On the soliton solutions of a family of Tzitzeica equations

TL;DR

This paper investigates soliton solutions of a family of Tzitzeica equations using dressing and Hirota methods, classifying solutions by eigenvalues and discussing their singularities and transformations.

Contribution

It introduces two types of soliton solutions derived via dressing method, linking eigenvalues to soliton types, and outlines construction of multi-soliton solutions for Tzitzeica equations.

Findings

Two types of soliton solutions identified based on eigenvalues.

Construction method for general multi-soliton solutions provided.

Discussion of singularities and variable transformations included.

Abstract

We analyze several types of soliton solutions to a family of Tzitzeica equations. To this end we use two methods for deriving the soliton solutions: the dressing method and Hirota method. The dressing method allows us to derive two types of soliton solutions. The first type corresponds to a set of 6 symmetrically situated discrete eigenvalues of the Lax operator $L$; to each soliton of the second type one relates a set of 12 discrete eigenvalues of $L$. We also outline how one can construct general $N$ soliton solution containing $N_1$ solitons of first type and $N_2$ solitons of second type, $N=N_1+N_2$. The possible singularities of the solitons and the effects of change of variables that relate the different members of Tzitzeica family equations are briefly discussed. All equations allow quasi-regular as well as singular soliton solutions.

Tables

Table 1. Table 1: Changes of variables that relate different members of Tzitzeica family equations.

	T1	T2	T3	T4
$A_{1,2}^{+}$, $s_0=0$	T1	T2	T3	T4
$A_{1,2}^{-}$, $s_0=0$	T2	T1	T4	T3
$A_{1,2}^{+}$, $s_0=1$	T3	T4	T1	T2
$A_{1,2}^{-}$, $s_0=1$	T4	T3	T2	T1

Table 2. Table 2: The set of signs η ν ( k ) superscript subscript 𝜂 𝜈 𝑘 \eta_{\nu}^{(k)} for each of the sectors Υ ν subscript Υ 𝜈 \Upsilon_{\nu} ( 89 ).

	${{\rm Y}}_0$	${{\rm Y}}_1$	${{\rm Y}}_2$	${{\rm Y}}_3$	${{\rm Y}}_4$	${{\rm Y}}_5$
${\eta }_{\nu }^{\left(1\right)}$	$-$	$-$	$-$	$+$	$+$	$+$
${\eta }_{\nu }^{\left(2\right)}$	$+$	$+$	$-$	$-$	$-$	$+$
${\eta }_{\nu }^{\left(3\right)}$	$-$	$+$	$+$	$+$	$-$	$-$

Equations

$$\begin{array}{rl}2\frac{{\partial }^2{\varphi }_1}{\partial \xi \partial \eta }& ={\mathrm{e}}^{2{\varphi }_1}-{\mathrm{e}}^{-4{\varphi }_1}2\frac{{\partial }^2{\varphi }_2}{\partial \xi \partial \eta }=-({\mathrm{e}}^{2{\varphi }_2}-{\mathrm{e}}^{-4{\varphi }_2})\end{array}$$

$$\begin{array}{r}2\frac{{\partial }^2\varphi }{\partial \xi \partial \eta }={ϵ}_1{c}_1^2{\mathrm{e}}^{2\varphi }+{ϵ}_2{c}_2^2{\mathrm{e}}^{-4\varphi }\end{array}$$

$$\begin{array}{rl}2\frac{{\partial }^2{\varphi }_3}{\partial \xi \partial \eta }& =-{\mathrm{e}}^{2{\varphi }_3}-{\mathrm{e}}^{-4{\varphi }_3}2\frac{{\partial }^2{\varphi }_4}{\partial \xi \partial \eta }={\mathrm{e}}^{2{\varphi }_4}+{\mathrm{e}}^{-4{\varphi }_4}.\end{array}$$

$$\displaystyle\vec{Y}^{\prime}=A\vec{Y},\qquad\vec{Y}^{\prime}=\left(\begin{array}[]{c}\xi^{\prime}\\ \eta^{\prime}\end{array}\right),\qquad\vec{Y}=\left(\begin{array}[]{c}\xi\\ \eta\end{array}\right),\qquad A=\left(\begin{array}[]{cc}a&b\\ c&d\end{array}\right)$$

$$\begin{split}A^{T}\sigma_{1}A=\pm\sigma_{1},\qquad\sigma_{1}=\left(\begin{array}[]{cc}0&1\\ 1&0\end{array}\right)\end{split}$$

$$\begin{array}{r}ac=bd=0,ad+bc=\pm 1.\end{array}$$

$$\begin{split}\mbox{1)}\quad A_{1}^{\pm}=\left(\begin{array}[]{cc}a&0\\ 0&\pm 1/a\end{array}\right),\qquad\mbox{2)}\quad A_{2}^{\pm}=\left(\begin{array}[]{cc}0&b\\ \pm 1/b&0\end{array}\right).\end{split}$$

$$\begin{array}{r}\varphi (\xi ,\eta )\to {\varphi }^{\prime }(\xi ,\eta )=\varphi (\xi ,\eta )-\ln c_0+s_0\frac{\pi \mathrm{i}}{2}\end{array}$$

