On the soliton solutions of a family of Tzitzeica equations
Corina N. Babalic, Radu Constantinescu, Vladimir S. Gerdjikov
aa r X i v : . [ n li n . S I] M a r JGSP (2012) 2–24 ON THE SOLITON SOLUTIONS OF A FAMILY OF TZITZEICAEQUATIONS
CORINA N. BABALIC , , RADU CONSTANTINESCU AND VLADIMIR S.GERDJIKOV Communicated by Metin Gürses
Abstract.
We analyze several types of soliton solutions to a family of Tzitzeicaequations. To this end we use two methods for deriving the soliton solutions: thedressing method and Hirota method. The dressing method allows us to derive twotypes of soliton solutions. The first type corresponds to a set of 6 symmetricallysituated discrete eigenvalues of the Lax operator L ; to each soliton of the secondtype one relates a set of 12 discrete eigenvalues of L . We also outline how onecan construct general N soliton solution containing N solitons of first type and N solitons of second type, N = N + N . The possible singularities of the solitonsand the effects of change of variables that relate the different members of Tzitzeicafamily equations are briefly discussed. All equations allow quasi-regular as well assingular soliton solutions. MSC:
Key words:
Tzitzeica equations, singular soliton solutions, Zakharov-Shabat dress-ing method, Hirota method
Contents N -soliton Solution for ¸T2 Equation 165 Hirota Method for Building 1-soliton Solution of ¸T2 Equation 17
1. Introduction
In the present paper we continue our investigations of the famous equation due tothe Romanian mathematician Gheorghe Tzitzeica , which we call now as Tzitzeica1 equation [27,28] and a closely related equation which we call Tzitzeica 2; in whatfollows we will denote them by ¸T1 and ¸T2. It was initially proposed as an equationdescribing special surfaces in differential geometry for which the ratio K/d isconstant, where K is the Gauss curvature of the surface and d is the distance fromthe origin to the tangent plane at the given point. Later on it turned out that theequation has wider importance, being nowadays used as an important evolutionaryequation in nonlinear dynamics.The explicit form of ¸T1 and ¸T2 equations is ∂ φ ∂ξ∂η = e φ − e − φ ∂ φ ∂ξ∂η = − (e φ − e − φ ) (1)i.e. ¸T1 and ¸T2 have different signs in the right hand sides. The transition betweenT1 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 forsome time it was believed, that it is the only known equation, allowing a finitenumber of higher integrals of motion [8]. Soon however, it was proved that infact, it possesses, like the other soliton equations, an infinite number of integralsof motion [30]. Next it was discovered that the equation has a hidden Z sym-metry, which becomes evident in its Lax representation [21, 22]. This importantdiscovery led Mikhailov to the notion of the reduction group and to the familyof 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 ofthe simple Lie algebras [9, 19, 23, 24]; ii) other classes of integrable NLEE mayalso possess such symmetries [7,9,12,13]; and iii) the expansions over the squaredsolutions and the theory of their recursion operators can be constructed [15,18,29].In previous papers [4, 5] we presented in the derivation of the soliton solutionsof ¸T1. Both versions of Tzitzeica equation allow Lax representation proposed by The name of the famous Romanian mathematician contains the Romanian letter ¸T, which maybe spelled as Tz. The factor 2 in eq. (1) can be easily removed, but is kept for historical reasons.
