phi }}_{m-1}\left(g_m\right)}{{\mathrm{\Phi }}_{m-1}\left(g_m\right)}}+v_{m-1}-{\displaystyle \frac{{\mathsf{\partial }}_x{\mathrm{\Psi }}_{m-1}(f_m)}{{\mathrm{\Psi }}_{m-1}\left(f_m\right)}}+v_{m-1}\\ \hfill =v+{\mathsf{\partial }}_{x}ln\left({\displaystyle \frac{{\mathrm{\Phi }}_{m-1}\left(g_m\right)}{{\mathrm{\Psi }}_{m-1}\left(f_m\right)}}\right),\end{array}$$

(55)

and proves the expression for recurrence relation for v.

From the lemma, we can express ${\mathrm{\Phi }}_{m-1}\left(g_m\right)$ and ${\mathrm{\Psi }}_{m-1}\left(f_m\right)$ in function of determinants A and B

$${\mathrm{\Phi }}_{m-1}\left(g_m\right)={\displaystyle \frac{B_m(g_1,\dots ,g_m)}{B_{m-1}(g_1,\dots ,g_{m-1})}},{\mathrm{\Psi }}_{m-1}\left(f_m\right)={\displaystyle \frac{B_m(f_1,\dots ,f_m)}{B_{m-1}(f_1,\dots ,f_{m-1})}}.$$

The relation (55) can be written as

$$v_m=v+{\mathsf{\partial }}_x\left(ln{\displaystyle \frac{g_1}{f_1}}\right)+\sum _{k=2}^m{\mathsf{\partial }}_x\left(ln\left({\displaystyle \frac{B_k(g_1,\dots ,g_k)A_{k-1}(f_1,\dots ,f_{k-1})}{B_{k-1}(g_1,\dots ,g_{k-1})A_k(f_1,\dots ,f_k)}}\right)\right).$$

which can be written as

$$v_m=v+{\mathsf{\partial }}_x\left(ln{\displaystyle \frac{g_1}{f_1}}\right)+\left({\mathsf{\partial }}_{x}ln\prod _{k=2}^m\left({\displaystyle \frac{B_k(g_1,\dots ,g_k)A_{k-1}(f_1,\dots ,f_{k-1})}{B_{k-1}(g_1,\dots ,g_{k-1})A_k(f_1,\dots ,f_k)}}\right)\right).$$

which can be reduced to

$$v_m=v+{\mathsf{\partial }}_x\left(ln{\displaystyle \frac{g_1}{f_1}}\right)+\left({\mathsf{\partial }}_{x}ln\left({\displaystyle \frac{B_m(g_1,\dots ,g_m)A_1\left(f_1\right)}{B_1\left(g_1\right)A_m(f_1,\dots ,f_m)}}\right)\right).$$

As $A_1\left(f_1\right)=f_1$ and $B_1\left(g_1\right)=g_1$, it is also equal to

$$v_m=v+{\mathsf{\partial }}_{x}ln\left({\displaystyle \frac{B_m(g_1,\dots ,g_m)}{A_m(f_1,\dots ,f_m)}}\right),$$

which is the first relation of the theorem.

2. To prove the second relation of the theorem, it is sufficient to simplify the expression

$$\begin{array}{c}\hfill {\mathsf{\partial }}_t{\mathrm{\Psi }}_m-{\lambda }^{-1}{w}_m{\mathrm{\Phi }}_m=0.\end{array}$$

(56)

Replacing ${\mathrm{\Psi }}_m$ and ${\mathrm{\Phi }}_m$ by their expressions

$$\begin{array}{c}\hfill {\mathrm{\Psi }}_m=\lambda {\mathrm{\Phi }}_{m-1}\left(G\right)-{\lambda }_m{\displaystyle \frac{{\mathrm{\Phi }}_{m-1}\left(g_m\right)}{{\mathrm{\Psi }}_{m-1}\left(f_m\right)}}{\mathrm{\Psi }}_{m-1}\left(F\right)\end{array}$$

(57)

$$\begin{array}{c}\hfill {\mathrm{\Phi }}_m=\lambda {\mathrm{\Psi }}_{m-1}\left(F\right)-{\lambda }_m{\displaystyle \frac{{\mathrm{\Psi }}_m\left(f_m\right)}{{\mathrm{\Phi }}_{m-1}\left(g_m\right)}}{\mathrm{\Phi }}_{m-1}\left(G\right)\end{array}$$

(58)

we get

$$\begin{array}{c}\lambda {\mathsf{\partial }}_t{\mathrm{\Phi }}_{m-1}\left(G\right)-{\displaystyle \frac{{\lambda }_m{\mathsf{\partial }}_t{\mathrm{\Phi }}_{m-1}\left(g_m\right){\mathrm{\Psi }}_{m-1}\left(F\right)}{{\mathrm{\Psi }}_{m-1}\left(f_m\right)}}-{\displaystyle \frac{{\lambda }_m{\mathrm{\Phi }}_{m-1}\left(f_m\right){\mathsf{\partial }}_t{\mathrm{\Psi }}_{m-1}\left(F\right)}{{\mathrm{\Psi }}_{m-1}\left(f_m\right)}}\hfill \\ +{\displaystyle \frac{{\lambda }_m{\mathrm{\Phi }}_{m-1}\left(g_m\right){\mathrm{\Psi }}_{m-1}\left(F\right){\mathsf{\partial }}_t{\mathrm{\Psi }}_{m-1}\left(f_m\right)}{{\left({\mathrm{\Psi }}_{m-1}\left(f_m\right)\right)}^2}}\hfill \\ -w_{m-1}\left({\mathrm{\Psi }}_{m-1}\left(F\right)-{\displaystyle \frac{{\lambda }_m{\mathrm{\Psi }}_{m-1}\left(f_m\right){\mathrm{\Phi }}_{m-1}\left(G\right)}{\lambda {\mathrm{\Phi }}_{m-1}\left(g_m\right)}}\right)=0.\hfill \end{array}$$

This can be rewritten as

$$\begin{array}{c}{w}_{m-1}{\left({\displaystyle \frac{{\mathrm{\Phi }}_{m-1}\left(g_m\right)}{{\mathrm{\Psi }}_{m-1}\left(f_m\right)}}\right)}^2\left({\mathrm{\Psi }}_{m-1}\left(F\right)-{\displaystyle \frac{{\lambda }_m{\mathrm{\Psi }}_{m-1}\left(f_m\right){\mathrm{\Phi }}_{m-1}\left(G\right)}{\lambda {\mathrm{\Phi }}_{m-1}\left(g_m\right)}}\right)\hfill \\ =w_m\left({\mathrm{\Psi }}_{m-1}\left(F\right)-{\displaystyle \frac{{\lambda }_m{\mathrm{\Psi }}_{m-1}\left(f_m\right){\mathrm{\Phi }}_{m-1}\left(G\right)}{\lambda {\mathrm{\Phi }}_{m-1}\left(g_m\right)}}\right)\hfill \end{array}$$

which proves the relation

$$w_m=w_{m-1}{\left({\displaystyle \frac{{\mathrm{\Phi }}_{m-1}\left(g_m\right)}{{\mathrm{\Psi }}_{m-1}\left(f_m\right)}}\right)}^2.$$

Using the expressions

$${\mathrm{\Phi }}_{m-1}\left(g_m\right)={\displaystyle \frac{B_m(g_1,\dots ,g_m)}{B_{m-1}(g_1,\dots ,g_{m-1})}},{\mathrm{\Psi }}_{m-1}\left(f_m\right)={\displaystyle \frac{B_m(f_1,\dots ,f_m)}{B_{m-1}(f_1,\dots ,f_{m-1})}},$$

$v_m$ can be rewritten as

$$w_m=w\times {\displaystyle \frac{g_1}{f_1}}{\left(\prod _{k=2}m{\displaystyle \frac{B_k(g_1,\dots ,g_k)A_k(f_1,\dots ,f_{k-1})}{B_{k-1}(g_1,\dots ,g_{k-1})A_k(f_1,\dots ,f_k)}}\right)}^2,$$

which can be simplified in

$$w_m=w\times {\left({\displaystyle \frac{B_m(g_1,\dots ,g_m)}{A_m(f_1,\dots ,f_m)}}\right)}^2,$$

and proves the second relation of the theorem.

The proof for ${\mathrm{\Phi }}_{m-1}$ is the same.

3. By construction $u_m=u-iln\left(w_m\right)$, and so

$$u_m=u-2iln\left({\displaystyle \frac{B_m(g_1,\dots ,g_m)}{A_m(f_1,\dots ,f_m)}}\right)=u-2iln\left({\displaystyle \frac{B_m}{A_m}}\right)$$

is a solution to the (SG) equation.

