On Tzitzeica equation and spectral properties of related Lax operators

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1 On Tzitzeica equation and spectral properties of 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 operator L. We prove that the continuous spectrum of L 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 of L. 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: (1.1) 2 2 φ ξ η = e2φ e 4φ, was discovered more than a century ago [15, 16] and was first used to analyze special surfaces in differential geometry for which the ratio K/d 4 is constant, see also [18, 17]. Here K is the Gauss curvature of the surface and d is 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 of L. We prove that it has analyticity properties with respect to λ in each of the sectors Ω ν, ν = Balkan Journal of Geometry and Its Applications, Vol.19, No.2, 2014, pp c Balkan Society of Geometers, Geometry Balkan Press 2014.

2 12 N. C. Babalic, R. Constantinescu, V. S. Gerdjikov 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 generic N-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 of L, see eq. (5.2) below. The theorem 5.1 demonstrates that the continuous spectrum of L is on the rays b ν (5.4) and is rotated with respect to the contour of the RHP on angle π/6. We prove that the poles of the dressing factors and their inverse are discrete eigenvalues of L. 2 Preliminaries We start with the Lax representation of Tzitzeica equation found by Mikhailov [10, 11]. (2.1) where (2.2) U 0 = V 0 = L 1 Ψ Ψ ξ (U 0 + λu 1 )Ψ(ξ, η, λ) = 0, L 2 Ψ Ψ η (V 0 + λ 1 V 1 )Ψ(ξ, η, λ) = 0, φ 1,ξ φ 2,ξ φ 3,ξ φ 1,η φ 2,η φ 3,η, U 1 =, V 1 = 0 e φ1 φ e φ 2 φ 3 e φ 1 φ e φ1 φ3 e φ1 φ e φ 2 φ 3 0,. It is easy to check that the compatibility conditions of L 1 and L 2 gives the equation: (2.3) 2 2 φ α ξ η = e2(φ α φ α+1 ) e 2(φ α 1 φ α ), α = 1, 2, 3, where α ± 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 a Z 3 -reduction of the form: (2.4) Q 1 Ψ(ξ, η, λ)q = Ψ(ξ, η, qλ), Q = We also impose two Z 2 -reductions, as follows. 1. The first Z 2 -reduction is q q 2, q = e 2πi/3. (2.5) Ψ (ξ, η, λ ) = Ψ(ξ, η, λ), U 0 = U 0, U 1 = U 1, V 0 = V 0, V 1 = V 1, i.e., the fields φ k = φ k are real functions.

3 On Tzitzeica equation and spectral properties of related Lax operators The second Z 2 -reduction (2.6) (2.7) A 1 0 Ψ (ξ, η, λ )A 0 = Ψ 1 (ξ, η, λ), A 0 = A 1 0 U k A 0 = ( 1) k+1 U k, A 1 0 V k A 0 = ( 1) k+1 V k, k = 1, 2. These conditions lead to:, (2.8) and (2.9) U 1 = φ 1 = φ 3 = φ, φ 2 = 0, 0 e φ e φ e 2φ 0 0 V 1 = 0 0 e 2φ e φ e φ 0. 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: (2.10) 2 2 φ ξ η = e2φ + e 4φ, which is obtained from (1.1) by replacing ξ iξ and η 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: (2.11) Lχ i χ ξ + (Q(ξ) λj) χ(ξ, λ) = 0, where we have replaced iλ by λ and (2.12) χ(ξ, λ) = f 0 e φh 1 Ψ(ξ, λ), Q(ξ) = 2 φ ξ (J J T ), J = diag (q, 1, q 2 ), J = , f 0 = 1 q 1 q q 2 1 q 3 The FAS of the Lax operators with Z n -reduction. The idea for the FAS for the generalized Zakharov-Shabat (GZS) system has been proposed by Shabat [14]. However for the GZS J is with real eigenvalues, while our Lax operator has complex eigenvalues. The Jost solutions of eq. (2.11) are defined by: (3.1) lim χ +(ξ, λ)e iλjξ = 11, ξ lim χ (ξ, λ)e iλjξ = 11. ξ

