3 The FAS of the Lax operators with Z

related Lax operators

N. C. Babalic, R. Constantinescu, V. S. Gerdjikov

Abstract.We discuss the Tzitzeica equation and the spectral properties associated with its Lax operatorL. We prove that the continuous spec- trum ofL is rotated with respect to the contour of the Riemann-Hilbert problem with angle π/6. We also show that the poles of the dressing factors and their inverses are discrete eigenvalues ofL.

M.S.C. 2010: 35C08, 35Q15, 35Q51, 47A10.

Key words: Tzitzeica equation; Riemann-Hilbert problem; resolvent of Lax operator.

1 Introduction

The famous Tzitzeica equation:

2 ∂²φ

∂ξ∂η =e^2φ−e^−4φ, (1.1)

was discovered more than a century ago [15, 16] and was first used to analyze special surfaces in differential geometry for which the ratioK/d⁴is constant, see also [18, 17].

HereKis the Gauss curvature of the surface anddis the distance from the origin to the tangent plane at the given point. At the end of 1970-is eq. (1.1) was established to have higher integrals of motion [4]. Next Zhiber and Shabat [22] proved that it is completely integrable Hamiltonian system. Finally Mikhailov constructed its Lax pair [10, 11] which possesses highly nontrivial symmetry, known today as the group of reductions. In fact along with the sine-Gordon eq., Tzitzeica equation (1.1) is one of the simplest representatives of the well known by now 2-dimensional Toda field theories [10, 11].

The present paper proposes a study of the Lax representation of (1.1) and of the spectral properties of the relevant Lax operator. In Section 2 we start with preliminaries concerning the well known facts about the Lax representation and the reductions, proposed by Mikhailov, used to pick it up from the generic Lax operators.

In the next Section 3 we construct the fundamental analytic solution ofL. We prove that it has analyticity properties with respect to λ in each of the sectors Ων, ν =

Balkan Journal of Geometry and Its Applications, Vol.19, No.2, 2014, pp. 11-22.∗

°c Balkan Society of Geometers, Geometry Balkan Press 2014.

0, . . . ,5, see eq. (3.4) and Figure 1. In the next Section 4, starting from a dressing factor inspired by Zakaharov-Shabat-Mikhailov [21, 11], we outline the construction of the genericN-soliton solutions of Tzitzeica eq. In Section 5 we analyze the spectral properties of the Lax pair. We construct the kernel of the resolvent in terms od the FAS ofL, see eq. (5.2) below. The theorem 5.1 demonstrates that the continuous spectrum ofLis on the raysbν(5.4) and is rotated with respect to the contour of the RHP on angleπ/6. We prove that the poles of the dressing factors and their inverse are discrete eigenvalues ofL.

2 Preliminaries

We start with the Lax representation of Tzitzeica equation found by Mikhailov [10, 11].

L1Ψ≡∂Ψ

∂ξ −(U0+λU1)Ψ(ξ, η, λ) = 0, L₂Ψ≡∂Ψ

∂η −(V₀+λ⁻¹V₁)Ψ(ξ, η, λ) = 0, (2.1)

where

(2.2)

U0=−



 φ1,ξ 0 0 0 φ2,ξ 0

0 0 φ3,ξ



, U1=



 0 e^φ1−φ2 0 0 0 e^φ2−φ3 e^φ1−φ3 0 0



,

V0=



 φ1,η 0 0 0 φ2,η 0

0 0 φ_3,η



, V1=



 0 0 e^φ1−φ3 e^φ1−φ2 0 0

0 e^φ2−φ3 0



.

It is easy to check that the compatibility conditions ofL1 andL2gives the equation:

2∂²φα

∂ξ∂η =e^{2(φα−φα+1)}−e^{2(φα−1−φα)}, α= 1,2,3, (2.3)

whereα±1 should be taken mod 3, which generalizes the Tzitzeica equation.

Following Mikhailov, we impose reductions of the Lax pair [10, 11]. We notice that the Lax pair above satisfies identically aZ3-reduction of the form:

Q⁻¹Ψ(ξ, η, λ)Q= Ψ(ξ, η, qλ), Q=



 1 0 0

0 q 0

0 0 q²



, q=e^2πi/3. (2.4)

We also impose twoZ2-reductions, as follows.