This completes the proof of the theorem. □

5. Explicit Solutions to the Sine-Gordon Equation

In the following, we choose particular generating functions to construct, by means of the Darboux transformation, explicit solutions to the sine-Gordon equation.

We take the functions of the $f_{ij}$ and $g_{ij}$ solutions of the system (18), (19) for elementary potentials $v(x,t)=0$ and $w(x,t)=1$.

We can indifferently choose $a_1,\dots ,a_{m+1}$ reals or $a_1={\alpha }_1+i{\beta }_1,\dots ,a_{m+1}={\alpha }_{m+1}+i{\beta }_{m+1}$ complex to get solutions depending on $m+1$ parameters.

We present different solutions of the sine-Gordon equation. If the coefficients $a_j$, $b_j$, $c_j$ are chosen different for all the values of j, the solutions given by the Darboux transformation are well defined. On the other hand, these solutions can present singularities for certain values of variables x and t.

Then, we can formulate the following result:

Theorem 3.

Let $f_{ij}$ and $g_{ij}$ be the functions defined by

$$\begin{array}{c}\hfill \begin{array}{c}{f}_{ij}=a_j^{i-1}\left(b_j{e}^{a_{j}x-c{j}^{-1}t}-c_j{e}^{-a_{j}x-c{j}^{-1}t}\right) if i is even\hfill \\ f_{ij}=a_j^{i-1}\left(b_j{e}^{a_{j}x+c{j}^{-1}t}+c_j{e}^{-a_{j}x-c{j}^{-1}t}\right) if i is odd;\hfill \\ g_{ij}=a_j^{i-1}\left(b_j{e}^{a_{j}x+c{j}^{-1}t}+c_j{e}^{-a_{j}x-c{j}^{-1}t}\right) if i is even\hfill \\ g_{ij}=a_j^{i-1}\left(b_j{e}^{a_{j}x-c{j}^{-1}t}-c_j{e}^{-a_{j}x-c{j}^{-1}t}\right) if i is odd.\hfill \end{array}\end{array}$$

(59)

Let $A_m$ and $B_m$ be the determinants defined by

$$\begin{array}{c}\hfill A=det{\left(f_{ij}\right)}_{\left\{1\le i\le m,1\le j\le m\right\}};B=det{\left(g_{ij}\right)}_{\left\{1\le i\le m,1\le j\le m\right\}},\end{array}$$

(60)

Then the function $u_m$ defined by

$$\begin{array}{c}\hfill u_m(x,t)=-2iln\left({\displaystyle \frac{B_{m+1}}{A_{m+1}}}\right)\end{array}$$

(61)

is a solution of the sine-Gordon Equation (17).

Proof. 

The functions $f_{ij}$ and $g_{ij}$ verify the relations (18), (19). So applying the multiple Darboux covariance property, we get this result. □

5.1. Case of Order 1

Example 1.

The function u defined by

$$\begin{array}{c}u(x,t)=-2iln\left(\left(b_1{\mathrm{e}}^{{\displaystyle \frac{{a_1}^{2}x+t}{a_1}}}-c_1{\mathrm{e}}^{-{\displaystyle \frac{{a_1}^{2}x+t}{a_1}}}\right)\right.\hfill \\ \times \left.{\left(b_1{\mathrm{e}}^{{\displaystyle \frac{{a_1}^{2}x+t}{a_1}}}+c_1{\mathrm{e}}^{-{\displaystyle \frac{{a_1}^{2}x+t}{a_1}}}\right)}^{-1}\right)\hfill \end{array}$$

is a solution of the sine-Gordon equation.

5.2. Case of Order 2

Example 2.

The function u defined by

$$\begin{array}{c}\hfill \begin{array}{c}u(x,t)=-2iln{\displaystyle \frac{n(x,t)}{d(x,t)}}\hfill \end{array}\end{array}$$

(62)

with

$$\begin{array}{c}n(x,t)=a_2{b}_1{\mathrm{e}}^{{\displaystyle \frac{\left(a_2+a_1\right)\left(a_{2}x{a}_1+t\right)}{a_1{a}_2}}}{b}_2+a_2{b}_1{\mathrm{e}}^{-{\displaystyle \frac{\left(-a_2+a_1\right)\left(-a_{2}x{a}_1+t\right)}{a_1{a}_2}}}{c}_2\hfill \\ -a_2{c}_1{\mathrm{e}}^{{\displaystyle \frac{\left(-a_2+a_1\right)\left(-a_{2}x{a}_1+t\right)}{a_1{a}_2}}}{b}_2-a_2{c}_1{\mathrm{e}}^{-{\displaystyle \frac{\left(a_2+a_1\right)\left(a_{2}x{a}_1+t\right)}{a_1{a}_2}}}{c}_2\hfill \\ -a_1{b}_1{\mathrm{e}}^{{\displaystyle \frac{\left(a_2+a_1\right)\left(a_{2}x{a}_1+t\right)}{a_1{a}_2}}}{b}_2-a_1{c}_1{\mathrm{e}}^{{\displaystyle \frac{\left(-a_2+a_1\right)\left(-a_{2}x{a}_1+t\right)}{a_1{a}_2}}}{b}_2\hfill \\ +a_1{b}_1{\mathrm{e}}^{{\displaystyle -\frac{\left(-a_2+a_1\right)\left(-a_{2}x{a}_1+t\right)}{a_1{a}_2}}}{c}_2+a_1{c}_1{\mathrm{e}}^{-{\displaystyle \frac{\left(a_2+a_1\right)\left(a_{2}x{a}_1+t\right)}{a_1{a}_2}}}{c}_2\hfill \end{array}$$

and

$$\begin{array}{c}d(x,t)=a_2{b}_1{\mathrm{e}}^{{\displaystyle \frac{\left(a_2+a_1\right)\left(a_{2}x{a}_1+t\right)}{a_1{a}_2}}}{b}_2+a_2{c}_1{\mathrm{e}}^{{\displaystyle \frac{\left(-a_2+a_1\right)\left(-a_{2}x{a}_1+t\right)}{a_1{a}_2}}}{b}_2\hfill \\ -a_2{b}_1{\mathrm{e}}^{-{\displaystyle \frac{\left(-a_2+a_1\right)\left(-a_{2}x{a}_1+t\right)}{a_1{a}_2}}}{c}_2-a_2{c}_1{\mathrm{e}}^{-{\displaystyle \frac{\left(a_2+a_1\right)\left(a_{2}x{a}_1+t\right)}{a_1{a}_2}}}{c}_2\hfill \\ -a_1{b}_1{\mathrm{e}}^{{\displaystyle \frac{\left(a_2+a_1\right)\left(a_{2}x{a}_1+t\right)}{a_1{a}_2}}}{b}_2-a_1{b}_1{\mathrm{e}}^{-{\displaystyle \frac{\left(-a_2+a_1\right)\left(-a_{2}x{a}_1+t\right)}{a_1{a}_2}}}{c}_2\hfill \\ +a_1{c}_1{\mathrm{e}}^{{\displaystyle \frac{\left(-a_2+a_1\right)\left(-a_{2}x{a}_1+t\right)}{a_1{a}_2}}}{b}_2+a_1{c}_1{\mathrm{e}}^{-{\displaystyle \frac{\left(a_2+a_1\right)\left(a_{2}x{a}_1+t\right)}{a_1{a}_2}}}{c}_2\hfill \end{array}$$

is a solution of the sine-Gordon equation.

Remark 5.

We could go on and give explicit solutions for higher orders, but the expressions of these solutions become too long for this paper.

Remark 6.

In all these solutions, for the different parameters $a_j$, $b_j$, $c_j$, we can indifferently choose real or complex parameters. We have limited ourselves to real parameters in this study.

6. Wronskian Representations of the Solutions to the Sine-Gordon Equation

In the following, we choose other particular generating functions to construct, by means of the Darboux transformation, explicit solutions to the sine-Gordon equation.

We take particular functions $f_j$ solutions of the system (18), (19) for elementary potentials $v(x,t)=0$ and $w(x,t)=1$.

Then, we can formulate the following result:

Theorem 4.

Let $f_j$ be the functions defined by

$$\begin{array}{c}\hfill \begin{array}{c}{f}_j(x,t)=sinh\left(a_{j}x+a{j}^{-1}t\right);\hfill \end{array}\end{array}$$

(63)

Let $W_m$ the Wronskian defined by

$$\begin{array}{c}\hfill W_m(f_1,\dots ,f_m)=det{\left({\mathsf{\partial }}_x^{i-1}{f}_j\right)}_{\left\{1\le i\le m,1\le j\le m\right\}},\end{array}$$

(64)

Then the function $u_m$ defined by

$$\begin{array}{c}\hfill u_m(x,t)=-2iln\left(\prod _{k=1}^m{a}_k{\displaystyle \frac{W_m(f_1,\dots ,f_m)(x,t)}{W_m({\mathsf{\partial }}_x\left(f_1\right),\dots ,{\mathsf{\partial }}_x\left(f_m\right))(x,t)}}\right)\end{array}$$

(65)

is a solution of the sine-Gordon Equation (17).