Mikhailov [21, 22]. This allows one to apply the dressing method of Zakharov-Shabat-Mikhailov [21, 32, 33] for calculating their soliton solutions. In fact allthese equations are particular examples of 2-dimensional Toda field theories (TFT)[9,19,21,23,24]. They all can be solved exactly using the inverse scattering method[10, 16, 31].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 ∂ φ∂ξ∂η = ǫ c e φ + ǫ c e − φ (2)where ǫ = ǫ = 1 and c and c are some positive real constants. Obviouslyequation ¸T1 (resp. equation ¸T2) is obtained from (2) by putting ǫ = 1 , ǫ = − , c = c = 1 (resp. ǫ = − , ǫ = 1 , c = c = 1 ). We will call ¸T3 and ¸T4 theequations ∂ φ ∂ξ∂η = − e φ − e − φ ∂ φ ∂ξ∂η = e φ + e − φ . (3)which follow from (2) with ǫ = ǫ = − , c = c = 1 and ǫ = ǫ = 1 , c = c = 1 respectively.The paper is organized as follows. In Section 2 we study a class of changes of vari-ables that interrelate different members of Tzitzeica family. We shall see that ¸T1 –¸T4 equations allow Lax representations so they can be solved exactly by the inversescattering method, [6,22]. In Section 3 the Zakharov-Shabat dressing method [33],adapted to systems with deep reductions, [21, 22] is used to construct their solitonsolutions. As a result we derive the soliton solutions of first and second types andanalyze their singularities. Indeed, we find that even the simplest one-soliton so-lutions of first type may have an infinite number of singularities for finite valuesof ξ, η . 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, seee.g. [11, 20, 25] and the references therein. At the same time, using an appropriatechange of variables we obtain a solution having singularities at only two pointswhich we call ‘quasi-regular’. In Section 4 we outline how the dressing formalismcan be extended to derive the N -soliton solution of the considered model with N solitons of first type and N solitons of second type, N = N + N . In Section5 we demonstrate how Hirota method can be applied for deriving the soliton so-lutions of Tzitzeica eqs. and show that it results compatible with the ones of thedressing method. In Section 6 we briefly outline the spectral properties of the Laxoperators L . We demonstrate that the resolvent of L has pole singularities that co-incide with the poles of the dressing factor and its inverse. We end by a discussionand conclusions.
2. Lorentz (Anti-)Invariance in 2-Dimensions
Obviously each of the TFT mentioned above can be viewed as a member of ahierarchy of NLEE which can be solved by applying the ISM to the correspondingLax operator. However the Lorentz invariance singles out the TFT models fromall the other members of NLEE in the hierarchy. Indeed, the TFT models allowchanges of variables which may drastically change, as we shall demonstrate below,the properties of the soliton solutions.
Let us now consider how simple linear change of variables ~Y ′ = A~Y , ~Y ′ = (cid:18) ξ ′ η ′ (cid:19) , ~Y = (cid:18) ξη (cid:19) , A = (cid:18) a bc d (cid:19) (4)affect the solutions of Tzitzeica eqs. Obviously this transformations have to pre-serve, up to a sign, ∂ ∂ξ∂η which means that A T σ A = ± σ , σ = (cid:18) (cid:19) (5)which is equivalent to the relations: ac = bd = 0 , ad + bc = ± . (6)These relations are satisfied in two cases1) A ± = (cid:18) a ± /a (cid:19) , A ± = (cid:18) b ± /b (cid:19) . (7)Here a and b can be, in general, arbitrary complex numbers. However, below wewill consider two cases: i) a and b – real and ii) a and b – purely imaginary.Second class of transformations involves shifts of the field φφ ( ξ, η ) → φ ′ ( ξ, η ) = φ ( ξ, η ) − ln c + s π ß2 (8)where c > is a real constant and s takes the values and . If s = 0 and c = c then ¸T1 goes into ∂ φ ′ ∂ξ∂η = c e φ ′ − c − e − φ ′ (9) T1 T2 T3 T4 A +1 , , s = 0 T1 T2 T3 T4 A − , , s = 0 T2 T1 T4 T3 A +1 , , s = 1 T3 T4 T1 T2 A − , , s = 1 T4 T3 T2 T1
Table 1.