1. The firstZ2-reduction is

Ψ^∗(ξ, η, λ^∗) = Ψ(ξ, η, λ),

U0=U₀^∗, U1=U₁^∗, V0=V₀^∗, V1=V₁^∗, (2.5)

i.e., the fieldsφk =φ^∗_k are real functions.

2. The secondZ2-reduction

A⁻¹₀ Ψ^†(ξ, η,−λ^∗)A0= Ψ⁻¹(ξ, η, λ), A0=



 0 0 1 0 1 0 1 0 0



, (2.6)

A⁻¹₀ U_k^†A0= (−1)^k+1Uk, A⁻¹₀ V_k^†A0= (−1)^k+1Vk, k= 1,2.

(2.7)

These conditions lead to:

φ1=−φ3=φ, φ2= 0, (2.8)

and

U1=



 0 e^φ 0

0 0 e^φ

e^−2φ 0 0



 V1=



 0 0 e^−2φ

e^φ 0 0

0 e^φ 0



. (2.9)

After the last reduction Tzitzeica equation acquires its classical form (1.1). There is another form of (1.1) which we will call Tzitzeica II:

2 ∂²φ

∂ξ∂η =−e^2φ+e^−4φ, (2.10)

which is obtained from (1.1) by replacingξ→iξandη→iη.

In what follows we will construct the fundamental analytic solutions (FAS) of the Lax pair. For the sake of convenience we will apply to the Lax pair a simple gauge transformation after which the new Lax operator takes the form:

Lχ≡i∂χ

∂ξ + (Q(ξ)−λJ)χ(ξ, λ) = 0, (2.11)

where we have replacediλbyλand (2.12)

χ(ξ, λ) =f0e^−φH1Ψ(ξ, λ), Q(ξ) =−2∂φ

∂ξ(J − J^T), J = diag (q,1, q²), J = 1

√3



 0 1 0 0 0 1 1 0 0



, f0= 1

√3



 q 1 q²

1 1 1

q² 1 q



.

3 The FAS of the Lax operators with Z

-reduction.

The idea for the FAS for the generalized Zakharov-Shabat (GZS) system has been proposed by Shabat [14]. However for the GZSJ is with real eigenvalues, while our Lax operator has complex eigenvalues.

The Jost solutions of eq. (2.11) are defined by:

ξ→−∞lim χ+(ξ, λ)e^iλJξ =11, lim

ξ→∞χ−(ξ, λ)e^iλJξ =11.

(3.1)

They satisfy the integral equations:

Y±(ξ, λ) =11 + Z _ξ

±∞

dye^{−iλJ(ξ−y)}Q(y)Y±(y, λ)e^iλJ(ξ−y), (3.2)

whereY±(ξ, λ) =χ±(ξ, λ)e^iλJξ. Unfortunately, with our choice forJ = diag (q,1, q²) this integral equations have no solutions. The reason is that the factorse^iλJ(ξ−y) in the kernel in (3.2) can not be made to decrease simultaneously.

Following the ideas of Caudrey, Beals and Coifman, see [3, 2, 8] we start with the Jost solutions for potentials on compact support, i.e. assume thatQ(ξ) = 0 for ξ <−L0 and ξ > L0. Then the integrals in (3.2) converge and one can prove the existence ofY±(ξ, λ).

Our next step will be to determine the continuous spectrum of L. As we shall show below, the continuous spectrum ofLconsists of those pointsλ, for which

Imλ(Jk−Jj) = Imλ(q^2−k−q^2−j) = 0.

(3.3)

It is easy to check that for each pair of indicesk6=j eq. (3.3) has a solution of the form argλ= constkj. The solutions for all choices of the pairs k, j fill up a pair of rayslν andlν+3 which are given by:

lν: arg(λ) =π(2ν+ 1)

6 , Ων: π(2ν+ 1)

6 ≤argλ≤ π(2ν+ 3)

6 ,

(3.4)

whereν= 0, . . . ,5, see Fig. 1.