Proof. 

The functions $f_j$ and $g_j$ defined by $f_j(x,t)=sinh\left(a_{j}x+a{j}^{-1}t\right)$ and $g_j(x,t)=cosh\left(a_{j}x+a{j}^{-1}t\right)$, verifying the relations (18) and (19). So applying the multiple Darboux covariance property, we get the expression of the solutions

$$\begin{array}{c}\hfill u_m(x,t)=-2iln\left({\displaystyle \frac{W_m(f_1,\dots ,f_{m+1})}{W_m(g_1,\dots ,g_{m+1})}}\right)\end{array}$$

(66)

because of the choices of the functions $f_j$ and $g_j$ of this section.

As $g_j$ is equal to ${\displaystyle \frac{{\mathsf{\partial }}_x{f}_j}{a_j}}$, it is then clear, that factorizing $a_1$, until $a_{m+1}$ respectively in column 1 until column $m+1$, we get the result

$$\begin{array}{c}\hfill u_m(x,t)=-2iln\left(\prod _{k=1}^m{a}_k{\displaystyle \frac{W_m(f_1,\dots ,f_m)}{W_m({\mathsf{\partial }}_x\left(f_1\right),\dots ,{\mathsf{\partial }}_x\left(f_m\right))}}\right)\end{array}$$

(67)

□

Remark 7.

We have the same result if we choose as generating functions the functions $f_j$ defined by $f_j(x,t)=cosh\left(a_{j}x+a{j}^{-1}t\right)$.

7. Explicit Wronskian Solutions to the Sine-Gordon-Equation

In this section, we present two types of solutions of the (SG) equation.

First at order m multi-soliton solutions depending on $m+1$ real parameters $a_1,\dots ,a_{m+1}$.

If we choose $a_1={\alpha }_1+i{\beta }_1,\dots ,a_{m+1}={\alpha }_{m+1}+i{\beta }_{m+1}$, we get at order m the multi-complexitons solutions depending on $m+1$ complex parameters.

In the following, we present the solutions with parameters $a_1,\dots ,a_{m+1}$, which can be seen as real or complex and so at the same time we present $m+1$ multi-solitons or $m+1$ multi-complexitons.

7.1. Case of Order 1

Example 3.

The function u defined by

$$\begin{array}{c}\hfill \begin{array}{c}u(x,t)=-2iln\left(sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right){\left(cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)\right)}^{-1}\right)\hfill \end{array}\end{array}$$

(68)

is a solution of the sine-Gordon equation.

7.2. Case of Order 2

Example 4.

The function u defined by

$$\begin{array}{c}\hfill \begin{array}{c}u(x,t)=-2iln{\displaystyle \frac{n(x,t)}{d(x,t)}}\hfill \end{array}\end{array}$$

(69)

with

$$n(x,t)=sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)cosh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)a_2-sinh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)a_1$$

and

$$d(x,t)=cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)sinh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)a_2-cosh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)a_1$$

is a solution of the sine-Gordon equation.

7.3. Case of Order 3

Example 5.

The function u defined by

$$\begin{array}{c}\hfill \begin{array}{c}u(x,t)=-2iln{\displaystyle \frac{n(x,t)}{d(x,t)}}\hfill \end{array}\end{array}$$

(70)

with

$$\begin{array}{c}n(x,t)=sinh{\displaystyle \frac{{a_1}^{2}x+t}{a_1}}cosh{\displaystyle \frac{{a_2}^{2}x+t}{a_2}}{a}_{2}sinh{\displaystyle \frac{{a_3}^{2}x+t}{a_3}}{a_3}^2\hfill \\ -sinh{\displaystyle \frac{{a_1}^{2}x+t}{a_1}}cosh{\displaystyle \frac{{a_3}^{2}x+t}{a_3}}{a}_{3}sinh{\displaystyle \frac{{a_2}^{2}x+t}{a_2}}{a_2}^2\hfill \\ -cosh{\displaystyle \frac{{a_1}^{2}x+t}{a_1}{a}_1}sinh{\displaystyle \frac{{a_2}^{2}x+t}{a_2}}sinh{\displaystyle \frac{{a_3}^{2}x+t}{a_3}}{a_3}^2\hfill \\ +cosh{\displaystyle \frac{{a_1}^{2}x+t}{a_1}{a}_1}sinh{\displaystyle \frac{{a_3}^{2}x+t}{a_3}}sinh{\displaystyle \frac{{a_2}^{2}x+t}{a_2}}{a_2}^2\hfill \\ +sinh{\displaystyle \frac{{a_1}^{2}x+t}{a_1}}{a_1}^{2}sinh{\displaystyle \frac{{a_2}^{2}x+t}{a_2}}cosh{\displaystyle \frac{{a_3}^{2}x+t}{a_3}}{a}_3\hfill \\ -sinh{\displaystyle \frac{{a_1}^{2}x+t}{a_1}}{a_1}^{2}sinh{\displaystyle \frac{{a_3}^{2}x+t}{a_3}}cosh{\displaystyle \frac{{a_2}^{2}x+t}{a_2}}{a}_2\hfill \end{array}$$

and

$$\begin{array}{c}d(x,t)=cosh{\displaystyle \frac{{a_1}^{2}x+t}{a_1}}sinh{\displaystyle \frac{{a_2}^{2}x+t}{a_2}}{a}_{2}cosh{\displaystyle \frac{{a_3}^{2}x+t}{a_3}}{a_3}^2\hfill \\ -cosh{\displaystyle \frac{{a_1}^{2}x+t}{a_1}}sinh{\displaystyle \frac{{a_3}^{2}x+t}{a_3}}{a}_{3}cosh{\displaystyle \frac{{a_2}^{2}x+t}{a_2}}{a_2}^2\hfill \\ -sinh{\displaystyle \frac{{a_1}^{2}x+t}{a_1}{a}_1}cosh{\displaystyle \frac{{a_2}^{2}x+t}{a_2}}cosh{\displaystyle \frac{{a_3}^{2}x+t}{a_3}}{a_3}^2\hfill \\ +sinh{\displaystyle \frac{{a_1}^{2}x+t}{a_1}{a}_1}cosh{\displaystyle \frac{{a_3}^{2}x+t}{a_3}}cosh{\displaystyle \frac{{a_2}^{2}x+t}{a_2}}{a_2}^2\hfill \\ +cosh{\displaystyle \frac{{a_1}^{2}x+t}{a_1}}{a_1}^{2}cosh{\displaystyle \frac{{a_2}^{2}x+t}{a_2}}sinh{\displaystyle \frac{{a_3}^{2}x+t}{a_3}}{a}_3\hfill \\ -cosh{\displaystyle \frac{{a_1}^{2}x+t}{a_1}}{a_1}^{2}cosh{\displaystyle \frac{{a_3}^{2}x+t}{a_3}}sinh{\displaystyle \frac{{a_2}^{2}x+t}{a_2}}{a}_2\hfill \end{array}$$

is a solution of the sine-Gordon equation.

7.4. Case of Order 4

Example 6.