Changes of variables that relate different members of Tzitzeicafamily equations. and similar expression for the ¸T2 equation for φ , but with opposite signs for theterms in the right hand side.If we now choose s = 1 and c = c then ¸T1 goes into ∂ φ ′ ∂ξ∂η = − c e φ ′ − c − e − φ ′ (10)which for c = 1 coincides with ¸T3 equation. We have listed the results of severalsuch transformations in Table 1. 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 solitonsolutions of only one of them, say the second equation in (1) ¸T2. It admits thefollowing Lax representation L Ψ( ξ, η, λ ) ≡ ß ∂ Ψ( ξ, η, λ ) ∂ξ + 2ß φ ξ H Ψ( ξ, η, λ ) + λ J Ψ( ξ, η, λ ) = 0 L Ψ( ξ, η, λ ) ≡ ß ∂ Ψ( ξ, η, λ ) ∂η + λ − V Ψ( ξ, η, λ ) = 0 (11)where H = − , J = V ( ξ, η ) = − φ e φ φ . (12)The reductions of the Lax pair for ¸T2 equation are similar but not the same as forthe well known ¸T1 eq. [4]:1. Z -reduction Q − Ψ( ξ, η, λ ) Q = Ψ( ξ, η, qλ ) , Q = q
00 0 q , q = e π ß / (13) which restricts H , J and V by Q − H Q = H , Q − J Q = q J , Q − V Q = q − V . (14)These conditions are satisfied identically.2. First Z -reduction Ψ ∗ ( ξ, η, − λ ∗ ) = Ψ( ξ, η, λ ) (15)i.e. H = H ∗ , J = J ∗ , V = V ∗ (16)which means that φ = φ ∗ .3. Second Z -reduction A − Ψ † ( ξ, η, λ ∗ ) A = Ψ − ( ξ, η, λ ) , A = (17)i.e. A − H † A = − H , A − J † A = J , A − V † A = V . (18)
3. The Dressing Method and Dressing Factors for ¸T2 Equation
Let us start with a Lax representation of the form L Ψ ≡ ß ∂ Ψ ∂ξ + λ J Ψ = 0 L Ψ ≡ ß ∂ Ψ ∂η + λ − J Ψ = 0 . (19)The fundamental solution Ψ ( ξ, η, λ ) , known also as the ‘naked’ solution, has aspotential the trivial solution of ¸T2 equation: φ ( ξ, η ) = 0 The ’dressed’ Lax pair, given by (11), admits the "dressed" fundamental solution Ψ( ξ, η, λ ) , with the potential the nontrivial solution φ ( ξ, η ) .The fundamental solutions Ψ and Ψ are related by the dressing factor u ( ξ, η, λ ) , Ψ( ξ, η, λ ) = u ( ξ, η, λ )Ψ ( ξ, η, λ ) u − ( λ ) , u + ( λ ) = lim ξ →∞ u ( ξ, η, λ ) (20)which means that u ( ξ, η, λ ) must satisfy ß ∂u∂ξ + 2ß φ ξ H u ( ξ, η, λ ) + λ [ J , u ( ξ, η, λ )] = 0ß ∂u∂η + 1 λ V u ( ξ, η, λ ) − λ u ( ξ, η, λ ) J = 0 . (21) Since both Lax pairs (the dressed (11) and the naked one (19)) satisfy the threereductions, then also the dressing factor must satisfy thema) Q − u ( ξ, η, λ ) Q = u ( ξ, η, qλ ) , b) u ∗ ( ξ, η, − λ ∗ ) = u ( ξ, η, λ ) c) A − u † ( ξ, η, λ ∗ ) A = u − ( ξ, η, λ ) (22)where A is defined by eq. (17). A natural anzatz for the dressing factor with simple poles in λ is [21] u ( ξ, η, λ ) = + 13 (cid:18) A λ − λ + Q − A Qλq − λ + Q − A Q λq − λ (cid:19) (23)where A ( ξ, η ) is a × degenerate matrix of the form A ( ξ, η ) = | n ( ξ, η ) ih m T ( ξ, η ) | ( A ) ij ( ξ, η ) = n i ( ξ, η ) m j ( ξ, η ) . (24)The first reduction (22a) on u ( x, t, λ ) is automatically satisfied by