$$\begin{array}{r}2\frac{{\partial }^2{\varphi }_1^{\prime }}{\partial \xi \partial \eta }=c_1^2{\mathrm{e}}^{2{\varphi }_1^{\prime }}-c_1^{-4}{\mathrm{e}}^{-4{\varphi }_1^{\prime }}\end{array}$$

$$\begin{array}{r}2\frac{{\partial }^2{\varphi }_1^{\prime }}{\partial \xi \partial \eta }=-c_1^2{\mathrm{e}}^{2{\varphi }_1^{\prime }}-c_1^{-4}{\mathrm{e}}^{-4{\varphi }_1^{\prime }}\end{array}$$

$$\begin{array}{rl}{L}_1\Psi (\xi ,\eta ,\lambda )& \equiv \mathrm{i}\frac{\partial \Psi (\xi ,\eta ,\lambda )}{\partial \xi }+2\mathrm{i}{\varphi }_{\xi }{H}_0\Psi (\xi ,\eta ,\lambda )+\lambda \mathcal{J}\Psi (\xi ,\eta ,\lambda )=0\\ L_2\Psi (\xi ,\eta ,\lambda )& \equiv \mathrm{i}\frac{\partial \Psi (\xi ,\eta ,\lambda )}{\partial \eta }+{\lambda }^{-1}{V}_1\Psi (\xi ,\eta ,\lambda )=0\end{array}$$

$$\begin{split}H_{0}=\left(\begin{array}[]{ccc}1&0&0\\ 0&0&0\\ 0&0&-1\end{array}\right),\quad\mathcal{J}=\left(\begin{array}[]{ccc}0&1&0\\ 0&0&1\\ 1&0&0\end{array}\right)\quad V_{1}(\xi,\eta)=\left(\begin{array}[]{ccc}0&0&\mathrm{e}^{-4\phi}\\ \mathrm{e}^{2\phi}&0&0\\ 0&\mathrm{e}^{2\phi}&0\end{array}\right).\end{split}$$

$$\begin{split}Q^{-1}\Psi(\xi,\eta,\lambda)Q=\Psi(\xi,\eta,q\lambda),\quad Q=\left(\begin{array}[]{ccc}1&0&0\\ 0&q&0\\ 0&0&q^{2}\end{array}\right),\quad q=e^{2\pi\mathrm{i\,}/3}\end{split}$$

$$Q^{-1}{H}_{0}Q=H_0,Q^{-1}\mathcal{J}Q=q\mathcal{J},Q^{-1}{V}_{1}Q=q^{-1}{V}_1.$$

$$\begin{array}{r}{\Psi }^{\ast }(\xi ,\eta ,-{\lambda }^{\ast })=\Psi (\xi ,\eta ,\lambda )\end{array}$$

$$\begin{array}{r}{H}_0=H_0^{\ast },J_1=J_1^{\ast },V_1=V_1^{\ast }\end{array}$$

$$\begin{split}A_{0}^{-1}\Psi^{\dagger}(\xi,\eta,\lambda^{*})A_{0}=\Psi^{-1}(\xi,\eta,\lambda),\qquad A_{0}=\left(\begin{array}[]{ccc}0&0&1\\ 0&1&0\\ 1&0&0\end{array}\right)\end{split}$$

$$\begin{array}{rl}{A}_0^{-1}{H}_0^{†}{A}_0& =-H_0,A_0^{-1}{\mathcal{J}}^{†}{A}_0=\mathcal{J},A_0^{-1}{V}_1^{†}{A}_0=V_1.\end{array}$$

$$\begin{array}{rl}{L}_{10}{\Psi }_0& \equiv \mathrm{i}\frac{\partial {\Psi }_0}{\partial \xi }+\lambda \mathcal{J}{\Psi }_0=0L_{20}{\Psi }_0\equiv \mathrm{i}\frac{\partial {\Psi }_0}{\partial \eta }+{\lambda }^{-1}{\mathcal{J}}^2{\Psi }_0=0.\end{array}$$

$$\begin{array}{r}\Psi (\xi ,\eta ,\lambda )=u(\xi ,\eta ,\lambda ){\Psi }_0(\xi ,\eta ,\lambda )u_{+}^{-1}(\lambda ),u_{+}(\lambda )=\underset{\xi \to \infty }{\lim }u(\xi ,\eta ,\lambda )\end{array}$$

$$\begin{array}{r}\mathrm{i}\frac{\partial u}{\partial \xi }+2\mathrm{i}{\varphi }_{\xi }{H}_{0}u(\xi ,\eta ,\lambda )+\lambda \left[\mathcal{J},u(\xi ,\eta ,\lambda )\right]=0\\ \mathrm{i}\frac{\partial u}{\partial \eta }+\frac{1}{\lambda }{V}_{1}u(\xi ,\eta ,\lambda )-\frac{1}{\lambda }u(\xi ,\eta ,\lambda ){\mathcal{J}}^2=0.\end{array}$$