The function u defined by

$$\begin{array}{c}\hfill \begin{array}{c}u(x,t)=-2iln{\displaystyle \frac{n(x,t)}{d(x,t)}}\hfill \end{array}\end{array}$$

(71)

with

$$\begin{array}{c}n(x,t)=sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)cosh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)a_{2}sinh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right){a_3}^{2}cosh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right){a_4}^3\hfill \\ -sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)cosh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)a_{2}sinh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right){a_4}^{2}cosh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right){a_3}^3\hfill \\ -sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)sinh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right){a_2}^{2}cosh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right)a_{3}cosh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right){a_4}^3\hfill \\ +sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)sinh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right){a_2}^{2}cosh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right)a_{4}cosh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right){a_3}^3\hfill \\ +sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)cosh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right){a_2}^{3}cosh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right)a_{3}sinh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right){a_4}^2\hfill \\ -sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)cosh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right){a_2}^{3}cosh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right)a_{4}sinh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right){a_3}^2\hfill \\ -cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)a_{1}sinh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)sinh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right){a_3}^{2}cosh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right){a_4}^3\hfill \\ +cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)a_{1}sinh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)sinh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right){a_4}^{2}cosh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right){a_3}^3\hfill \\ +cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)a_{1}sinh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right){a_2}^{2}sinh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right)cosh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right){a_4}^3\hfill \\ -cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)a_{1}sinh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right){a_2}^{2}sinh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right)cosh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right){a_3}^3\hfill \\ -cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)a_{1}cosh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right){a_2}^{3}sinh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right)sinh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right){a_4}^2\hfill \\ +cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)a_{1}cosh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right){a_2}^{3}sinh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right)sinh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right){a_3}^2\hfill \\ +sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right){a_1}^{2}sinh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)cosh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right)a_{3}cosh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right){a_4}^3\hfill \\ -sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right){a_1}^{2}sinh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)cosh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right)a_{4}cosh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right){a_3}^3\hfill \\ -sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right){a_1}^{2}cosh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)a_{2}sinh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right)cosh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right){a_4}^3\hfill \\ +sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right){a_1}^{2}cosh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)a_{2}sinh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right)cosh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right){a_3}^3\hfill \\ +sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right){a_1}^{2}cosh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right){a_2}^{3}sinh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right)cosh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right)a_4\hfill \\ -sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right){a_1}^{2}cosh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right){a_2}^{3}sinh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right)cosh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right)a_3\hfill \\ -cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right){a_1}^{3}sinh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)cosh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right)a_{3}sinh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right){a_4}^2\hfill \\ +cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right){a_1}^{3}sinh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)cosh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right)a_{4}sinh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right){a_3}^2\hfill \\ +cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right){a_1}^{3}cosh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)a_{2}sinh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right)sinh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right){a_4}^2\hfill \\ -cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right){a_1}^{3}cosh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)a_{2}sinh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right)sinh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right){a_3}^2\hfill \\ -cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right){a_1}^{3}sinh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right){a_2}^{2}sinh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right)cosh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right)a_4\hfill \\ +cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right){a_1}^{3}sinh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right){a_2}^{2}sinh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right)cosh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right)a_3\hfill \end{array}$$

and

$$\begin{array}{c}d(x,t)=cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)sinh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)a_{2}cosh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right){a_3}^{2}sinh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right){a_4}^3\hfill \\ -cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)sinh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)a_{2}cosh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right){a_4}^{2}sinh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right){a_3}^3\hfill \\ -cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)cosh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right){a_2}^{2}sinh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right)a_{3}sinh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right){a_4}^3\hfill \\ +cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)cosh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right){a_2}^{2}sinh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right)a_{4}sinh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right){a_3}^3\hfill \\ +cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)sinh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right){a_2}^{3}sinh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right)a_{3}cosh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right){a_4}^2\hfill \\ -cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)sinh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right){a_2}^{3}sinh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right)a_{4}cosh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right){a_3}^2\hfill \\ -sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)a_{1}cosh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)cosh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right){a_3}^{2}sinh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right){a_4}^3\hfill \\ +sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)a_{1}cosh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)cosh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right){a_4}^{2}sinh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right){a_3}^3\hfill \\ +sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)a_{1}cosh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right){a_2}^{2}cosh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right)sinh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right){a_4}^3\hfill \\ -sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)a_{1}cosh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right){a_2}^{2}cosh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right)sinh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right){a_3}^3\hfill \\ -sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)a_{1}sinh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right){a_2}^{3}cosh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right)cosh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right){a_4}^2\hfill \\ +sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)a_{1}sinh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right){a_2}^{3}cosh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right)cosh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right){a_3}^2\hfill \\ +cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right){a_1}^{2}cosh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)sinh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right)a_{3}sinh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right){a_4}^3\hfill \\ -cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right){a_1}^{2}cosh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)sinh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right)a_{4}sinh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right){a_3}^3\hfill \\ -cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right){a_1}^{2}sinh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)a_{2}cosh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right)sinh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right){a_4}^3\hfill \\ +cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right){a_1}^{2}sinh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)a_{2}cosh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right)sinh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right){a_3}^3\hfill \\ +cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right){a_1}^{2}sinh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right){a_2}^{3}cosh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right)sinh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right)a_4\hfill \\ -cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right){a_1}^{2}sinh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right){a_2}^{3}cosh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right)sinh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right)a_3\hfill \\ -sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right){a_1}^{3}cosh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)sinh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right)a_{3}cosh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right){a_4}^2\hfill \\ +sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right){a_1}^{3}cosh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)sinh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right)a_{4}cosh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right){a_3}^2\hfill \\ +sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right){a_1}^{3}sinh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)a_{2}cosh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right)cosh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right){a_4}^2\hfill \\ -sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right){a_1}^{3}sinh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right)a_{2}cosh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right)cosh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right){a_3}^2\hfill \\ -sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right){a_1}^{3}cosh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right){a_2}^{2}cosh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right)sinh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right)a_4\hfill \\ +sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right){a_1}^{3}cosh\left({\displaystyle \frac{{a_2}^{2}x+t}{a_2}}\right){a_2}^{2}cosh\left({\displaystyle \frac{{a_4}^{2}x+t}{a_4}}\right)sinh\left({\displaystyle \frac{{a_3}^{2}x+t}{a_3}}\right)a_3\hfill \end{array}$$

is a solution of the sine-Gordon equation.

Remark 8.

In all these solutions, for the different parameters $a_j$, we can indifferently choose real or complex parameters. We have limited ourselves to real parameters in this study.

8. Generalized Wronskian Representations of the Solutions to the Sine-Gordon Equation

We choose in this section the same generating functions $f_j$, as defined previously as solutions of the system (18), (19) for elementary potentials $v(x,t)=0$ and $w(x,t)=1$. Then, we can formulate the following result:

Theorem 5.

Let $f_j$ be the functions defined by

$$\begin{array}{c}\hfill \begin{array}{c}{f}_j(x,t)=sinh\left(a_{j}x+a{j}^{-1}t\right);\hfill \end{array}\end{array}$$

(72)

Let $W_m$ the Wronskian defined by

$$\begin{array}{c}\hfill W_m(f_1,\dots ,f_m)=det{\left({\mathsf{\partial }}_x^{i-1}{f}_j\right)}_{\left\{1\le i\le m,1\le j\le m\right\}},\end{array}$$

(73)

Then, the function $u_m$ defined by

$$\begin{array}{c}{u}_m(x,t)=-2iln\left(\prod _{j=1}^k{a}_j^{i_j+1}\right.\hfill \\ \left.\times {\displaystyle \frac{W_m(f_1,{\mathsf{\partial }}_{a_1}{f}_1\dots ,{\mathsf{\partial }}_{a_1}^{i_1}{f}_1,\dots f_k,{\mathsf{\partial }}_{a_k}{f}_k\dots ,{\mathsf{\partial }}_{a_k}^{i_k}{f}_k)}{W_m({\mathsf{\partial }}_x\left(f_1\right),{\mathsf{\partial }}_{a_1}{\mathsf{\partial }}_x\left(f_1\right)\dots ,{\mathsf{\partial }}_{a_1}^{i_1}{\mathsf{\partial }}_x\left(f_1\right),\dots {\mathsf{\partial }}_x\left(f_k\right),{\mathsf{\partial }}_{a_k}{\mathsf{\partial }}_x\left(f_k\right)\dots ,{\mathsf{\partial }}_{a_k}^{i_k}{\mathsf{\partial }}_x\left(f_k\right))}}\right)\hfill \end{array}$$

(74)

with

$$k+\sum _{j=1}^k{i}_j=m$$

is a solution of the sine-Gordon Equation (17).

Proof. 

The columns of the Wronskian $W_{m+1}$ of the numerator of the solution $u_m$ are divided into k groups; the first group consisting of the columns $C_1,\dots ,C_{i_1+1}$, until the last group consists of the columns

$C_{i_1+\dots +i_{k-1}+k},\dots ,C_{i_1+\dots +i_k+k}$.

Each column in a group j consists of function $f_j$ depending on $a_j$.

Precisely, the group of columns of order l consists of the columns

$C_{i_1+\dots +i_{l-1}+l},\dots ,C_{i_1+\dots +i_l+l}$.

The column $\widetilde{C_j}=C_{i_1+\dots +i_l+l+j}$ is the $m+1$ dimensional vector defined by

$$\widetilde{C_j}=\left(\begin{array}{c}f(a_l+(j-1)h)\\ {\mathsf{\partial }}_{x}f(a_l+(j-1)h)\\ ⋮\\ {\mathsf{\partial }}_x^{m-1}f(a_l+(j-1)h)\end{array}\right)$$

We first carry out a limited development of each of its terms at order l. Then, we combine each of the columns of this group in such a way to keep only (the notation $a=o\left(h^l\right)$, meaning that ${lim}_{t\to 0}{\displaystyle \frac{a}{h^l}}=0$) is the derivative with respect to $a_l$ at order k in the $k+1$ column.