the anzatz (23).The second reduction (22b) leads to η j − k n k m j λ − λ ∗ , = − ρ j − k n ∗ k m ∗ j λ + λ ∗ , . (25)Here and below j − k is understood modulo and η = λ , η = λλ , η = λ ρ = λ ∗ , , ρ = − λλ ∗ , ρ = λ . (26)In addition we must have λ ∗ , = − λ and λ ∗ , n ∗ i m ∗ i = − λ n i m i , λ ∗ n ∗ i m ∗ i +1 = λ n i m i +1 , n ∗ i m ∗ i +2 = − n i m i +2 (27)where again all matrix indices are understood modulo 3. These relations can berewritten as arg n i + arg m i = π − λ , arg n i + arg m i +1 = − arg λ arg n i + arg m i +2 = π , arg λ = (2 k + 1) π , k = 0 , , ..., . (28) So we can consider with no limitations that λ = − λ ∗ and A = − A ∗ . Morespecifically we will assume that the vector h m T ( ξ, η ) | is real, while the vector | n ( ξ, η ) i has purely imaginary components.The third reduction (22c) on u ( x, t, λ ) can be put in the form u ( ξ, η, λ ) A − u † ( ξ, η, λ ∗ ) A = . (29)Let us now multiply (29) by λ − λ , take the limit λ → λ and take into accounteq. (14). This gives m k = n ∗ − k λ − λ ∗ ,
31 3 X k =1 κ s − k m s m ∗ − s (30)where κ = λ ∗ , , κ = λ , κ = λ λ ∗ . (31)Thus, taking into account that λ = ß ρ , ρ – real and m k = m ∗ k , we obtain n = 2 λ m ∗ λ m ∗ m + | λ | | m | + λ , ∗ m ∗ m = 2ß ρ m m m − m n = 2 λ m ∗ λ , ∗ m ∗ m + λ | m | + | λ | m ∗ m = 2ß ρ m n = 2 λ m ∗ | λ | m ∗ m + λ , ∗ | m | + λ m ∗ m = 2ß ρ m m . (32)In order to obtain the vectors | n i and h m T | in terms of ξ and η we first impose thelimit λ → λ in equation (21). We obtain that the residue A must satisfy ß ∂A ∂ξ + 2ß φ ξ H A + λ [ J , A ] = 0ß ∂A ∂η + λ − V A − λ − A J = 0 . (33)Since A = | n ih m T | we find that (33) is satisfied if ß ∂ | n i ∂ξ + 2ß φ ξ H | n i + λ J | n i = 0 , ß ∂ h m T | ∂ξ − λ h m T |J = 0ß ∂ | n i ∂η + λ − V | n i = 0 , ß ∂ h m T | ∂η − λ − h m T |J = 0 (34) i.e. | n i = Ψ( ξ, η, ß ρ ) | n i , h m T | = h m T | (Ψ ) − ( ξ, η, ß ρ ) (35)which means that | n ( ξ, η ) i is an eigenfunction for the "dressed" Lax pair L , L ,while h m T ( ξ, η ) | is an eigenfunction for the "naked" Lax pair L , L .From (19), using direct calculation we obtain Ψ ( ξ, η, λ ) = f e ß λJξ +ß λ − J η f − f = 1 √ q q q q , f − = 1 √ q q q q , J = diag ( q , , q ) (36)which means that h m T | = h m T | f e ρ Jξ − ρ − J η f − . (37)Using the notations h m T | √ f = ( µ , µ , µ ) (38)we obtain the following explicit forms for the components of vector h m T ( ξ, η ) | m ( ξ, η ) = q µ e −X e − ßΩ + µ e X + qµ e −X e ßΩ m ( ξ, η ) = µ e −X e − ßΩ + µ e X + µ e −X e ßΩ m ( ξ, η ) = qµ e −X e − ßΩ + µ e X + q µ e −X e ßΩ (39)where X = 12 (cid:18) ρ ξ − ηρ (cid:19) , Ω = √ (cid:18) ρ ξ + ηρ (cid:19) . (40)For µ , = µ ∗ , = | µ | e ß α and µ , = µ ∗ , we can rewrite m i from (39) as thefollowing real-valued functions m ( ξ, η ) = µ e X + 2 | µ | e −X cos (cid:18) Ω − α + 2 π (cid:19) m ( ξ, η ) = µ e X + 2 | µ | e −X cos (Ω − α ) m ( ξ, η ) = µ e X + 2 | µ | e −X cos (cid:18) Ω − α − π (cid:19) . (41) The components of the vector | n i in (32) become n = 2ß ρ m m m − m , n = 2ß ρ m , n = 2ß ρ m m . (42)In order to obtain the solution of ¸T2 equation we impose the limit λ → in (21)with the result: φ ξ − = − ∂u∂ξ u − ( ξ, η, (43)where u ( ξ, η,