4 14 N. C. Babalic, R. Constantinescu, V. S. Gerdjikov They satisfy the integral equations: (3.2) Y ± (ξ, λ) = 11 + ξ ± dye iλj(ξ y) Q(y)Y ± (y, λ)e iλj(ξ y), where Y ± (ξ, λ) = χ ± (ξ, λ)e iλjξ. Unfortunately, with our choice for J = diag (q, 1, q 2 ) this integral equations have no solutions. The reason is that the factors e 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 that Q(ξ) = 0 for ξ < L 0 and ξ > L 0. Then the integrals in (3.2) converge and one can prove the existence of Y ± (ξ, λ). Our next step will be to determine the continuous spectrum of L. As we shall show below, the continuous spectrum of L consists of those points λ, for which (3.3) Im λ(j k J j ) = Im λ(q 2 k q 2 j ) = 0. It is easy to check that for each pair of indices k j eq. (3.3) has a solution of the form arg λ = const kj. The solutions for all choices of the pairs k, j fill up a pair of rays l ν and l ν+3 which are given by: (3.4) l ν : arg(λ) = π(2ν + 1), Ω ν : 6 π(2ν + 1) 6 arg λ π(2ν + 3), 6 where ν = 0,..., 5, see Fig. 1. Thus the analyticity regions of the FAS are the 6 sectors Ω ν, ν = 0,..., 6 split up by the set of rays l ν, ν = 0,..., 5, see Fig. 1. Now we will outline how one can construct a FAS in each of these sectors. Obviously, if Im λα(j) = 0 on the rays l ν l ν+3, then Im λα(j) > 0 for λ Ω ν Ω ν+1 Ω ν+2 and Im λα(j) < 0 for λ Ω ν 1 Ω ν 2 Ω ν 3 ; of course all indices here are understood modulo 6. As a result the factors e iλj(ξ y) will decay exponentially if Im α(j) < 0 and ξ y > 0 or if Im α(j) > 0 and ξ y < 0. In eq. (3.5) below we have listed the signs of Im α(j) for each of the sectors Ω ν. To each ray one can relate the root satisfying Im λα(j) = 0, i.e. (3.5) l 0, ±(e 1 e 2 ) Ω 0 α 1 < 0, α 2 > 0 α 3 > 0 l 1, ±(e 1 e 3 ) Ω 1 α 1 > 0, α 2 > 0 α 3 < 0 l 2, ±(e 2 e 3 ) Ω 2 α 1 < 0, α 2 < 0 α 3 < 0. There are two fundamental regions: Ω 0 and Ω 1. The transition from Ω 0 and Ω 1 to the other sectors is realized by the automorphism C 0 : (3.6) C 0 Ω ν Ω ν+2, C 0 l ν l ν+2, ν = 0, 1, 2. The next step is to construct the set of integral equations for FAS which will be analytic in Ω ν. 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 factors e iλ(j k J j)(ξ y) decrease for ξ, y ±, x (3.7) Xkj(ξ, ν λ) = δ kj + i ɛ kj dye iλ(j k J j )(ξ y) h Q kp (y)xpj(y, ν λ), p=1

5 On Tzitzeica equation and spectral properties of related Lax operators 15 l 1 λ b 1 b 0 l l 0 + b 2 b 5 + l l 5 b 3 b 4 l 4 Figure 1: The contour of the RHP with Z 3 -symmetry fills up the rays l ν, 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 signs ɛ kj for each of the sectors Ω ν are collected in the table 1, see also [19, 7, 9]. We also assume that for k = j ɛ kk = 1. The solution of the integral equations (3.7) will be the FAS of L in the sector Ω ν. The asymptotics of X ν (x, λ) and X ν 1 (x, λ) along the ray l ν can be written in the form [8, 9]: (3.8) lim x eiλjx X ν (x, λe i0 )e iλjx = S ν + (λ), λ l ν, lim x eiλjx X ν (x, λe i0 )e iλjx = Tν (λ)d ν + (λ), λ l ν, lim x eiλjx X ν 1 (x, λe i0 )e iλjx = Sν (λ), λ l ν, lim x eiλjx X ν 1 (x, λe i0 )e iλjx = T ν + (λ)dν (λ), λ l ν, where the matrices S ± ν and T ± ν belong to su(2) subgroups of sl(3). More specifically from the integral equations (3.7) we find: (3.9) S + 0 (λ) = 11 + s+ 0;21 E 21, T 0 (λ) = 11 + τ 0;12 E 12, S 0 (λ) = 11 + s+ 0;12 E 12, T + 0 (λ) = 11 + τ + 0;21 E 21, D 0 + (λ) = d+ 0;1 E d + E 22 + E 33, D0 (λ) = 1 0;1 d E 11 + d 0;1 E 22 + E 33. 0;1