This is done in the following way.

From the second column to the last one, we perform the following operations. In columns 2 to the last one, we subtract the first one (multiply by a certain coefficient) in such a way to remove the first term of the limited development. We start over for columns 3 until the last one to eliminate the second term of the limited development. So we repeat this procedure until the next to last column to obtain the following:

$$(\widetilde{C_1},\dots ,\widetilde{C_{l+1}})$$

$$=\left(\begin{array}{cccc}f\left(a_l\right)& h{\mathsf{\partial }}_{a_l}f\left(a_l\right)+o\left(h\right)& \dots & h^l{\mathsf{\partial }}_{a_l}^{l}f\left(a_l\right)+o\left(h^l\right)\\ {\mathsf{\partial }}_{x}f\left(a_l\right)& h{\mathsf{\partial }}_x{\mathsf{\partial }}_{a_l}f\left(a_l\right)+o\left(h\right)& \dots & h^l{\mathsf{\partial }}_x{\mathsf{\partial }}_{a_l}^{l}f\left(a_l\right)+o\left(h^l\right)\\ ⋮\\ {\mathsf{\partial }}_x^{m-1}f\left(a_l\right)& h{\mathsf{\partial }}_x^{m-1}{\mathsf{\partial }}_{a_l}f\left(a_l\right)+o\left(h\right)& \dots & h^l{\mathsf{\partial }}_x^{m-1}{\mathsf{\partial }}_{a_l}^{l}f\left(a_l\right)+o\left(h^l\right)\end{array}\right)$$

This operation is performed for each of the k groups of columns of the numerator.

This procedure is repeated for the denominator.

We factorize $h^j$ in each column of rank $j+1$ of each group l of columns for $1\le l\le i_l+1$.

Then, we simplify the numerator and the denominator of the function $u_m$.

Finally, we go to the limit when h tends to 0 to find

$$\begin{array}{c}\hfill \begin{array}{c}{u}_m(x,t)=-2iln\left(\prod _{j=1}^k{a}_k^{i_j+1}\right.\hfill \\ \left.\times {\displaystyle \frac{W_m(f_1,{\mathsf{\partial }}_{a_1}{f}_1\dots ,{\mathsf{\partial }}_{a_1}^{i_1}{f}_1,\dots f_k,{\mathsf{\partial }}_{a_k}{f}_k\dots ,{\mathsf{\partial }}_{a_k}^{i_k}{f}_k)}{W_m({\mathsf{\partial }}_x\left(f_1\right),{\mathsf{\partial }}_{a_1}{\mathsf{\partial }}_x\left(f_1\right)\dots ,{\mathsf{\partial }}_{a_1}^{i_1}{\mathsf{\partial }}_x\left(f_1\right),\dots {\mathsf{\partial }}_x\left(f_k\right),{\mathsf{\partial }}_{a_k}{\mathsf{\partial }}_x\left(f_k\right)\dots ,{\mathsf{\partial }}_{a_k}^{i_k}{\mathsf{\partial }}_x\left(f_k\right))}}\right)\hfill \end{array}\end{array}$$

(75)

□

Remark 9.

We have the same result if we choose as generating functions the functions $f_j$ defined by $f_j(x,t)=cosh\left(a_{j}x+a{j}^{-1}t\right)$.

9. Application to the Construction of the $\mathit{k}$-Negaton-$\mathit{l}$-Positon-$\mathit{n}$-Soliton Solutions to the Sine-Gordon Equation

From the previous section, we get the explicit expression of the k-negaton-l-positon-n-soliton solutions to the sine-Gordon equation.

We consider $a_1,\dots ,a_k$ real numbers, $a_{k+1}=i{\alpha }_{k+l}$ pure imaginary numbers, and $a_{k+1+1},\dots ,a_{k+l+n}$ real numbers.

Theorem 6.

Let $f_j$ be the functions defined by

$$\begin{array}{c}\hfill \begin{array}{c}{f}_j(x,t)=sinh\left(a_{j}x+a{j}^{-1}t\right);\hfill \end{array}\end{array}$$

(76)

with $a_1,\dots ,a_k\in \mathbf{R}$, $a_{k+1},\dots ,a_{k+l}\in i\mathbf{R}$, $a_{k+l+1},\dots ,a_{k+l+n}\in \mathbf{R}$.

Let $W_m$ be the Wronskian defined by

$$\begin{array}{c}\hfill W_m(f_1,\dots ,f_m)=det{\left({\mathsf{\partial }}_x^{i-1}{f}_j\right)}_{\left\{1\le i\le m,1\le j\le m\right\}},\end{array}$$

(77)

Then the function $u_m$ defined by

$$\begin{array}{c}\hfill \begin{array}{c}{u}_m(x,t)=-2iln\left(\prod _{j=1}^k{a}_j^2\prod _{j=k+1}^{k+l}{a}_j^2{(-1)}^l\prod _{j=k+l+1}^{k+l+n}{a}_j\right.\hfill \\ \left.\times {\displaystyle \frac{W_m(f_1,{\mathsf{\partial }}_{a_1}{f}_1,f_2,{\mathsf{\partial }}_{a_2}{f}_2,\dots ,f_{k+l+n},{\mathsf{\partial }}_{a_{k+l+n}}{f}_{k+l+n})}{W_m({\mathsf{\partial }}_x{f}_1,{\mathsf{\partial }}_x{\mathsf{\partial }}_{a_1}{f}_1,{\mathsf{\partial }}_x{f}_2,{\mathsf{\partial }}_x{\mathsf{\partial }}_{a_2}{f}_2,\dots ,{\mathsf{\partial }}_x{f}_{k+l+n},{\mathsf{\partial }}_x{\mathsf{\partial }}_{a_{k+l+n}}{f}_{k+l+n})}}\right)\hfill \end{array}\end{array}$$

(78)

with

$$2k+2l+n=m$$

is a k-negaton-l-positon-n-soliton solution to the sine-Gordon Equation (17).

In the following, we do not give 1 to 2 soliton solutions which have been constructed previously.

9.1. Case of 1-Negaton Solution

Example 7.

We take here $k=1$, $l=0$, $n=0$. Here $a_1$ is real.

The function u defined by

$$\begin{array}{c}\hfill \begin{array}{c}u(x,t)=-2iln{\displaystyle \frac{n(x,t)}{d(x,t)}}\hfill \end{array}\end{array}$$

(79)

with

$$n(x,t)=-{a_1}^{2}x+t+sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)a_1$$

and

$$d(x,t)=sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)a_1+{a_1}^{2}x-t$$

is a solution to the sine-Gordon equation.

Remark 10.

We get here the 1-negaton solution to the (SG) equation.

Remark 11.

This solution is singular; the singularities are determined by the equations

$$sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)a_1=-{a_1}^{2}x+t$$

and

$$sinh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)cosh\left({\displaystyle \frac{{a_1}^{2}x+t}{a_1}}\right)a_1={a_1}^{2}x-t$$

9.2. Case of 1-Positon Solution

Example 8.

We take here $k=0$, $l=1$, $n=0$. Here, the parameter is a pure imaginary number.

The function u defined by

$$\begin{array}{c}\hfill \begin{array}{c}u(x,t)=-2iln{\displaystyle \frac{n(x,t)}{d(x,t)}}\hfill \end{array}\end{array}$$

(80)

with

$$n(x,t)={a_1}^{2}x+t+sin\left({\displaystyle \frac{-{a_1}^{2}x+t}{a_1}}\right)cos\left({\displaystyle \frac{-{a_1}^{2}x+t}{a_1}}\right)a_1$$

and

$$d(x,t)=sin\left({\displaystyle \frac{-{a_1}^{2}x+t}{a_1}}\right)cos\left({\displaystyle \frac{-{a_1}^{2}x+t}{a_1}}\right)a_1-{a_1}^{2}x-t$$

is a solution to the sine-Gordon equation.

Remark 12.

Here, we obtain the 1-positon solution to the (SG) equation.

Remark 13.

This solution is also singular; the singularities are determined by the equations

$$sin\left({\displaystyle \frac{-{a_1}^{2}x+t}{a_1}}\right)cos\left({\displaystyle \frac{-{a_1}^{2}x+t}{a_1}}\right)a_1=-{a_1}^{2}x-t$$

and

$$sin\left({\displaystyle \frac{-{a_1}^{2}x+t}{a_1}}\right)cos\left({\displaystyle \frac{-{a_1}^{2}x+t}{a_1}}\right)a_1={a_1}^{2}x+t$$

9.3. Case of the 2-Negaton Solution

Example 9.

Here, we choose $k=2$, $l=0$, $n=0$; there are two blocks of order 2 of derivatives with respect to two different parameters; $a_1$ and $a_2$ are real.