$$Q^{-1}u(\xi ,\eta ,\lambda )Q$$

$$A_0^{-1}{u}^{†}(\xi ,\eta ,{\lambda }^{\ast })A_0$$

$$\begin{array}{r}u(\xi ,\eta ,\lambda )=\text{leavevmode}11+\frac{1}{3}\left(\frac{A_1}{\lambda -{\lambda }_1}+\frac{Q^{-1}{A}_{1}Q}{\lambda q^2-{\lambda }_1}+\frac{Q^{-2}{A}_1{Q}^2}{\lambda q-{\lambda }_1}\right)\end{array}$$

$$\begin{array}{r}{A}_1(\xi ,\eta )=|n(\xi ,\eta )⟩⟨m^T(\xi ,\eta )|(A_1{)}_{ij}(\xi ,\eta )=n_i(\xi ,\eta )m_j(\xi ,\eta ).\end{array}$$

$$\begin{array}{r}\frac{{\eta }_{j-k}{n}_k{m}_j}{{\lambda }^3-{\lambda }_1^{\ast ,3}}=-\frac{{\rho }_{j-k}{n}_k^{\ast }{m}_j^{\ast }}{{\lambda }^3+{\lambda }_1^{\ast ,3}}.\end{array}$$

$${\eta }_0$$

$${\rho }_0$$

$${\lambda }_1^{\ast ,3}=-{\lambda }_1^3$$

$$\begin{array}{r}{\lambda }_1^{\ast ,2}{n}_i^{\ast }{m}_i^{\ast }=-{\lambda }_1^2{n}_i{m}_i,{\lambda }_1^{\ast }{n}_i^{\ast }{m}_{i+1}^{\ast }={\lambda }_1{n}_i{m}_{i+1},n_i^{\ast }{m}_{i+2}^{\ast }=-n_i{m}_{i+2}\end{array}$$

$$\begin{array}{rl}\arg n_i+\arg m_i& =\frac{\pi }{2}-2\arg {\lambda }_1,\arg n_i+\arg m_{i+1}=-\arg {\lambda }_1\\ \arg n_i+\arg m_{i+2}& =\frac{\pi }{2},\arg {\lambda }_1=(2k+1)\frac{\pi }{6},k=0,1,...,5.\end{array}$$

Full text

MSC: 35Q51, 35Q53, 37K40

Key words: Tzitzeica equations, singular soliton solutions, Zakharov-Shabat dressing method, Hirota method

\comm

Communicated by Metin Gürses

1. 1 Introduction

2. 2 Lorentz (Anti-)Invariance in 2-Dimensions

3. 2.1 Changes of Variables and the Lorentz (Anti-)Invariance

4. 2.2 The Lax Representation of Ţ2 Equation

5. 3 The Dressing Method and Dressing Factors for Ţ2 Equation

6. 3.1 One Soliton Solution of First Type

7. 3.2 The Singularity Properties of the Soliton Solutions

8. 3.3 One Soliton Solutions of Second Type

9. 4 The Generic $N$-soliton Solution for Ţ2 Equation

10. 5 Hirota Method for Building 1-soliton Solution of Ţ2 Equation

11. 6 The Spectral Properties of the Dressed Lax Operator

12. 7 Conclusions

1 Introduction

In the present paper we continue our investigations of the famous equation due to the Romanian mathematician Gheorghe Tzitzeica 111The name of the famous Romanian mathematician contains the Romanian letter Ţ, which may be spelled as Tz. The factor 2 in eq. (1) can be easily removed, but is kept for historical reasons. , which we call now as Tzitzeica 1 equation [27, 28] and a closely related equation which we call Tzitzeica 2; in what follows we will denote them by Ţ1 and Ţ2. It was initially proposed as an equation describing special surfaces in differential geometry for which the ratio $K/d^{4}$ is constant, where $K$ is the Gauss curvature of the surface and $d$ is the distance from the origin to the tangent plane at the given point. Later on it turned out that the equation has wider importance, being nowadays used as an important evolutionary equation in nonlinear dynamics.

The explicit form of Ţ1 and Ţ2 equations is

[TABLE]

i.e. Ţ1 and Ţ2 have different signs in the right hand sides. The transition between T1 and T2 can be performed by several simple changes of variables (see below), some of which substantially modify the singularity properties of their solutions.

Tzitzeica equations attracted a lot of attention at the end of the ’70-ies when for some time it was believed, that it is the only known equation, allowing a finite number of higher integrals of motion [8]. Soon however, it was proved that in fact, it possesses, like the other soliton equations, an infinite number of integrals of motion [30]. Next it was discovered that the equation has a hidden $\mathbb{Z}_{3}$ symmetry, which becomes evident in its Lax representation [22, 21]. This important discovery led Mikhailov to the notion of the reduction group and to the family of two-dimensional Toda field theories (TFT) related to the $sl(n)$ algebras [21]. Soon after it was established that: i) 2-dimensional TFT can be related to any of the simple Lie algebras [23, 24, 9, 19]; ii) other classes of integrable NLEE may also possess such symmetries [9, 12, 13, 7]; and iii) the expansions over the squared solutions and the theory of their recursion operators can be constructed [18, 29, 15].