Thus the analyticity regions of the FAS are the 6 sectors Ων, ν = 0, . . . ,6 split up by the set of rays lν,ν = 0, . . . ,5, see Fig. 1. Now we will outline how one can construct a FAS in each of these sectors.

Obviously, if Imλα(J) = 0 on the rays lν ∪lν+3, then Imλα(J) > 0 for λ ∈ Ων ∪Ων+1 ∪Ων+2 and Imλα(J) < 0 for λ ∈ Ων−1∪Ων−2∪Ων−3; of course all indices here are understood modulo 6. As a result the factorse^{−iλJ(ξ−y)} will decay exponentially if Imα(J)<0 and ξ−y > 0 or if Imα(J)>0 and ξ−y < 0. In eq.

(3.5) below we have listed the signs of Imα(J) for each of the sectors Ων. To each ray one can relate the root satisfying Imλα(J) = 0, i.e.

(3.5)

l0, ±(e1−e2) Ω0 α1<0, α2>0 α3>0 l₁, ±(e₁−e₃) Ω₁ α₁>0, α₂>0 α₃<0 l2, ±(e2−e3) Ω2 α1<0, α2<0 α3<0.

There are two fundamental regions: Ω0 and Ω1. The transition from Ω0 and Ω1 to the other sectors is realized by the automorphismC0:

(3.6) C0Ων ≡Ων+2, C0lν ≡lν+2, ν = 0,1,2.

The next step is to construct the set of integral equations for FAS which will be analytic in Ων. They are different from the integral equations for the Jost solutions (3.2) because for each choice of the matrix element (k, j) we specify the lower limit of the integral so that all exponential factorse^{iλ(Jk−Jj)(ξ−y)}decrease forξ, y→ ±∞, (3.7) X_kj^ν (ξ, λ) =δkj+i

Z _x

²kj∞

dye^{−iλ(Jk−Jj)(ξ−y)} Xh

p=1

Qkp(y)X_pj^ν(y, λ),

λ

×

⊗

×

⊗

×

⊗

+

⊕

+

⊕

+

⊕

+

⊕ +

⊕

+

⊕

l

l

l

l

l

l

b

b

b

b

b

b

Figure 1: The contour of the RHP withZ3-symmetry fills up the rayslν,ν= 1, . . . ,6.

By × and ⊗ (resp. by + and ⊕) we have denoted the locations of the discrete eigenvalues corresponding to a soliton of first type (resp. of second type).

where the signs²kj for each of the sectors Ων are collected in the table 1, see also [19, 7, 9]. We also assume that fork=j ²kk=−1.

The solution of the integral equations (3.7) will be the FAS ofLin the sector Ων. The asymptotics ofX^ν(x, λ) and X^ν−1(x, λ) along the raylν can be written in the form [8, 9]:

(3.8)

x→−∞lim e^iλJxX^ν(x, λeⁱ⁰)e^−iλJx=S_ν⁺(λ), λ∈lν,

x→∞lim e^iλJxX^ν(x, λeⁱ⁰)e^−iλJx=T_ν⁻(λ)D⁺_ν(λ), λ∈lν,

x→−∞lim e^iλJxX^ν−1(x, λe⁻ⁱ⁰)e^−iλJx=S_ν⁻(λ), λ∈lν,

x→∞lim e^iλJxX^ν−1(x, λe⁻ⁱ⁰)e^−iλJx=T_ν⁺(λ)D⁻_ν(λ), λ∈lν,

where the matricesS_ν^± andT_ν^± belong tosu(2) subgroups ofsl(3). More specifically from the integral equations (3.7) we find:

(3.9)

S₀⁺(λ) =11 +s⁺_0;21E21, T₀⁻(λ) =11 +τ_0;12⁻ E12, S₀⁻(λ) =11 +s⁺_0;12E12, T₀⁺(λ) =11 +τ_0;21⁺ E21, D₀⁺(λ) =d⁺_0;1E11+ 1

d⁺_0;1E22+E33, D₀⁻(λ) = 1

d⁻_0;1E11+d⁻_0;1E22+E33.