The function u defined by

$$\begin{array}{c}\hfill \begin{array}{c}u(x,t)=-2iln{\displaystyle \frac{n(x,t)}{d(x,t)}}\hfill \end{array}\end{array}$$

(81)

with

$$\begin{array}{c}n(x,t)=-(-4{a_2}^4{a_1}^2(cosh(\frac{{a_2}^{2}x+t}{a_2}){)}^2-4(cosh(\frac{{a_1}^{2}x+t}{a_1}){)}^2{a_2}^2{a_1}^4\hfill \\ -sinh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^{5}cosh(\frac{{a_1}^{2}x+t}{a_1})a_{1}sinh(\frac{{a_2}^{2}x+t}{a_2})cosh(\frac{{a_2}^{2}x+t}{a_2})\hfill \\ -6sinh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^3{a_1}^{3}cosh(\frac{{a_1}^{2}x+t}{a_1})sinh(\frac{{a_2}^{2}x+t}{a_2})cosh(\frac{{a_2}^{2}x+t}{a_2})\hfill \\ -sinh(\frac{{a_1}^{2}x+t}{a_1}){a_1}^{5}cosh(\frac{{a_1}^{2}x+t}{a_1})sinh(\frac{{a_2}^{2}x+t}{a_2})cosh(\frac{{a_2}^{2}x+t}{a_2})a_2-{a_2}^2{a_1}^{4}xt\hfill \\ -{a_1}^2{a_2}^{4}tx+4{a_2}^4{a_1}^2(cosh(\frac{{a_2}^{2}x+t}{a_2}){)}^2(cosh(\frac{{a_1}^{2}x+t}{a_1}){)}^2\hfill \\ +4(cosh(\frac{{a_1}^{2}x+t}{a_1}){)}^2{a_2}^2{a_1}^4(cosh(\frac{{a_2}^{2}x+t}{a_2}){)}^2\hfill \\ -sinh(\frac{{a_1}^{2}x+t}{a_1}){a_1}^{5}cosh(\frac{{a_1}^{2}x+t}{a_1})t-{a_2}^{5}tsinh(\frac{{a_2}^{2}x+t}{a_2})cosh(\frac{{a_2}^{2}x+t}{a_2})\hfill \\ -{a_2}^4{t}^2-{a_1}^4{t}^2+{a_1}^{6}xt+{a_2}^{6}tx-{a_1}^{4}tsinh(\frac{{a_2}^{2}x+t}{a_2})cosh(\frac{{a_2}^{2}x+t}{a_2})a_2\hfill \\ +sinh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^{6}cosh(\frac{{a_1}^{2}x+t}{a_1})a_{1}x-2sinh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^4{a_1}^{3}cosh(\frac{{a_1}^{2}x+t}{a_1})x\hfill \\ +2sinh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^2{a_1}^{3}cosh(\frac{{a_1}^{2}x+t}{a_1})t+cosh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^2{a_1}^{5}sinh(\frac{{a_1}^{2}x+t}{a_1})x\hfill \\ +{a_1}^{6}xsinh(\frac{{a_2}^{2}x+t}{a_2})cosh(\frac{{a_2}^{2}x+t}{a_2})a_2+{a_2}^5{a_1}^{2}xsinh(\frac{{a_2}^{2}x+t}{a_2})cosh(\frac{{a_2}^{2}x+t}{a_2})\hfill \\ -2{a_2}^3{a_1}^{4}xsinh(\frac{{a_2}^{2}x+t}{a_2})cosh(\frac{{a_2}^{2}x+t}{a_2})+2{a_2}^3{a_1}^{2}tsinh(\frac{{a_2}^{2}x+t}{a_2})cosh(\frac{{a_2}^{2}x+t}{a_2})\hfill \\ -sinh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^{4}cosh(\frac{{a_1}^{2}x+t}{a_1})a_{1}t-{a_1}^6{x}^2{a_2}^2-{a_2}^6{a_1}^2{x}^2+2{a_2}^4{a_1}^4{x}^2\hfill \\ +2{a_2}^2{a_1}^2{t}^2){a_1}^{-1}{a_2}^{-1}\hfill \end{array}$$

and

$$\begin{array}{c}d(x,t)=-(-4(cosh(\frac{{a_1}^{2}x+t}{a_1}){)}^2{a_2}^4{a_1}^2-4{a_2}^2{a_1}^4(cosh(\frac{{a_2}^{2}x+t}{a_2}){)}^2\hfill \\ -sinh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^{5}cosh(\frac{{a_1}^{2}x+t}{a_1})a_{1}sinh(\frac{{a_2}^{2}x+t}{a_2})cosh(\frac{{a_2}^{2}x+t}{a_2})\hfill \\ -6sinh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^3{a_1}^{3}cosh(\frac{{a_1}^{2}x+t}{a_1})sinh(\frac{{a_2}^{2}x+t}{a_2})cosh(\frac{{a_2}^{2}x+t}{a_2})\hfill \\ -sinh(\frac{{a_1}^{2}x+t}{a_1}){a_1}^{5}cosh(\frac{{a_1}^{2}x+t}{a_1})sinh(\frac{{a_2}^{2}x+t}{a_2})cosh(\frac{{a_2}^{2}x+t}{a_2})a_2\hfill \\ -{a_2}^2{a_1}^{4}xt-{a_1}^2{a_2}^{4}tx+4{a_2}^4{a_1}^2(cosh(\frac{{a_2}^{2}x+t}{a_2}){)}^2(cosh(\frac{{a_1}^{2}x+t}{a_1}){)}^2\hfill \\ +4(cosh(\frac{{a_1}^{2}x+t}{a_1}){)}^2{a_2}^2{a_1}^4(cosh(\frac{{a_2}^{2}x+t}{a_2}){)}^2\hfill \\ +sinh(\frac{{a_1}^{2}x+t}{a_1}){a_1}^{5}cosh(\frac{{a_1}^{2}x+t}{a_1})t+{a_2}^{5}tsinh(\frac{{a_2}^{2}x+t}{a_2})cosh(\frac{{a_2}^{2}x+t}{a_2})\hfill \\ -{a_2}^4{t}^2-{a_1}^4{t}^2+{a_1}^{6}xt+{a_2}^{6}tx+{a_1}^{4}tsinh(\frac{{a_2}^{2}x+t}{a_2})cosh(\frac{{a_2}^{2}x+t}{a_2})a_2\hfill \\ -sinh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^{6}cosh(\frac{{a_1}^{2}x+t}{a_1})a_{1}x+2sinh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^4{a_1}^{3}cosh(\frac{{a_1}^{2}x+t}{a_1})x\hfill \\ -2sinh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^2{a_1}^{3}cosh(\frac{{a_1}^{2}x+t}{a_1})t-cosh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^2{a_1}^{5}sinh(\frac{{a_1}^{2}x+t}{a_1})x\hfill \\ -{a_1}^{6}xsinh(\frac{{a_2}^{2}x+t}{a_2})cosh(\frac{{a_2}^{2}x+t}{a_2})a_2-{a_2}^5{a_1}^{2}xsinh(\frac{{a_2}^{2}x+t}{a_2})cosh(\frac{{a_2}^{2}x+t}{a_2})\hfill \\ +2{a_2}^3{a_1}^{4}xsinh(\frac{{a_2}^{2}x+t}{a_2})cosh(\frac{{a_2}^{2}x+t}{a_2})-2{a_2}^3{a_1}^{2}tsinh(\frac{{a_2}^{2}x+t}{a_2})cosh(\frac{{a_2}^{2}x+t}{a_2})\hfill \\ +sinh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^{4}cosh(\frac{{a_1}^{2}x+t}{a_1})a_{1}t-{a_1}^6{x}^2{a_2}^2-{a_2}^6{a_1}^2{x}^2+2{a_2}^4{a_1}^4{x}^2\hfill \\ +2{a_2}^2{a_1}^2{t}^2){a_1}^{-1}{a_2}^{-1}\hfill \end{array}$$

is a solution to the sine-Gordon equation.

Remark 14.

Here, we obtain the 2-negaton solution to the (SG) equation.

9.4. Case of 1-Negaton-1-Positon Solution

Example 10.

Here, we choose $k=1$, $l=1$, $n=0$; there are two blocks of order 2 of derivatives with respect to two different parameters; one parameter is real the other parameter is a pure imaginary number.