In previous papers [4, 5] we presented in the derivation of the soliton solutions of Ţ1. Both versions of Tzitzeica equation allow Lax representation proposed by Mikhailov [22, 21]. This allows one to apply the dressing method of Zakharov-Shabat-Mikhailov [33, 21, 32] for calculating their soliton solutions. In fact all these equations are particular examples of 2-dimensional Toda field theories (TFT) [21, 9, 23, 24, 19]. They all can be solved exactly using the inverse scattering method [10, 31, 16].

In the present paper we start with the analysis of a more general class of equations, which we call Tzitzeica family equations. Their general form is

[TABLE]

where $\epsilon_{1}^{2}=\epsilon_{2}^{2}=1$ and $c_{1}$ and $c_{2}$ are some positive real constants. Obviously equation Ţ1 (resp. equation Ţ2) is obtained from (2) by putting $\epsilon_{1}=1$, $\epsilon_{2}=-1$, $c_{1}=c_{4}=1$ (resp. $\epsilon_{1}=-1$, $\epsilon_{2}=1$, $c_{1}=c_{4}=1$). We will call Ţ3 and Ţ4 the equations

[TABLE]

which follow from (2) with $\epsilon_{1}=\epsilon_{2}=-1$, $c_{1}=c_{4}=1$ and $\epsilon_{1}=\epsilon_{2}=1$, $c_{1}=c_{4}=1$ respectively.

The paper is organized as follows. In Section 2 we study a class of changes of variables that interrelate different members of Tzitzeica family. We shall see that Ţ1 – Ţ4 equations allow Lax representations so they can be solved exactly by the inverse scattering method, [22, 6]. In Section 3 the Zakharov-Shabat dressing method [33], adapted to systems with deep reductions, [21, 22] is used to construct their soliton solutions. As a result we derive the soliton solutions of first and second types and analyze their singularities. Indeed, we find that even the simplest one-soliton solutions of first type may have an infinite number of singularities for finite values of $\xi,\eta$. Such singularities are characteristic also for other soliton-type equations, e.g. for Liouville equation [1, 2, 25, 26], for sinh-Gordon equation and others, see e.g. [25, 11, 20] and the references therein. At the same time, using an appropriate change of variables we obtain a solution having singularities at only two points which we call ‘quasi-regular’. In Section 4 we outline how the dressing formalism can be extended to derive the $N$-soliton solution of the considered model with $N_{1}$ solitons of first type and $N_{2}$ solitons of second type, $N=N_{1}+N_{2}$. In Section 5 we demonstrate how Hirota method can be applied for deriving the soliton solutions of Tzitzeica eqs. and show that it results compatible with the ones of the dressing method. In Section 6 we briefly outline the spectral properties of the Lax operators $L$. We demonstrate that the resolvent of $L$ has pole singularities that coincide with the poles of the dressing factor and its inverse. We end by a discussion and conclusions.

2 Lorentz (Anti-)Invariance in 2-Dimensions

Obviously each of the TFT mentioned above can be viewed as a member of a hierarchy of NLEE which can be solved by applying the ISM to the corresponding Lax operator. However the Lorentz invariance singles out the TFT models from all the other members of NLEE in the hierarchy. Indeed, the TFT models allow changes of variables which may drastically change, as we shall demonstrate below, the properties of the soliton solutions.

2.1 Changes of Variables and the Lorentz (Anti-)Invariance

Let us now consider how simple linear change of variables

[TABLE]

affect the solutions of Tzitzeica eqs. Obviously this transformations have to preserve, up to a sign, $\frac{\partial^{2}}{\partial\xi\partial\eta}$ which means that

[TABLE]

which is equivalent to the relations:

[TABLE]

These relations are satisfied in two cases

[TABLE]

Here $a$ and $b$ can be, in general, arbitrary complex numbers. However, below we will consider two cases: i) $a$ and $b$ – real and ii) $a$ and $b$ – purely imaginary.

Second class of transformations involves shifts of the field $\phi$

[TABLE]

where $c_{0}>0$ is a real constant and $s_{0}$ takes the values [math] and $1$. If $s_{0}=0$ and $c_{0}=c_{1}$ then Ţ1 goes into

[TABLE]

and similar expression for the Ţ2 equation for $\phi_{2}$, but with opposite signs for the terms in the right hand side.

If we now choose $s_{0}=1$ and $c_{0}=c_{1}$ then Ţ1 goes into

[TABLE]

which for $c_{1}=1$ coincides with Ţ3 equation. We have listed the results of several such transformations in Table 1.

2.2 The Lax Representation of Ţ2 Equation

Since different members of Tzitzeica family are related by changes of variables (see Table 1), then it will be enough to consider the Lax representation and soliton solutions of only one of them, say the second equation in (1) Ţ2. It admits the following Lax representation

[TABLE]

where

[TABLE]

The reductions of the Lax pair for Ţ2 equation are similar but not the same as for the well known Ţ1 eq. [4]:

$\mathbb{Z}_{3}$-reduction

[TABLE]

which restricts $H_{0}$, $\mathcal{J}$ and $V_{1}$ by

[TABLE]

These conditions are satisfied identically. 2. 2.

First $\mathbb{Z}_{2}$-reduction

[TABLE]

i.e.