(k, j) (1,2) (1,3) (2,3) (2,1) (3,2) (3,1)

Ω₀ − + + + − −

Ω1 − + − + − +

Ω2 − + − + − +

Ω3 + + + − − −

Ω4 − + − + − +

Ω5 − + − + + −

Table 1: The set of signs²kj for each of the sectors Ων.

and

(3.10)

S⁺₁(λ) =11 +s⁺_1;31E31, T₁⁻(λ) =11 +τ_1;13⁻ E13, S₁⁻(λ) =11 +s⁺_1;13E13, T₁⁺(λ) =11 +τ_1;31⁺ E31, D⁺₁(λ) =d⁺_1;1E11+E22+ 1

d⁺_1;1E33, D₁⁻(λ) = 1

d⁻_1;1E11+E22+d⁻_1;1E33, where byEkjwe mean a 3×3 matrix with matrix elements (Ekj)mn=δumδjn.

The corresponding factors for the asymptotics ofX^ν(x, λeⁱ⁰) forν >1 are obtained from eqs. (3.9), (3.10) by applying the automorphismC0. If we consider potential on finite support, then we can define not only the Jost solutions Ψ_±(x, λ) but also the scattering matrix T(λ) =χ−(x, λ)χ⁻¹₊ (x, λ). The factors S_ν^±(λ), T_ν^±(λ) andD_ν^±(λ) provide an analog of the Gauss decomposition of the scattering matrix with respect to theν-ordering, i.e.:

(3.11) Tν(λ) =T_ν⁻(λ)D_ν⁺(λ) ˆS_ν⁺(λ) =T_ν⁺(λ)D⁻_ν(λ) ˆS_ν⁻(λ), λ∈lν.

TheZn-symmetry imposes the following constraints on the FAS and on the scat- tering matrix and its factors:

(3.12) C0X^ν(x, λω)C₀⁻¹=X^ν−2(x, λ), C0Tν(λω)C₀⁻¹=Tν−2(λ), C0S_ν^±(λω)C₀⁻¹=S_ν−2^± (λ), C0D_ν^±(λω)C₀⁻¹=D^±_ν−2(λ), where the index ν −2 should be taken modulo 6. Consequently we can view as independent only the data on two of the rays, e.g. onl0 and l1; all the rest will be recovered using the reduction conditions.

If in addition we impose theZ2-symmetry, then we will have also:

(3.13)

a) K₀⁻¹(X^ν(x,−λ^∗))^†K0= ˆX^N+1−ν(x, λ), K₀⁻¹(S_ν^±(−λ^∗))K0= ˆS_N^∓_+1−ν(λ), b) K₀⁻¹(X^ν(x, λ^∗))^∗K0= ˆX^ν(x, λ), K₀⁻¹(S^±_ν(λ^∗))K0= ˆS_N^∓_+1−ν(λ), where by ‘hat’ we denote the inverse matrix. Analogous relations hold true forT_ν^±(λ) andD^±_ν(λ). One can prove also thatD⁺_ν(λ) (resp. D_ν⁻(λ)) allows analytic extension forλ∈Ων (resp. forλ∈Ων−1. Another important fact is thatD⁺_ν(λ) =D⁻_ν+1(λ) for allλ∈Ων.

The next important step is the possibility to reduce the solution of the ISP for the GZSs to a (local) RHP. More precisely, we have:

(3.14)

X^ν(x, η, λ) =X^ν−1(x, η, λ)Gν(x, η, λ), λ∈lν, Gν(x, η, λ) =e^{iλJξ−λ−1V2t}G0,ν(λ)e^{−iλJξ+λ−1V2t}, G0,ν(λ) = ˆS_ν⁻S_ν⁺(λ)

¯¯

¯t=0. The collection of all these relations forν = 0,1, . . . ,5 together with

(3.15) lim

λ→∞X^ν(x, η, λ) =11,

can be viewed as a local RHP posed on the collection of rays Σ ≡ {lν}^2N_ν=1 with canonical normalization. Rather straightforwardly we can prove that ifX^ν(x, λ) is a solution of the RHP thenχ^ν(x, λ) =X^ν(x, λ)e^−iλJξ is a FAS ofLwith potential

(3.16) Q(ξ, t) = lim

λ→∞λ³

J−X^ν(ξ, η, λ)JXˆ^ν(ξ, η, λ)´ .