The function u defined by

$$\begin{array}{c}\hfill \begin{array}{c}u(x,t)=-2iln{\displaystyle \frac{n(x,t)}{d(x,t)}}\hfill \end{array}\end{array}$$

(82)

with

$$\begin{array}{c}n(x,t)=({a_2}^4{t}^2+{a_1}^4{t}^2+sinh(\frac{{a_1}^{2}x+t}{a_1}){a_1}^{5}cosh(\frac{{a_1}^{2}x+t}{a_1})t\hfill \\ +{a_2}^{5}tsin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})\hfill \\ +sinh(\frac{{a_1}^{2}x+t}{a_1}){a_1}^{5}cosh(\frac{{a_1}^{2}x+t}{a_1})sin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})a_2\hfill \\ +sinh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^{5}cosh(\frac{{a_1}^{2}x+t}{a_1})a_{1}sin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})\hfill \\ -6sinh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^3{a_1}^{3}cosh(\frac{{a_1}^{2}x+t}{a_1})sin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})\hfill \\ +sinh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^{6}cosh(\frac{{a_1}^{2}x+t}{a_1})a_{1}x+sinh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^{4}cosh(\frac{{a_1}^{2}x+t}{a_1})a_{1}t\hfill \\ +2sinh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^4{a_1}^{3}cosh(\frac{{a_1}^{2}x+t}{a_1})x+2sinh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^2{a_1}^{3}cosh(\frac{{a_1}^{2}x+t}{a_1})t\hfill \\ -{a_2}^5{a_1}^{2}xsin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})-2{a_2}^3{a_1}^{4}xsin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})\hfill \\ +2{a_2}^3{a_1}^{2}tsin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})+sinh(\frac{{a_1}^{2}x+t}{a_1}){a_1}^{5}cosh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^{2}x\hfill \\ -{a_1}^{6}xsin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})a_2+{a_1}^{4}tsin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})a_2\hfill \\ -{a_1}^6{x}^2{a_2}^2-{a_1}^{6}xt-2{a_2}^4{a_1}^4{x}^2+2{a_2}^2{a_1}^2{t}^2+{a_2}^{6}tx-{a_2}^6{a_1}^2{x}^2\hfill \\ +4{a_2}^4{a_1}^2(cos(\frac{-{a_2}^{2}x+t}{a_2}){)}^2-4(cosh(\frac{{a_1}^{2}x+t}{a_1}){)}^2{a_2}^2{a_1}^4-{a_2}^2{a_1}^{4}xt\hfill \\ +{a_1}^2{a_2}^{4}tx-4{a_2}^4{a_1}^2(cos(\frac{-{a_2}^{2}x+t}{a_2}){)}^2(cosh(\frac{{a_1}^{2}x+t}{a_1}){)}^2\hfill \\ +4(cosh(\frac{{a_1}^{2}x+t}{a_1}){)}^2{a_2}^2{a_1}^4(cos(\frac{-{a_2}^{2}x+t}{a_2}){)}^2){a_1}^{-1}{a_2}^{-1}\hfill \end{array}$$

and

$$\begin{array}{c}d(x,t)=({a_2}^4{t}^2+{a_1}^4{t}^2-sinh(\frac{{a_1}^{2}x+t}{a_1}){a_1}^{5}cosh(\frac{{a_1}^{2}x+t}{a_1})t\hfill \\ -{a_2}^{5}tsin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})\hfill \\ +sinh(\frac{{a_1}^{2}x+t}{a_1}){a_1}^{5}cosh(\frac{{a_1}^{2}x+t}{a_1})sin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})a_2\hfill \\ +sinh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^{5}cosh(\frac{{a_1}^{2}x+t}{a_1})a_{1}sin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})\hfill \\ -6sinh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^3{a_1}^{3}cosh(\frac{{a_1}^{2}x+t}{a_1})sin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})\hfill \\ -sinh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^{6}cosh(\frac{{a_1}^{2}x+t}{a_1})a_{1}x-sinh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^{4}cosh(\frac{{a_1}^{2}x+t}{a_1})a_{1}t\hfill \\ -2sinh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^4{a_1}^{3}cosh(\frac{{a_1}^{2}x+t}{a_1})x\hfill \\ -2sinh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^2{a_1}^{3}cosh(\frac{{a_1}^{2}x+t}{a_1})t+{a_2}^5{a_1}^{2}xsin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})\hfill \\ +2{a_2}^3{a_1}^{4}xsin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})-2{a_2}^3{a_1}^{2}tsin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})\hfill \\ -sinh(\frac{{a_1}^{2}x+t}{a_1}){a_1}^{5}cosh(\frac{{a_1}^{2}x+t}{a_1}){a_2}^{2}x+{a_1}^{6}xsin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})a_2\hfill \\ -{a_1}^{4}tsin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})a_2-{a_1}^6{x}^2{a_2}^2-{a_1}^{6}xt-2{a_2}^4{a_1}^4{x}^2+2{a_2}^2{a_1}^2{t}^2+\hfill \\ {a_2}^{6}tx-{a_2}^6{a_1}^2{x}^2+4(cosh(\frac{{a_1}^{2}x+t}{a_1}){)}^2{a_2}^4{a_1}^2-4{a_2}^2{a_1}^4(cos(\frac{-{a_2}^{2}x+t}{a_2}){)}^2-{a_2}^2{a_1}^{4}xt+\hfill \\ {a_1}^2{a_2}^{4}tx-4{a_2}^4{a_1}^2(cos(\frac{-{a_2}^{2}x+t}{a_2}){)}^2(cosh(\frac{{a_1}^{2}x+t}{a_1}){)}^2\hfill \\ +4(cosh(\frac{{a_1}^{2}x+t}{a_1}){)}^2{a_2}^2{a_1}^4(cos(\frac{-{a_2}^{2}x+t}{a_2}){)}^2){a_1}^{-1}{a_2}^{-1}\hfill \end{array}$$

is a solution to the sine-Gordon equation.

Remark 15.

Here, we obtain the 1-negaton-1-positon solution to the (SG) equation.

9.5. Case of 2-Positon Solution

Example 11.

Here, we take $k=0$, $l=2$, $n=0$. Here, the parameters are pure imaginary numbers.

The function u defined by

$$\begin{array}{c}\hfill \begin{array}{c}u(x,t)=-2iln{\displaystyle \frac{n(x,t)}{d(x,t)}}\hfill \end{array}\end{array}$$

(83)

with

$$\begin{array}{c}n(x,t)=6sin(\frac{-{a_1}^{2}x+t}{a_1}){a_2}^3{a_1}^{3}cos(\frac{-{a_1}^{2}x+t}{a_1})sin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})\hfill \\ +sin(\frac{-{a_1}^{2}x+t}{a_1}){a_1}^{5}cos(\frac{-{a_1}^{2}x+t}{a_1})sin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})a_2\hfill \\ +sin(\frac{-{a_1}^{2}x+t}{a_1}){a_2}^{5}cos(\frac{-{a_1}^{2}x+t}{a_1})a_{1}sin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})\hfill \\ -2{a_2}^4{a_1}^4{x}^2-2{a_2}^2{a_1}^2{t}^2+{a_2}^{6}tx+{a_2}^6{a_1}^2{x}^2\hfill \\ +4{a_2}^4{a_1}^2(cos(\frac{-{a_2}^{2}x+t}{a_2}){)}^2(cos(\frac{-{a_1}^{2}x+t}{a_1}){)}^2\hfill \\ +4(cos(\frac{-{a_1}^{2}x+t}{a_1}){)}^2{a_2}^2{a_1}^4(cos(\frac{-{a_2}^{2}x+t}{a_2}){)}^2\hfill \\ -4{a_2}^4{a_1}^2(cos(\frac{-{a_2}^{2}x+t}{a_2}){)}^2-4(cos(\frac{-{a_1}^{2}x+t}{a_1}){)}^2{a_2}^2{a_1}^4\hfill \\ +{a_1}^{4}tsin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})a_2\hfill \\ -2sin(\frac{-{a_1}^{2}x+t}{a_1}){a_2}^2{a_1}^{3}cos(\frac{-{a_1}^{2}x+t}{a_1})t\hfill \\ +{a_2}^5{a_1}^{2}xsin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})\hfill \\ -2{a_2}^3{a_1}^{4}xsin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})\hfill \\ -2{a_2}^3{a_1}^{2}tsin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})\hfill \\ +cos(\frac{-{a_1}^{2}x+t}{a_1}){a_2}^2{a_1}^{5}sin(\frac{-{a_1}^{2}x+t}{a_1})x\hfill \\ +{a_1}^{6}xsin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})a_2\hfill \\ +sin(\frac{-{a_1}^{2}x+t}{a_1}){a_2}^{6}cos(\frac{-{a_1}^{2}x+t}{a_1})a_{1}x\hfill \\ +sin(\frac{-{a_1}^{2}x+t}{a_1}){a_2}^{4}cos(\frac{-{a_1}^{2}x+t}{a_1})a_{1}t\hfill \\ -2sin(\frac{-{a_1}^{2}x+t}{a_1}){a_2}^4{a_1}^{3}cos(\frac{-{a_1}^{2}x+t}{a_1})x\hfill \\ +{a_1}^{6}xt+{a_1}^6{x}^2{a_2}^2-{a_1}^2{a_2}^{4}tx-{a_2}^2{a_1}^{4}xt+{a_1}^4{t}^2\hfill \\ +{a_2}^4{t}^2+sin(\frac{-{a_1}^{2}x+t}{a_1}){a_1}^{5}cos(\frac{-{a_1}^{2}x+t}{a_1})t\hfill \\ +{a_2}^{5}tsin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})\hfill \end{array}$$