[TABLE]

which means that $\phi=\phi^{*}$. 3. 3.

Second $\mathbb{Z}_{2}$-reduction

[TABLE]

i.e.

[TABLE]

3 The Dressing Method and Dressing Factors for Ţ2 Equation

Let us start with a Lax representation of the form

[TABLE]

The fundamental solution $\Psi_{0}(\xi,\eta,\lambda)$, known also as the ‘naked’ solution, has as potential the trivial solution of Ţ2 equation: $\phi_{0}(\xi,\eta)=0$

The ’dressed’ Lax pair, given by (11), admits the "dressed" fundamental solution $\Psi(\xi,\eta,\lambda)$, with the potential the nontrivial solution $\phi(\xi,\eta)$.

The fundamental solutions $\Psi$ and $\Psi_{0}$ are related by the dressing factor $u(\xi,\eta,\lambda)$,

[TABLE]

which means that $u(\xi,\eta,\lambda)$ must satisfy

[TABLE]

Since both Lax pairs (the dressed (11) and the naked one (19)) satisfy the three reductions, then also the dressing factor must satisfy them

[TABLE]

where $A_{0}$ is defined by eq. (17).

3.1 One Soliton Solution of First Type

A natural anzatz for the dressing factor with simple poles in $\lambda$ is [21]

[TABLE]

where $A_{1}(\xi,\eta)$ is a $3\times 3$ degenerate matrix of the form

[TABLE]

The first reduction (22a) on $u(x,t,\lambda)$ is automatically satisfied by the anzatz (23). The second reduction (22b) leads to

[TABLE]

Here and below $j-k$ is understood modulo $3$ and

[TABLE]

In addition we must have

[TABLE]

and

[TABLE]

where again all matrix indices are understood modulo 3. These relations can be rewritten as

[TABLE]

So we can consider with no limitations that $\lambda_{1}=-\lambda_{1}^{*}$ and $A_{1}=-A_{1}^{*}$. More specifically we will assume that the vector $\langle m^{T}(\xi,\eta)|$ is real, while the vector $|n(\xi,\eta)\rangle$ has purely imaginary components.

The third reduction (22c) on $u(x,t,\lambda)$ can be put in the form

[TABLE]

Let us now multiply (29) by $\lambda-\lambda_{1}$, take the limit $\lambda\to\lambda_{1}$ and take into account eq. (14). This gives

[TABLE]

where

[TABLE]

Thus, taking into account that $\lambda_{1}=\mathrm{i\,}\rho_{1}$, $\rho_{1}$ – real and $m_{k}=m_{k}^{*}$, we obtain

[TABLE]

In order to obtain the vectors $|n\rangle$ and $\langle m^{T}|$ in terms of $\xi$ and $\eta$ we first impose the limit $\lambda\to\lambda_{1}$ in equation (21). We obtain that the residue $A_{1}$ must satisfy

[TABLE]

Since $A_{1}=|n\rangle\langle m^{T}|$ we find that (33) is satisfied if

[TABLE]

i.e.

[TABLE]

which means that $|n(\xi,\eta)\rangle$ is an eigenfunction for the "dressed" Lax pair $L_{1}$, $L_{2}$, while $\langle m^{T}(\xi,\eta)|$ is an eigenfunction for the "naked" Lax pair $L_{10}$, $L_{20}$.

From (19), using direct calculation we obtain

[TABLE]

which means that

[TABLE]

Using the notations

[TABLE]

we obtain the following explicit forms for the components of vector $\langle m^{T}(\xi,\eta)|$

[TABLE]

where

[TABLE]

For $\mu_{0,1}=\mu_{0,3}^{*}=|\mu_{01}|\mathrm{e}^{\mathrm{i\,}\alpha_{0}}$ and $\mu_{0,2}=\mu_{0,2}^{*}$ we can rewrite $m_{i}$ from (39) as the following real-valued functions

[TABLE]

The components of the vector $|n\rangle$ in (32) become

[TABLE]

In order to obtain the solution of Ţ2 equation we impose the limit $\lambda\to 0$ in (21) with the result:

[TABLE]

where

[TABLE]

which means that

[TABLE]

or

[TABLE]

After introducing $m_{i}$ from (41) we obtain the 1-soliton solution of the first type for $\lambda_{1}=\mathrm{i\,}\rho_{1}$

[TABLE]

where $\tilde{\Omega}_{1}=\Omega_{1}-\alpha_{01}$. We observe that this is not a traveling wave solution. Only if we take the limit $\mu_{02}\to 0$ we obtain a traveling wave solution of the form

[TABLE]

The solution is singular and it blows up for $\frac{\sqrt{3}}{2}(\rho_{1}\xi+\rho_{1}^{-1}\eta)-\alpha_{01}=(2k+1)\pi/2$, $k=0,\pm 1,\dots$. For $\alpha_{01}\rightarrow\alpha_{01}+\frac{\pi}{2}$ ($m_{10},m_{20},m_{30}\in\mathbb{C}$ and they are purely imaginary), the solution (48) becomes

[TABLE]

The above solution is also singular and it blows up for $\frac{\sqrt{3}}{2}(\rho_{1}\xi+\rho_{1}^{-1}\eta)+\alpha_{0}=k\pi$, $k=0,\pm 1,\dots$.