6 16 N. C. Babalic, R. Constantinescu, V. S. Gerdjikov (k, j) (1,2) (1,3) (2,3) (2,1) (3,2) (3,1) Ω Ω Ω Ω Ω Ω Table 1: The set of signs ɛ kj for each of the sectors Ω ν. and (3.10) S + 1 (λ) = 11 + s+ 1;31 E 31, T 1 (λ) = 11 + τ 1;13 E 13, S 1 (λ) = 11 + s+ 1;13 E 13, T + 1 (λ) = 11 + τ + 1;31 E 31, D 1 + (λ) = d+ 1;1 E 11 + E d + E 33, D1 (λ) = 1 1;1 d E 11 + E 22 + d 1;1 E 33, 1;1 where by E kj we mean a 3 3 matrix with matrix elements (E kj ) mn = δ um δ jn. The corresponding factors for the asymptotics of X ν (x, λe i0 ) for ν > 1 are obtained from eqs. (3.9), (3.10) by applying the automorphism C 0. 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) and D ν ± (λ) provide an analog of the Gauss decomposition of the scattering matrix with respect to the ν-ordering, i.e.: (3.11) T ν (λ) = T ν (λ)d + ν (λ)ŝ+ ν (λ) = T + ν (λ)d ν (λ)ŝ ν (λ), λ l ν. The Z n -symmetry imposes the following constraints on the FAS and on the scattering matrix and its factors: (3.12) C 0 X ν (x, λω)c0 1 = X ν 2 (x, λ), C 0 T ν (λω)c0 1 = T ν 2 (λ), C 0 S ν ± (λω)c0 1 = S ν 2 ± (λ), C 0D 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. on l 0 and l 1 ; all the rest will be recovered using the reduction conditions. If in addition we impose the Z 2 -symmetry, then we will have also: (3.13) a) K0 1 (Xν (x, λ )) K 0 = ˆX N+1 ν (x, λ), K0 1 (S± ν ( λ ))K 0 = Ŝ N+1 ν (λ), b) K 1 0 (Xν (x, λ )) K 0 = ˆX ν (x, λ), K 1 0 (S± ν (λ ))K 0 = Ŝ N+1 ν (λ), where by hat we denote the inverse matrix. Analogous relations hold true for T ν ± (λ) and D ν ± (λ). One can prove also that D ν + (λ) (resp. Dν (λ)) allows analytic extension for λ Ω ν (resp. for λ Ω ν 1. Another important fact is that D ν + (λ) = Dν+1 (λ) for all λ Ω ν.

7 On Tzitzeica equation and spectral properties of related Lax operators 17 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)g ν (x, η, λ), λ l ν, G ν (x, η, λ) = e iλjξ λ 1 V 2 t G 0,ν (λ)e iλjξ+λ 1 V 2 t, G 0,ν (λ) = Ŝ ν S ν + (λ). t=0 The collection of all these relations for ν = 0, 1,..., 5 together with (3.15) lim λ Xν (x, η, λ) = 11, can be viewed as a local RHP posed on the collection of rays Σ {l ν } 2N ν=1 with canonical normalization. Rather straightforwardly we can prove that if X ν (x, λ) is a solution of the RHP then χ ν (x, λ) = X ν (x, λ)e iλjξ is a FAS of L with potential ( (3.16) Q(ξ, t) = lim λ J X ν (ξ, η, λ)j ˆX ) ν (ξ, η, λ). 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 onesoliton solutions, see below. The dressing method that we will use below [21, 11, 10] allows us also to find how the spectral properties of L change due to the dressing. Let us consider dressing factor of the following form: (4.1) u(ξ, η, λ) = s=0 ( N1 l=1 Q s A l Q s λ λ l q s + N r=n 1+1 Q s A r Q s λ λ r q s + N r=n 1+1 ) Q s A rq s λ (λ r)q s, with 3N 1 + 6N 2 poles and λ p is real if p 1, N 1 and complex if p N 1 + 1, N 1 + N 2. Then we write down the residues A k (ξ, η) as degenerate matrices of the form: (4.2) A k (ξ, η) = n k (ξ, η) m T k (ξ, η), (A k ) ij (ξ, η) = n ki (ξ, η)m kj (ξ, η). Thus u(ξ, η, λ) has 9 poles located at λ 1 q k with λ 1 real and λ 2 q k, λ 2q k, with k = 0, 1, 2 and λ 2 complex. From the second Z 2 -reduction, A 1 0 u (ξ, η, λ )A 0 = u 1 (ξ, η, λ), after taking the limit λ λ k, we obtain algebraic equation for n k in terms of m T k : (4.3) ν = M 1 µ. Below for simplicity we write down the matrix M for N 1 = N 2 = 1: (4.4) ν = n 1 n 2 n 2, µ = A 0 m 1 A 0 m 2 A 0 m 2, M = A B B B D E B E D,