0) = − λ ( A + Q − A Q + Q − A Q )= (cid:18) − λ A ,jk (cid:19) δ jk (44)which means that φ ξ = − ∂u ∂ξ u = − ∂∂ξ ln u (45)or φ ( ξ, η ) = − ln (cid:12)(cid:12)(cid:12)(cid:12) − n m λ (cid:12)(cid:12)(cid:12)(cid:12) = ln (cid:12)(cid:12)(cid:12)(cid:12) m m − m m (cid:12)(cid:12)(cid:12)(cid:12) . (46)After introducing m i from (41) we obtain the 1-soliton solution of the first type for λ = ß ρ φ s ( ξ, η ) = 12 ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | µ | e − X (cid:16) ( ˜Ω ) − (cid:17) − | µ | µ cos( ˜Ω ) + µ e X | µ | e − X cos ( ˜Ω ) + 4 | µ | µ cos( ˜Ω ) + µ e X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (47)where ˜Ω = Ω − α . We observe that this is not a traveling wave solution. Onlyif we take the limit µ → we obtain a traveling wave solution of the form φ ( ξ, η ) = ß π "
32 tan √
32 ( ρ ξ + ρ − η ) − α ! + 12 . (48) The solution is singular and it blows up for √ ( ρ ξ + ρ − η ) − α = (2 k + 1) π/ , k = 0 , ± , . . . . For α → α + π ( m , m , m ∈ C and they are purelyimaginary), the solution (48) becomes φ ( ξ, η ) = ß π "
32 cot √
32 ( ρ ξ + ρ − η ) − α ! + 12 . (49)The above solution is also singular and it blows up for √ ( ρ ξ + ρ − η ) + α = kπ , k = 0 , ± , . . . . Remark 1
It is easy to check, that the real parts of φ ( ξ, η ) in eqs. (48) and (49)are in fact solutions to ¸T4 equation. In order to get ‘quasi-regular’ solutions of ¸T2 equation, we can apply the changesof variables A +1 with a = ß or A +2 with b = ß . This gives the following solutionsexpressed in terms of hyperbolic functions φ ( ξ, η ) = 12 ln "
32 tanh √
32 ( ρ ξ − ρ − η ) − α ! − (50)and φ ( ξ, η ) = 12 ln "
32 coth √
32 ( ρ ξ − ρ − η ) − α ! − (51)which are singular at the points for which tanh √
32 ( ρ ξ − ρ − η ) − α ! = ± √ or coth √
32 ( ρ ξ − ρ − η ) − α ! = ± √ respectively. These solutions have also been found by Mikhailov in [21]. As com-pared 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 tocall them ‘quasi-regular’. Here we will discuss the types of singularities of the one-soliton solutions and howthey are influenced by the changes of variables. As we already mentioned, thesingularities 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 particularlyinterested in the properties of the ‘naked’ Lax pair and its fundamental solution Ψ ( ξ, η, λ ) . This comes from the fact, that the soliton solution is constructed as arational function of the elements of Ψ ( ξ, η, λ ) .Let us start with the change of variables A +1 . Here