Remark 1

It is easy to check, that the real parts of $\phi(\xi,\eta)$ in eqs. (48) and (49) are in fact solutions to Ţ4 equation.

In order to get ‘quasi-regular’ solutions of Ţ2 equation, we can apply the changes of variables $A_{1}^{+}$ with $a=\mathrm{i\,}$ or $A_{2}^{+}$ with $b=\mathrm{i\,}$. This gives the following solutions expressed in terms of hyperbolic functions

[TABLE]

and

[TABLE]

which are singular at the points for which

[TABLE]

or

[TABLE]

respectively. These solutions have also been found by Mikhailov in [21]. As compared with the previous solutions, that have an infinite number of singularities, these ones have singularities at only two points. That is why we took the liberty to call them ‘quasi-regular’.

3.2 The Singularity Properties of the Soliton Solutions

Here we will discuss the types of singularities of the one-soliton solutions and how they are influenced by the changes of variables. As we already mentioned, the singularities in the soliton solutions are not rare, see [11, 20].

Let us first see how the changes of variables affect the Lax representation (11) and, as a consequence, how they affect the fundamental solution. We will be particularly interested in the properties of the ‘naked’ Lax pair and its fundamental solution $\Psi_{0}(\xi,\eta,\lambda)$. This comes from the fact, that the soliton solution is constructed as a rational function of the elements of $\Psi_{0}(\xi,\eta,\lambda)$.

Let us start with the change of variables $A_{1}^{+}$. Here the situation is simple; we readily get

[TABLE]

In other words this change of variables leaves invariant the compatibility of the Lax pair, so obviously it will map a solution of Ţ2 into a solution of Ţ2. However now we have to keep in mind, that the change of variables must be extended also to the spectral parameter $\lambda\to\lambda/a$ and, of course to the discrete eigenvalues of $L_{1,2}$: $\lambda_{1}\to\lambda_{1}/a$ and therefore $\rho_{1}\to\rho_{1}/a$.

In particular, from eq. (40) we see, that

[TABLE]

i.e. $X_{1}$ and $\Omega_{1}$ are invariant with respect to $A_{1}^{+}$ transformations provided

[TABLE]

Now it is a bit more interesting to analyze the changes $A_{2}^{+}$.

[TABLE]

Let us apply a gauge transformation, i.e. change from $\Psi(\xi^{\prime\prime},\eta^{\prime\prime},\lambda)$ to

[TABLE]

where $H_{0}$ and $A_{0}$ are defined in eqs. (16) and (17) respectively. This gives us

[TABLE]

So the $A_{2}^{+}$ change is equivalent to interchanging the Lax operators $L_{1}$ and $L_{2}$, which again preserves their compatibility condition. Applied to $X_{1}$ and $\Phi_{1}$ these transformations lead to

[TABLE]

Of course, analyzing the fundamental solutions we have to pay attention also whether the parameters $a$ and $b$ are real or purely imaginary. In addition we have to take into account, that $\lambda_{1}$ could be purely imaginary as above, but for other cases it could also be real. It is precisely this choice of the parameters $a$, $b$ and $\lambda_{1}$ that may change the singularity properties of the solutions.

3.3 One Soliton Solutions of Second Type

Our anzatz for the dressing factor is

[TABLE]

which obviously satisfies the $\mathbb{Z}_{3}$-reduction and the first $\mathbb{Z}_{2}$-reduction.

In order to find how the components of the vector $|n\rangle$ are expressed in terms of the vector $|m^{T}\rangle$ we use the same procedure as in the $3$-poles case. First we rewrite the dressing factor in the following form.

[TABLE]

where

[TABLE]

with

[TABLE]

We insert the dressing factor $u(\xi,\eta,\lambda)$ into The second $\mathbb{Z}_{2}$-reduction, we multiply by $\lambda-\lambda_{1}$, take the limit $\lambda\to\lambda_{1}$ and obtain

[TABLE]

After direct calculation we obtain

[TABLE]

where

[TABLE]

We rewrite the result in a matrix form

[TABLE]

where

[TABLE]

The result is

[TABLE]

where

[TABLE]

[TABLE]

From the above equations we obtain $|n\rangle$ in terms of $\langle m^{T}|$

[TABLE]

In this case we choose a general form for the poles: $\lambda_{1}=\rho_{1}\mathrm{e}^{\mathrm{i\,}\beta_{1}}$; without restrictions we can choose $0<\beta_{1}<\frac{\pi}{6}$ and determine the expressions of $\langle m^{T}|$ as

[TABLE]

where

[TABLE]

We determine the 1-soliton solution for the second kind of solitons using exactly the same technique

[TABLE]

4 The Generic $N$-soliton Solution for Ţ2 Equation

Let us consider the dressing factor of the following form

[TABLE]

with $3N_{1}+6N_{2}$ complex poles from which $N_{1}$ are purely imaginary, satisfying the following condition: $\lambda_{p}=-\lambda_{p}^{*}$.