4 The dressing method and the N -soliton solutions

There are several methods for effective calculations of soliton solutions for Tzitzeica eq., see [12, 20, 13]. It is also well known that Tzitzeica eq. has two types of one- soliton solutions, see below. The dressing method that we will use below [21, 11, 10]

allows us also to find how the spectral properties ofLchange due to the dressing.

Let us consider dressing factor of the following form:

u(ξ, η, λ) =11 + X2

s=0

Ã_N X1

l=1

Q^−sAlQ^s λ−λlq^s +

XN

r=N1+1

Q^−sArQ^s λ−λrq^s +

XN

r=N1+1

Q^−sA^∗_rQ^s λ−(λ^∗_r)q^s

! , (4.1)

with 3N1+ 6N2 poles andλp is real ifp∈1, N1and complex ifp∈N1+ 1, N1+N2. Then we write down the residuesAk(ξ, η) as degenerate matrices of the form:

Ak(ξ, η) =|nk(ξ, η)ihm^T_k(ξ, η)|, (Ak)ij(ξ, η) =nki(ξ, η)mkj(ξ, η).

(4.2)

Thusu(ξ, η, λ) has 9 poles located atλ1q^kwithλ1real andλ2q^k,λ^∗₂q^k, withk= 0,1,2 andλ2 complex. From the second Z2-reduction,A⁻¹₀ u^†(ξ, η,−λ^∗)A0 =u⁻¹(ξ, η, λ), after taking the limitλ→λk, we obtain algebraic equation for|nkiin terms ofhm^T_k|:

|νi=M⁻¹|µi.

(4.3)

Below for simplicity we write down the matrixMforN1=N2= 1:

|νi=



 |n1i

|n2i

|n^∗₂i



, |µi=



 A0|m1i A0|m2i A0|m^∗₂i



, M=



 A B B^∗

B^∗ D E

B E^∗ D^∗



, (4.4)

(4.5)

A= 1

2λ³₁diag (Q⁽¹⁾, Q⁽²⁾, Q⁽³⁾), B= 1

λ³₁+λ³₂diag (P⁽¹⁾, P⁽²⁾, P⁽³⁾), D= 1

2λ³₂diag (P⁽¹⁾, P⁽²⁾, P⁽³⁾), E= 1

λ³₂+λ^∗,3₂ diag (K⁽¹⁾, K⁽²⁾, K⁽³⁾), Q^(j)=hm^T₁|Λ^(j)₁₁(λl, λ1)|m1i, K^(j)=hm^∗,T₂ |Λ^(j)₁₂(λ1, λ^∗₂)|m1i,

P^(j)=hm^T₂|Λ^(j)₂₁(λ2, λ1)|mli, with

(4.6) Λ^(j)_lp =−λlλpE1+j,3−j+λ²_lE2+j,2−j+λ²_pE3+j,1−j, j = 1,2,3.

For example, in order to obtain the 2-soliton solution of the Tzitzeica equation we take the limitλ→ 0 in the equations satisfied by the dressing factoru(ξ, η, λ) and integrate. The result is:

(4.7) φNs(ξ, η) =−1 2ln

¯¯

¯¯1−n1,1m1,1

λ1

−n2,1m2,1

λ2

−n^∗_2,1m^∗_2,1 λ^∗₂

¯¯

¯¯. The above formulae can be easily generalized for anyN1 andN2.

For the sake of brevity we skip the details, which allow one to obtain the explicit form of theN-soliton solutions. We just mention that along with the explicit expres- sions for the vectors|nkiin terms ofhmj|that follow from (4.3)–(4.6) and take into account that|mjiare solutions of the ‘naked’ Lax operator with vanishing potential φ= 0.

5 The resolvent of the Lax operator

The FAS can be used to construct the kernel of the resolvent of the Lax operatorL.