8 18 N. C. Babalic, R. Constantinescu, V. S. Gerdjikov (4.5) with A = 1 2λ 3 diag (Q (1), Q (2), Q (3) ), B = 1 D = 1 2λ 3 diag (P (1), P (2), P (3) ), E = 2 1 λ diag (P (1), P (2), P (3) ), λ3 2 1 λ diag (K (1), K (2), K (3) ), λ,3 2 Q (j) = m T 1 Λ (j) 11 (λ l, λ 1 ) m 1, K (j) = m,t 2 Λ (j) 12 (λ 1, λ 2) m 1, P (j) = m T 2 Λ (j) 21 (λ 2, λ 1 ) m l, (4.6) Λ (j) lp = λ lλ p E 1+j,3 j + λ 2 l E 2+j,2 j + λ 2 pe 3+j,1 j, j = 1, 2, 3. For example, in order to obtain the 2-soliton solution of the Tzitzeica equation we take the limit λ 0 in the equations satisfied by the dressing factor u(ξ, η, λ) and integrate. The result is: (4.7) φ Ns (ξ, η) = 1 2 ln 1 n 1,1m 1,1 n 2,1m 2,1 n 2,1m 2,1 λ 1 λ 2. The above formulae can be easily generalized for any N 1 and N 2. For the sake of brevity we skip the details, which allow one to obtain the explicit form of the N-soliton solutions. We just mention that along with the explicit expressions for the vectors n k in terms of m j that follow from (4.3) (4.6) and take into account that m j are 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 operator L. In this section by χ ν (ξ, λ) we will denote: λ 2 (5.1) χ ν (ξ, λ) = u(ξ, λ)χ ν 0(ξ, λ), where χ ν 0(ξ, λ) is a regular FAS and u(ξ, λ) is a dressing factor of general form (4.1). Remark 5.1. The dressing factor u(ξ, λ) has 3N 1 +6N 2 simple poles located at λ l q p, λ r q p and λ rq p where l = 1,..., N 1, r = 1,..., N 2 and p = 0, 1, 2. Its inverse u 1 (ξ, λ) has also 3N 1 +6N 2 poles located λ l q p, λ r q p and λ rq p. In what follows for brevity we will denote them by λ j, λ j for j = 1,..., 3N 1 + 6N 2. Let us introduce (5.2) R ν (ξ, ξ, λ) = 1 i χν (ξ, λ)θ ν (ξ ξ )ˆχ ν (ξ, λ), (5.3) Θ ν (ξ ξ ) = diag ( ) η ν (1) θ(η ν (1) (ξ ξ )), η ν (2) θ(η ν (2) (ξ ξ )), η ν (3) θ(η ν (3) (ξ ξ )), where θ(ξ ξ ) is the step-function and η (k) ν = ±1, see the table 2. Theorem 5.1. Let Q(ξ) be a Schwartz-type function and let λ ± j of the dressing factor u(ξ, λ) (4.1). Then be the simple zeroes

9 On Tzitzeica equation and spectral properties of related Lax operators 19 Υ 0 Υ 1 Υ 2 Υ 3 Υ 4 Υ 5 η ν (1) η ν (2) η ν (3) Table 2: The set of signs η (k) ν for each of the sectors Υ ν (5.4). 1. The functions R ν (ξ, ξ, λ) are analytic for λ Υ ν where (5.4) b ν : arg λ = π(ν + 1), Υ ν : 3 π(ν + 1) 3 arg λ π(ν + 2), 3 having pole singularities at ±λ ± j ; 2. R ν (ξ, ξ, λ) is a kernel of a bounded integral operator for λ Υ ν ; 3. R ν (ξ, ξ, λ) is uniformly bounded function for λ b ν and provides a kernel of an unbounded integral operator; 4. R ν (ξ, ξ, λ) satisfy the equation: (5.5) L(λ)R ν (ξ, ξ, λ) = 11δ(ξ ξ ). Idea of the proof. 1. First we shall prove that R ν (ξ, ξ, λ) has no jumps on the rays l ν. From Section 3 we know that X ν (ξ, λ) and therefore also χ ν (ξ, λ) are analytic for λ Ω ν. So we have to show that the limits of R ν (ξ, ξ, λ) for λ l ν from Υ ν and Υ ν 1 are equal. Let show that for ν = 0. From the asymptotics (3.8) and from the RHP (3.14) we have: (5.6) χ 0 (ξ, λ) = χ 1 (ξ, λ)g 1 (λ), G 1 (λ) = Ŝ+ 1 (λ)s 1 (λ), λ l 1, where G 1 (λ) belongs to an SL(2) subgroup of SL(3) and is such that it commutes with Θ 1 (ξ ξ ). Thus we conclude that (5.7) R 1 (ξ, ξ, λe +i0 ) = R 1 (ξ, ξ, λe i0 ), λ l 1. Analogously we prove that R ν (ξ, ξ, λe +i0 ) has no jumps on the other rays l ν. The jumps on the rays b ν appear because of two reasons: first, because of the functions Θ ν (ξ ξ ) and second, it is easy to check that for λ b ν the kernel R ν (ξ, ξ, λ) oscillates for ξ, ξ tending to ±. Thus on these lines the resolvent is unbounded integral operator. 2. Assume that λ Υ ν and consider the asymptotic behavior of R ν (ξ, ξ, λ) for ξ, ξ. From equations (3.8) we find that (5.8) R ν ij(ξ, ξ, λ) = n Xip(ξ, ν λ)e iλj p(ξ ξ ) Θ ν;pp (ξ ξ ) ˆX pj(ξ ν, λ). p=1

10 20 N. C. Babalic, R. Constantinescu, V. S. Gerdjikov Due to the fact that χ ν (ξ, λ) has the special triangular asymptotics for ξ and λ Υ ν and for the correct choice of Θ ν (ξ ξ ) (5.3) we check that the right hand side of (5.8) falls off exponentially for ξ and arbitrary choice of ξ. All other possibilities are treated analogously. 3. For λ b ν the arguments of 2) can not be applied because the exponentials in the right hand side of (5.8) Im λ = 0 only oscillate. Thus we conclude that R ν (ξ, ξ, λ) for λ b ν is only a bounded function and thus the corresponding operator R(λ) is an unbounded integral operator. 4. The proof of eq. (5.5) follows from the fact that L(λ)χ ν (ξ, λ) = 0 and (5.9) Θ(ξ ξ ) ξ = 11δ(ξ ξ ), which concludes the proof. Lemma 5.2. The poles of R ν (ξ, ξ, λ) coincide with the poles of the dressing factors u(ξ, λ) and its inverse u 1 (ξ, λ). Proof. The proof follows immediately from the definition of R ν (ξ, ξ, 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 6N N 2 discrete eigenvalues; for N 1 = N 2 = 1 they are shown on Figure 1. 6 Conclusions We have constructed the FAS of L which satisfy a RHP on the set of rays l ν. We also constructed the resolvent of the Lax operator and proved that its continuous spectrum fills up the rays b ν rather than l ν. From Figure 1 we see that the eigenvalues corresponding to the solitons of first type lay on the continuous spectrum of L. This explains why the solitons of first type are singular functions. Using the explicit form of the resolvent R ν (ξ, ξ, 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 operator L, but with different M- operators; usually they are taken to be polynomial in λ. So in deriving their soliton solutions we will need to change only the η-dependence of the vectors m k. Similarly one can construct the N-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 the N-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