Then we write down the residues $A_{k}(\xi,\eta)$ as degenerate $3\times 3$ matrices of the form

[TABLE]

From the second $\mathbb{Z}_{2}$-reduction, $A_{0}^{-1}u^{\dagger}(\xi,\eta,\lambda^{*})A_{0}=u^{-1}(\xi,\eta,\lambda)$, after taking the limit $\lambda\to\lambda_{k}$, we obtain algebraic equation for $|n_{k}\rangle$ in terms of $\langle m_{k}^{T}|$

[TABLE]

Below, for simplicity, we write down the matrix $\mathcal{M}$ for $N_{1}=N_{2}=1$

[TABLE]

[TABLE]

with

[TABLE]

In order to obtain the 2-soliton solution of the Tzitzeica equation we take the limit $\lambda\to 0$ in the equations satisfied by the dressing factor $u(\xi,\eta,\lambda)$ and integrate to get

[TABLE]

The above formulae can be easily generalized for any $N_{1}$ and $N_{2}$.

5 Hirota Method for Building 1-soliton Solution of Ţ2 Equation

There are many methods for deriving the soliton solutions; we have demonstrated two of the most used: the dressing method and the Hirota method [17, 3, 6]. Both methods give the same results both for the kinks and for the breathers.

We build the Hirota bilinear form of Ţ2 eq. using the substitution

[TABLE]

Introducing it into the second equation in (1) and decoupling in the bilinear dispersion relation and the soliton-phase constraint, we obtain the following system

[TABLE]

We impose that:

[TABLE]

where $z(\xi,\eta)=e^{k\xi-\omega\eta}$, $k$ - the wave number, $\omega$ - the angular frequency.

Using a software for analytical computation like MATHEMATICA, we obtain that

[TABLE]

where the dispersion relation is $\omega=\frac{3}{k}$.

Using the above results our 1-soliton solution for Ţ2 acquires the following form

[TABLE]

This solution coincide with the one obtained by Mikhailov in [21] for $k=\sqrt{3}\rho_{1}$. In this very direct manner, Hirota method gives immediately the 1-soliton solution of the first type, which we have obtained also in (50) through the dressing method, as a particular case of a more general form (47).

One can also use the standard Hirota technique to derive $N$-soliton solution of first type each parametrized with real eigenvalue $\rho_{k}$ and a vector $(\mu_{k,1},\mu_{k,2},\mu_{k,3})$ with $\mu_{k,2}=0$. We believe, that using Hirota method one can derive also more complicated cases of one- and $N$-soliton solutions. To this end, however one needs a more complicated ansatz for the functions $f(\xi,\eta)$ and $g(\xi,\eta)$ which would solve equation (82) but could not be reduced to functions of $z(\xi,\eta)$ only.

Of course, the equation (82) can be solved in a more general case, but the only one solution we were able to obtain by now, using the well known ansatz (83), was (84), which corresponds to the soliton solutions of first type. To find $g(\xi,\eta)$ and $f(\xi,\eta)$ corresponding to the second type of soliton solutions is still an open problem for us and it will be tackled in a next paper. A possible approach could be to start from the second type solitons given by the dressing factor method and, on this basis, to guess the ansatz which should be imposed to obtain $g(\xi,\eta)$ and $f(\xi,\eta)$ verifying (82).

6 The Spectral Properties of the Dressed Lax Operator

Here we shall demonstrate that each dressing adds to the discrete spectrum of $L$ sets of discrete eigenvalues.

In our previous paper we showed that the Lax operator has a set of 6 fundamental analytic solutions. We will denote them by $\chi_{\nu}(\xi,\eta,\lambda)$ where $\nu=0,\dots,5$ denotes the number of the sector $\Omega_{\nu}\equiv\frac{(2\nu+1)\pi}{6}\leq\arg\lambda\leq\frac{(2\nu+3)\pi}{6}$, i.e. those are the sectors closed by the rays $(l_{\nu},l_{\nu+1})$. The dressing factor for solitons of first type (23) obviously has simple poles located at $|\lambda_{1}|\mathrm{e}^{2\pi\mathrm{i\,}k/3}$, $k=0,1,2$. The inverse of this dressing factor has also simple poles located at $|\lambda_{1}|e^{\pi\mathrm{i\,}(2k+1)/3}$, $k=0,1,2$. Each dressing factor for soliton of second type (59) has 6 simple poles located at $|\lambda_{2}|\mathrm{e}^{\mathrm{i\,}\beta_{1}+2\pi\mathrm{i\,}k/3}$ and $|\lambda_{2}|\mathrm{e}^{-\mathrm{i\,}\beta_{1}+\pi\mathrm{i\,}(2k+1)/3}$, $k=0,1,2$. The inverse of this dressing factor has also 6 simple poles located at $|\lambda_{2}|\mathrm{e}^{\mathrm{i\,}\beta_{1}+\pi\mathrm{i\,}(2k+1)/3}$ and $|\lambda_{2}|\mathrm{e}^{-\mathrm{i\,}\beta_{1}+2\pi\mathrm{i\,}k/3}$, $k=0,1,2$.