In this section byχ^ν(ξ, λ) we will denote:

χ^ν(ξ, λ) =u(ξ, λ)χ^ν₀(ξ, λ), (5.1)

whereχ^ν₀(ξ, λ) is a regular FAS andu(ξ, λ) is a dressing factor of general form (4.1).

Remark 5.1. The dressing factoru(ξ, λ) has 3N1+ 6N2simple poles located atλlq^p, λrq^pandλ^∗_rq^p wherel= 1, . . . , N1,r= 1, . . . , N2andp= 0,1,2. Its inverseu⁻¹(ξ, λ) has also 3N1+6N2poles located−λlq^p,−λrq^pand−λ^∗_rq^p. In what follows for brevity we will denote them byλj,−λj forj= 1, . . . ,3N1+ 6N2.

Let us introduce

(5.2) R^ν(ξ, ξ⁰, λ) = 1

iχ^ν(ξ, λ)Θν(ξ−ξ⁰) ˆχ^ν(ξ⁰, λ), Θν(ξ−ξ⁰) = diag

³

η_ν⁽¹⁾θ(η⁽¹⁾_ν (ξ−ξ⁰)), η⁽²⁾_ν θ(η⁽²⁾_ν (ξ−ξ⁰)), η⁽³⁾_ν θ(η_ν⁽³⁾(ξ−ξ⁰))

´ , (5.3)

whereθ(ξ−ξ⁰) is the step-function andην^(k)=±1, see the table 2.

Theorem 5.1. Let Q(ξ)be a Schwartz-type function and letλ^±_j be the simple zeroes of the dressing factoru(ξ, λ)(4.1). Then

Υ0 Υ1 Υ2 Υ3 Υ4 Υ5

η⁽¹⁾ν − − − + + +

η⁽²⁾ν + + − − − +

η⁽³⁾ν − + + + − −

Table 2: The set of signsην^(k)for each of the sectors Υν (5.4).

1. The functionsR^ν(ξ, ξ⁰, λ)are analytic for λ∈Υν where bν: argλ=π(ν+ 1)

3 , Υν: π(ν+ 1)

3 ≤argλ≤ π(ν+ 2)

3 ,

(5.4)

having pole singularities at±λ^±_j;

2. R^ν(ξ, ξ⁰, λ)is a kernel of a bounded integral operator forλ∈Υν;

3. R^ν(ξ, ξ⁰, λ) is uniformly bounded function for λ∈ bν and provides a kernel of an unbounded integral operator;

4. R^ν(ξ, ξ⁰, λ)satisfy the equation:

(5.5) L(λ)R^ν(ξ, ξ⁰, λ) =11δ(ξ−ξ⁰).

Idea of the proof. 1. First we shall prove that R^ν(ξ, ξ⁰, λ) has no jumps on the rayslν. From Section 3 we know thatX^ν(ξ, λ) and therefore alsoχ^ν(ξ, λ) are analytic forλ∈Ων. So we have to show that the limits ofR^ν(ξ, ξ⁰, λ) forλ→lν

from Υν and Υν−1 are equal. Let show that for ν = 0. From the asymptotics (3.8) and from the RHP (3.14) we have:

χ⁰(ξ, λ) =χ¹(ξ, λ)G1(λ), G1(λ) = ˆS₁⁺(λ)S₁⁻(λ), λ∈l1, (5.6)

where G1(λ) belongs to an SL(2) subgroup of SL(3) and is such that it com- mutes with Θ1(ξ−ξ⁰). Thus we conclude that

R1(ξ, ξ⁰, λe⁺ⁱ⁰) =R1(ξ, ξ⁰, λe⁻ⁱ⁰), λ∈l1. (5.7)

Analogously we prove thatRν(ξ, ξ⁰, λe⁺ⁱ⁰) has no jumps on the other rayslν. The jumps on the rays bν appear because of two reasons: first, because of the functions Θν(ξ−ξ⁰) and second, it is easy to check that for λ∈bν the kernel Rν(ξ, ξ⁰, λ) oscillates forξ, ξ⁰ tending to±∞. Thus on these lines the resolvent is unbounded integral operator.