11 On Tzitzeica equation and spectral properties of related Lax operators 21 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 , Romanian Ministry of Education. References [1] N. C. Babalic, R. Constantinescu, V. S. Gerdjikov, On the properties of the soliton solutions of Tzitzeica equation, Vol. 24, p.607, paper presented in TIM 13 conference (unpublished). [2] R. Beals and R. Coifman, Inverse scattering and evolution equations, Commun. Pure & Appl. Math. 38 (1985), [3] P. Caudrey, The inverse problem for the third order equation u xxx + q(x)u x + r(x)u = iζ 3 u, Phys. Lett. A 79A (1980) ; P. Caudrey, The inverse problem for a general n n spectral equation, Physica D 6 (1982), [4] R. K. Dodd, R. K. Bullough, Polynomial conserved densities for the sine-gordon equations, Proc. Roy. Soc. London Ser. A, 352 (1977), [5] V. S. Gerdjikov, Z N reductions and new integrable versions of derivative nonlinear Schrödinger equations, In: Nonlinear evolution equations: integrability and spectral methods (Eds.: A. P. Fordy, A. Degasperis, M. Lakshmanan), Manchester University Press 1981, [6] V. S. Gerdjikov, Derivative nonlinear Schrödinger equations with Z N and D N reductions, Romanian Journal of Physics, 58, 5-6 (2013), [7] V. S. Gerdjikov, Algebraic and analytic aspects of n-wave type equations, Contemporary Mathematics, nlin.si/ , 301 (2002), [8] V. S. Gerdjikov and A. Yanovski, Completeness of the eigenfunctions for the Caudrey-Beals-Coifman system, J. Math. Phys. 35 (1994), [9] V. S. Gerdjikov, A. B. Yanovski, On soliton equations with Z h and D h reductions: conservation laws and generating operators, J. Geom. Symmetry Phys. 31 (2013), [10] A. V. Mikhailov, Integrability of a two-dimensional generalization of the Toda chain, Pis ma Zh. Eksp. Teor. Fiz. 30 (1979), [11] A. V. Mikhailov, The reduction problem and the inverse scattering method, Physica D 3 (1981), [12] R. Hirota and D. Takahashi, Ultradiscretization of the Tzitzeica equation, Glasgow Mathematical Journal 47 Issue A (2005), 77 85; DOI: /S [13] O. V. Kaptsov, Yu. V. Shanko. Trilinear representation and the Moutard transformation for the Tzitzeica equation, arxiv:solv-int/ v1 21 Apr [14] A. Shabat, The inverse scattering problem for a system of differential equations (in Russian), Functional Annal. & Appl. 9 (1975), A. Shabat, The inverse scattering problem (in Russian), Diff. Equations 15 (1979), [15] G. Tzitzeica, Sur une nouvelle classe de surfaces, C. R. Acad. Sc., 150 (1910), [16] G. Tzitzeica, Sur une nouvelle classe de surfaces, C. R. Acad. Sc., 150 (1910),

12 22 N. C. Babalic, R. Constantinescu, V. S. Gerdjikov [17] C. Udriste, V. Arsinte, C. Cipu, Tzitzeica and sine-gordon solitons, Balkan J. Geom. Appl. 16, 1 (2011), [18] C. Udriste, N. Bila, Symmetry group of Titeica surfaces PDE, Balkan J. Geom. Appl. 4, 2 (1999), [19] T. Valchev, On the Kaup-Kupershmidt equation. Completeness relations for the squared solutions, (Eds: I. Mladenov and M. de Leon), Softex, Sofia, 2008, [20] R. Willox, On a generalized Tzitzeica equation, Glasgow Math. Journal 47, Issue A (2005), ; DOI: /S [21] V. Zakharov, A. B. Shabat, A scheme for integrating nonlinear evolution equations of Mathematical Physics by the inverse scattering method I, Funkts. Anal. Prilozhen 8 (1974), [22] A. V. Zhiber, A. B. Shabat, Klein-Gordon equations with a nontrivial group, Soviet Physics Dokl. 24, 8 (1979), ; A. V. Zhiber and A. B. Shabat, The Klein-Gordon equation with nontrivial group (in Russian), Dokl. Akad. Nauk SSSR 247, 4-6 (1979), Authors addresses: Nicoleta Corina Babalic Department of Physics, University of Craiova, 13 A.I. Cuza Str., RO , Craiova, Romania; Dept. of Theoretical Physics, National Institute of Physics and Nuclear Engineering, 407 Atomistilor Str., RO , Magurele, Bucharest, Romania. b coryna@yahoo.com Radu Constantinescu Department of Physics, University of Craiova, 13 A.I. Cuza Str., RO , Craiova, Romania. rconsta@yahoo.com Vladimir Stefanov Gerdjikov Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences 72 Tzarigradsko Str., 1784 Sofia, Bulgaria. gerjikov@inrne.bas.bg