The FAS can be used to construct the kernel of the resolvent of the Lax operator $L$. In this section by $\chi^{\nu}(\xi,\lambda)$ we will denote

[TABLE]

where $\chi_{0}^{\nu}(\xi,\lambda)$ is a regular FAS and $u(\xi,\lambda)$ is a dressing factor of general form (74). Let us introduce

[TABLE]

[TABLE]

where $\theta(\xi-\xi^{\prime})$ is the step-function and $\eta_{\nu}^{(k)}=\pm 1$, see the table 2.

Then the following theorem holds true [4]:

Theorem 2

Let $Q(\xi)$ be a Schwartz-type function and let $\lambda_{j}^{\pm}$ be the simple zeroes of the dressing factor $u(\xi,\lambda)$ (74). Then

The functions $R^{\nu}(\xi,\xi^{\prime},\lambda)$ are analytic for $\lambda\in\Upsilon_{\nu}$ where

[TABLE]

having pole singularities at $\pm\lambda_{j}^{\pm}$; 2. 2.

$R^{\nu}(\xi,\xi^{\prime},\lambda)$* is a kernel of a bounded integral operator for $\lambda\in\Upsilon_{\nu}$;* 3. 3.

$R^{\nu}(\xi,\xi^{\prime},\lambda)$* is uniformly bounded function for $\lambda\in b_{\nu}$ and provides a kernel of an unbounded integral operator;* 4. 4.

$R^{\nu}(\xi,\xi^{\prime},\lambda)$* satisfy the equation*

[TABLE]

Remark 3

The dressing factor $u(\xi,\lambda)$ has $3N_{1}+6N_{2}$ simple poles located at $\lambda_{l}q^{p}$, $\lambda_{r}q^{p}$ and $\lambda_{r}^{*}q^{p}$ where $l=1,\dots,N_{1}$, $r=1,\dots,N_{2}$ and $p=0,1,2$. Its inverse $u^{-1}(\xi,\lambda)$ has also $3N_{1}+6N_{2}$ poles located $-\lambda_{l}q^{p}$, $-\lambda_{r}q^{p}$ and $-\lambda_{r}^{*}q^{p}$. In what follows for brevity we will denote them by $\lambda_{j}$, $-\lambda_{j}$ for $j=1,\dots,3N_{1}+6N_{2}$.

It remains to show that the poles of $R^{\nu}(\xi,\xi^{\prime},\lambda)$ coincide with the poles of the dressing factors $u(\xi,\lambda)$ and its inverse $u^{-1}(\xi,\lambda)$.

The proof follows immediately from the definition of $R^{\nu}(\xi,\xi^{\prime},\lambda)$ and from eq. (86), taking into account that the limiting value $u_{-}(\lambda)$ commutes with the corresponding matrix $\Theta_{\nu}(\xi-\xi^{\prime})$.

Thus we have established that dressing by the factor $u(\xi,\lambda)$, we in fact add to the discrete spectrum of the Lax operator $6N_{1}+12N_{2}$ discrete eigenvalues; for $N_{1}=N_{2}=1$ they are shown on Figure 1.

7 Conclusions

Shortly before finishing this paper we became aware of the fact, that appropriate combination of changes of variables, considered in Section 2 can take each member of Tzitzeica family (2) into one of its 4 versions that we introduced. Let us demonstrate how this can be done of the equation

[TABLE]

where $c_{1}$ and $c_{2}$ are real positive constants. Now we shall use somewhat more general change of variables. First we apply the transformation (8) with $s_{0}=0$ and $\phi^{\prime}=\phi+\ln(c_{1}/c_{2})$. Then we change $\xi\to\xi^{\prime}/k$, $\eta\to\eta^{\prime}/k$ where $k$ is also real positive constant taken to be $k=\sqrt[3]{c_{1}^{2}c_{2}}$. Easy calculation shows that as a result eq. (91) goes into Ţ2 for $\phi^{\prime}(\xi,\eta)$. Using in addition Table 1 we can transform each member of Tzitzeica family into Ţ2 and then solve it using the results above.

Let us consider the soliton solutions Tzitzeica eq. in a small neighborhood around the singularities, where $\phi_{\rm as}(\xi,\eta)$ tends to $\infty$. Then the second term in the Ţ2 equation can be neglected and the asymptotically we get

[TABLE]

Similarly if in a small neighborhood around the singularity $\phi_{\rm as}^{\prime}(\xi,\eta)$ tends to $-\infty$ we have

[TABLE]

In both cases we find equations, equivalent to the Liouville equation. Thus the asymptotical behavior of the solutions of Tzitzeica equation around the singularities must be the same as the singularities of Liouville equation [26].

Acknowledgements

One of us (VSG) is grateful to professor A. V. Mikhailov and professor A. K. Pogrebkov for useful discussions. This work has been supported in part by a joint project between the Bulgarian academy of sciences and the Romanian academy of sciences. One of the authors (CNB) acknowledges the support of the strategic grant POSDRU/159/1.5/S/133255, Project ID 133255 (2014), co-financed by the European Social Fund within the Sectorial Operational Program Human Resources Development 2007-2013, and also the support of the project IDEI, PN-II-ID-PCE-2011-3-0083 (MECTS).