2. Assume that λ ∈Υν and consider the asymptotic behavior of R^ν(ξ, ξ⁰, λ) for ξ, ξ⁰→ ∞. From equations (3.8) we find that

R^ν_ij(ξ, ξ⁰, λ) = Xn

p=1

X_ip^ν(ξ, λ)e^{−iλJp(ξ−ξ0)}Θν;pp(ξ−ξ⁰) ˆX_pj^ν(ξ⁰, λ).

(5.8)

Due to the fact thatχν(ξ, λ) has the special triangular asymptotics for ξ→ ∞ andλ∈Υνand for the correct choice of Θν(ξ−ξ⁰) (5.3) we check that the right hand side of (5.8) falls off exponentially forξ→ ∞ and arbitrary choice ofξ⁰. All other possibilities are treated analogously.

3. For λ ∈ bν the arguments of 2) can not be applied because the exponentials in the right hand side of (5.8) Imλ= 0 only oscillate. Thus we conclude that R^ν(ξ, ξ⁰, λ) for λ∈bν is only a bounded function and thus the corresponding operatorR(λ) is an unbounded integral operator.

4. The proof of eq. (5.5) follows from the fact thatL(λ)χν(ξ, λ) = 0 and

(5.9) ∂Θ(ξ−ξ⁰)

∂ξ =11δ(ξ−ξ⁰),

which concludes the proof. ¤

Lemma 5.2. The poles of R^ν(ξ, ξ⁰, λ)coincide with the poles of the dressing factors u(ξ, λ) and its inverseu⁻¹(ξ, λ).

Proof. The proof follows immediately from the definition of R^ν(ξ, ξ⁰, λ) and from

Remark 5.1. ¤

Thus we have established that dressing by the factor u(ξ, λ), we in fact add to the discrete spectrum of the Lax operator 6N1+ 12N2discrete eigenvalues; forN1= N2= 1 they are shown on Figure 1.

6 Conclusions

We have constructed the FAS ofL which satisfy a RHP on the set of rays lν. We also constructed the resolvent of the Lax operator and proved that its continuous spectrum fills up the raysbνrather thanlν. From Figure 1 we see that the eigenvalues corresponding to the solitons of first type lay on the continuous spectrum ofL. This explains why the solitons of first type are singular functions.

Using the explicit form of the resolvent R^ν(ξ, ξ⁰, λ) and the contour integration method one can derive the completeness relation of the FAS. One can derive also the soliton solutions of the other NLEE in Tzitzeica hierarchy [5, 6]. These equations also have Lax representation with the same Lax operatorL, but with different M- operators; usually they are taken to be polynomial inλ. So in deriving their soliton solutions we will need to change only theη-dependence of the vectorsmk.

Similarly one can construct theN-soliton solutions also of the Tzitzeica-II equation and analyze the spectral properties of the relevant Lax operator. This, along with the details of calculating theN-soliton solutions will be published elsewhere.

Acknowledgements. One of us (V. S. G.) is grateful for the warm hospitality during his visits to Craiova and Timisoara Universities, Romania, where the paper was finished. This work is supported by the ICTP - SEENET-MTP project Cosmology and Strings PRJ-09” and by the bilateral agreement between the Bulgarian Academy of Sciences and the National Romanian Academy. V. S. G. is grateful also to Professor

C. Udriste and to Professor V. Balan for giving him the chance to participate in the DGDS-2013 conference. C. N. Babalic also acknowledges the support of the project IDEI, PN-II-ID-PCE-53-2011, Romanian Ministry of Education.

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Authors’ addresses:

Nicoleta Corina Babalic

Department of Physics, University of Craiova, 13 A.I. Cuza Str., RO-200585, Craiova, Romania;

Dept. of Theoretical Physics, National Institute of Physics and Nuclear Engineering, 407 Atomistilor Str., RO-077125, Magurele, Bucharest, Romania.

E-mail: b [email protected] Radu Constantinescu

Department of Physics, University of Craiova, 13 A.I. Cuza Str., RO-200585, Craiova, Romania.

E-mail: [email protected] Vladimir Stefanov Gerdjikov

Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences

72 Tzarigradsko Str., 1784 Sofia, Bulgaria.

E-mail: [email protected]