and

$$\begin{array}{c}d(x,t)=6sin(\frac{-{a_1}^{2}x+t}{a_1}){a_2}^3{a_1}^{3}cos(\frac{-{a_1}^{2}x+t}{a_1})sin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})\hfill \\ +sin(\frac{-{a_1}^{2}x+t}{a_1}){a_1}^{5}cos(\frac{-{a_1}^{2}x+t}{a_1})sin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})a_2\hfill \\ +sin(\frac{-{a_1}^{2}x+t}{a_1}){a_2}^{5}cos(\frac{-{a_1}^{2}x+t}{a_1})a_{1}sin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})\hfill \\ -2{a_2}^4{a_1}^4{x}^2-2{a_2}^2{a_1}^2{t}^2+{a_2}^{6}tx+{a_2}^6{a_1}^2{x}^2\hfill \\ +4{a_2}^4{a_1}^2(cos(\frac{-{a_2}^{2}x+t}{a_2}){)}^2(cos(\frac{-{a_1}^{2}x+t}{a_1}){)}^2\hfill \\ +4(cos(\frac{-{a_1}^{2}x+t}{a_1}){)}^2{a_2}^2{a_1}^4(cos(\frac{-{a_2}^{2}x+t}{a_2}){)}^2\hfill \\ -4(cos(\frac{-{a_1}^{2}x+t}{a_1}){)}^2{a_2}^4{a_1}^2-4{a_2}^2{a_1}^4(cos(\frac{-{a_2}^{2}x+t}{a_2}){)}^2\hfill \\ -{a_1}^{4}tsin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})a_2\hfill \\ +2sin(\frac{-{a_1}^{2}x+t}{a_1}){a_2}^2{a_1}^{3}cos(\frac{-{a_1}^{2}x+t}{a_1})t\hfill \\ -{a_2}^5{a_1}^{2}xsin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})\hfill \\ +2{a_2}^3{a_1}^{4}xsin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})\hfill \\ +2{a_2}^3{a_1}^{2}tsin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})\hfill \\ -cos(\frac{-{a_1}^{2}x+t}{a_1}){a_2}^2{a_1}^{5}sin(\frac{-{a_1}^{2}x+t}{a_1})x\hfill \\ -{a_1}^{6}xsin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})a_2\hfill \\ -sin(\frac{-{a_1}^{2}x+t}{a_1}){a_2}^{6}cos(\frac{-{a_1}^{2}x+t}{a_1})a_{1}x\hfill \\ -sin(\frac{-{a_1}^{2}x+t}{a_1}){a_2}^{4}cos(\frac{-{a_1}^{2}x+t}{a_1})a_{1}t\hfill \\ +2sin(\frac{-{a_1}^{2}x+t}{a_1}){a_2}^4{a_1}^{3}cos(\frac{-{a_1}^{2}x+t}{a_1})x+{a_1}^{6}xt\hfill \\ +{a_1}^6{x}^2{a_2}^2-{a_1}^2{a_2}^{4}tx-{a_2}^2{a_1}^{4}xt+{a_1}^4{t}^2+{a_2}^4{t}^2\hfill \\ -sin(\frac{-{a_1}^{2}x+t}{a_1}){a_1}^{5}cos(\frac{-{a_1}^{2}x+t}{a_1})t\hfill \\ -{a_2}^{5}tsin(\frac{-{a_2}^{2}x+t}{a_2})cos(\frac{-{a_2}^{2}x+t}{a_2})\hfill \end{array}$$

is a solution to the sine-Gordon equation.

Remark 16.

Here, we obtain the 2-positon solution to the (SG) equation.

Remark 17.

One could thus continue this study and build solutions of a higher order, in particular multi-negaton, multi-negaton-soliton, multi-positon, multi-positon-soliton, multi-negaton-positon, and multi-negaton-positon-soliton, but the explicit expressions become too long to be presented here.

9.6. Case of Complexiton Solutions

Example 12.

We take here $k=1$, $l=0$, $n=0$. Here, the parameter is a complex number $a_i={\alpha }_i+i{\beta }_i$.

The function u defined by

$$\begin{array}{c}u(x,t)=-2iln{\displaystyle \frac{n(x,t)}{d(x,t)}}\end{array}$$

(84)

with

$$\begin{array}{c}n(x,t)=-x{{\alpha }_1}^2-2ix{\alpha }_1{\beta }_1+x{{\beta }_1}^2+t+sinh(\frac{x{{\alpha }_1}^2+2ix{\alpha }_1{\beta }_1-x{{\beta }_1}^2+t}{{\alpha }_1+i{\beta }_1})\hfill \\ cosh(\frac{x{{\alpha }_1}^2+2ix{\alpha }_1{\beta }_1-x{{\beta }_1}^2+t}{{\alpha }_1+i{\beta }_1}){\alpha }_1+isinh(\frac{x{{\alpha }_1}^2+2ix{\alpha }_1{\beta }_1-x{{\beta }_1}^2+t}{{\alpha }_1+i{\beta }_1})\hfill \\ cosh(\frac{x{{\alpha }_1}^2+2ix{\alpha }_1{\beta }_1-x{{\beta }_1}^2+t}{{\alpha }_1+i{\beta }_1}){\beta }_1\hfill \end{array}$$

and

$$\begin{array}{c}d(x,t)=sinh(\frac{x{{\alpha }_1}^2+2ix{\alpha }_1{\beta }_1-x{{\beta }_1}^2+t}{{\alpha }_1+i{\beta }_1})cosh(\frac{x{{\alpha }_1}^2+2ix{\alpha }_1{\beta }_1-x{{\beta }_1}^2+t}{{\alpha }_1+i{\beta }_1}){\alpha }_1\hfill \\ +isinh(\frac{x{{\alpha }_1}^2+2ix{\alpha }_1{\beta }_1-x{{\beta }_1}^2+t}{{\alpha }_1+i{\beta }_1})cosh(\frac{x{{\alpha }_1}^2+2ix{\alpha }_1{\beta }_1-x{{\beta }_1}^2+t}{{\alpha }_1+i{\beta }_1}){\beta }_1+x{{\alpha }_1}^2\hfill \\ +2ix{\alpha }_1{\beta }_1-x{{\beta }_1}^2-t\hfill \end{array}$$

being solutions to the sine-Gordon equation.

Remark 18.

Here, we obtain the 1-complexiton solution to the (SG) equation.

We could go on but the solutions become too long. We do not give the explicit expression of 2-complexiton solution to the (SG) equation because it takes more than 10 pages.

10. Conclusions

Many exact solutions to the sine-Gordon equation in terms of elliptic functions were given in [17]. Some of these solutions can be written in terms of elementary functions [18,19].

In [20], elliptic function solutions of some special nonlinear evolution equations like the two-dimensional Sine-Gordon equation are constructed.

Solutions to the sine-Gordon equation with initial data can be written explicitly in terms of Jacobi theta functions [21]. Some exact solutions to the sine-Gordon equations can be obtained via the Darboux or Bäcklund transformations [22] from already known exact solutions.

Some explicit solutions were given in [3] in terms of ${tan}^{-1}$.

Other types of solutions has been given in [23] for the first and second in terms of Jacobi functions.

We have given a complete description of the method of construction of solutions to the sine-Gordon equation by means Darboux transformations. These results were not proved, even in [9,10]. With some generating functions, we construct explicit solutions to the (SG) equation for the first orders.

With particular generating solutions, we give another representation of the solutions to the (SG) equation in terms of Wronskians. We give some explicit solutions.

Using derivations, we generalize the previous solutions and get solutions in terms of generalized Wronskians. We also derived some particular explicit solutions to the (SG) equation. We construct, in particular, a 1-negaton solution, 2-negaton solution, and a 1-negaton-1-positon solution.

It would be interesting to continue the study with the case of multi-negaton, multi-negaton-soliton, multi-positon, multi-positon-soliton, multi-negaton-positon, multi-negaton-positon-soliton, or multi-complexitons.