the situation is simple; wereadily get L ( λ ) → a L ( λ/a ) L ( λ ) → aL ( aλ )Ψ ( ξ ′ , η ′ , λ ′ ) → Ψ (cid:18) aξ, ηa , λa (cid:19) . (52)In other words this change of variables leaves invariant the compatibility of theLax pair, so obviously it will map a solution of ¸T2 into a solution of ¸T2. Howevernow we have to keep in mind, that the change of variables must be extended also tothe spectral parameter λ → λ/a and, of course to the discrete eigenvalues of L , : λ → λ /a and therefore ρ → ρ /a .In particular, from eq. (40) we see, that X ′ ( ξ ′ , η ′ , λ ′ ) = 12 (cid:18) λ ′ ξ ′ + η ′ λ ′ (cid:19) = X ( ξ, η, λ )Ω ′ ( ξ ′ , η ′ , λ ′ ) = 12 (cid:18) λ ′ ξ ′ + η ′ λ ′ (cid:19) = Ω ( ξ, η, λ ) (53)i.e. X and Ω are invariant with respect to A +1 transformations provided λ ′ = λ a . (54)Now it is a bit more interesting to analyze the changes A +2 . L ′′ ( λ ) ≡ b (cid:18) ß ∂∂η ′′ + 2ß φ η ′′ H + λ J (cid:19) Ψ( ξ ′′ , η ′′ , λ ) = 0 L ′′ ( λ ) ≡ b (cid:18) ß ∂∂ξ ′′ + 1 λb V ( ξ ′′ , η ′′ ) (cid:19) Ψ( ξ ′′ , η ′′ , λ ) = 0 . (55) Let us apply a gauge transformation, i.e. change from Ψ( ξ ′′ , η ′′ , λ ) to ˜Ψ( ξ ′′ , η ′′ , λ ) A e φH Ψ( ξ ′′ , η ′′ , λ ′′ ) (56)where H and A are defined in eqs. (16) and (17) respectively. This gives us L ′′ ( λ ′′ ) = L ( λ ) , L ′′ ( λ ′′ ) = L ( λ ) , λ ′′ = 1 bλ . (57)So the A +2 change is equivalent to interchanging the Lax operators L and L ,which again preserves their compatibility condition. Applied to X and Φ thesetransformations lead to Ψ ′′ ( ξ ′ , η ′ , λ ′′ ) = A Ψ ( ξ, η, λ ) A . (58)Of course, analyzing the fundamental solutions we have to pay attention alsowhether the parameters a and b are real or purely imaginary. In addition we have totake into account, that λ could be purely imaginary as above, but for other casesit could also be real. It is precisely this choice of the parameters a , b and λ thatmay change the singularity properties of the solutions. Our anzatz for the dressing factor is u ( ξ, η, λ ) = + 13 (cid:18) A λ − λ + Q − A Qλq − λ + Q − A Q λq − λ (cid:19) − (cid:18) A ∗ λ + λ ∗ + Q − A ∗ Qλq + λ ∗ + Q − A ∗ Q λq + λ ∗ (cid:19) (59)which obviously satisfies the Z -reduction and the first Z -reduction.In order to find how the components of the vector | n i are expressed in terms of thevector | m T i we use the same procedure as in the -poles case. First we rewrite thedressing factor in the following form. u ( ξ, η, λ ) = + 1 λ − λ A ( ξ, η, λ ) − λ + λ , ∗ A ∗ ( ξ, η, − λ ∗ ) (60)where A ( ξ, η, λ ) = η n m η n m η n m η n m η n m η n m η n m η n m η n m A ∗ ( ξ, η, − λ ∗ ) = ρ n ∗ m ∗ ρ n ∗ m ∗ ρ n ∗ m ∗ ρ n ∗ m ∗ ρ n ∗ m ∗ ρ n ∗ m ∗ ρ n ∗ m ∗ ρ n ∗ m ∗ ρ n ∗ m ∗ (61) with η = λ , η = λλ , η = λ ρ = λ ∗ , , ρ = − λλ ∗ , ρ = λ . (62)We insert the dressing factor u ( ξ, η, λ ) into The second Z -reduction, we multiplyby λ − λ , take the limit λ → λ and obtain h m T | A = h m T | A " − λ − λ ∗ , A † ( λ ∗ ) + 12 λ A T ( − λ ) (63)After direct calculation we obtain m = ζ K n ∗ + c P n m = ζ K n ∗ + c P n m = ζ K n ∗ + c P n (64)where K = λ ∗ , m m ∗ + λ λ ∗ m m ∗ + λ m m ∗ , P = 2 m m − m K = λ m m ∗ + λ ∗ , m m ∗ + λ λ ∗ m m ∗ , P = m K = λ λ ∗ m m ∗ + λ m m ∗ + λ ∗ , m m ∗ , P = m ζ = − λ − λ ∗ , , c = 12 λ . (65)We rewrite the result in a matrix form | µ i = m m m m ∗ m ∗ m ∗ , | ν i = n n n n ∗ n ∗ n ∗ , | µ i = M| ν i (66)where M = c P ζ K c P ζ K
00 0 c P ζ K ζ K ∗ c P ∗ ζ K ∗ c P ∗
00 0 ζ K ∗ c P ∗ (67) The result is | ν i = M − | ν iM − = − c ∗ ˜ P ∗ ζ ˜ K − c ∗ ˜ P ∗ ζ ˜ K
00 0 − c ∗ ˜ P ∗ ζ ˜ K ζ ∗ ˜ K ∗ − c ˜ P ζ ∗ ˜ K ∗ − c ˜ P
00 0 ζ ∗ ˜ K ∗ − c ˜ P (68)where ˜ P ∗ s = P ∗ s d s , ˜ P s = P s d s , ˜ K s = K s d s , ˜ K ∗ s = K ∗ s d d = ζ ζ ∗ K K ∗ − c c ∗ P P ∗ d = ζ ζ ∗ K K ∗ − c c ∗ P P ∗ d = ζ ζ ∗ K K ∗ − c c ∗ P P ∗ . (69)From the above equations we obtain | n i in terms of h m T | n = 1 d ( − c ∗ P ∗ m + ζ K m ∗ ) n = 1 d ( − c ∗ P ∗ m + ζ K m ∗ ) n = 1 d ( − c ∗ P ∗ m + ζ K m ∗ ) . (70)In this case we choose a general form for the poles: λ = ρ e ß β ; without restric-tions we can choose < β < π and determine the expressions of h m T | as m = q µ e ß X −Y + µ e ß X −Y + qµ e ß X −Y m = µ e ß X −Y + µ e ß X −Y + µ e ß X −Y m = qµ e ß X −Y + µ e ß X −Y + q µ e ß X −Y (71)where X = − (cid:18) ξρ + ηρ (cid:19) cos (cid:18) β − π (cid:19) , Y = − (cid:18) ξρ − ηρ (cid:19) sin (cid:18) β − π (cid:19) X = − (cid:18) ξρ + ηρ (cid:19) cos ( β ) , Y = − (cid:18) ξρ − ηρ (cid:19) sin ( β ) X = − (cid:18) ξρ + ηρ (cid:19) cos (cid:18) β + 2 π (cid:19) , Y = − (cid:18) ξρ − ηρ (cid:19) sin (cid:18) β + 2 π (cid:19) . (72)We determine the 1-soliton solution for the second kind of solitons using exactlythe same technique Φ = −
12 ln (cid:12)(cid:12)(cid:12)(cid:12) − λ n m − λ ∗ n ∗ m ∗ (cid:12)(cid:12)(cid:12)(cid:12) . (73)
4. The Generic N -soliton Solution for ¸T2 Equation Let us consider the dressing factor of the following form u ( ξ, η, λ ) = + 13 X s =0 N X l =1 Q − s A l Q s λq s − λ l + N X r = N +1 (cid:18) Q − s A r Q s q s λ − λ r − Q − s A ∗ r Q s λq s + λ ∗ r (cid:19) (74)with N + 6 N complex poles from which N are purely imaginary, satisfyingthe following condition: λ p = − λ ∗ p .Then we write down the residues A k ( ξ, η ) as degenerate × matrices of the form A k ( ξ, η ) = | n k ( ξ, η ) ih m Tk ( ξ, η ) | , ( A k ) ij ( ξ, η ) = n ki ( ξ, η ) m kj ( ξ, η ) . (75)From the second Z -reduction, A − u † ( ξ, η, λ ∗ ) A = u − ( ξ, η, λ ) , after taking thelimit λ → λ k , we obtain algebraic equation for | n k i in terms of h m Tk || ν i = M − | µ i . (76)Below, for simplicity, we write down the matrix M for N = N = 1 | ν i = | n i| n i| n ∗ i , | µ i = A | m i A | m i A | m ∗ i , M = A B B ∗ B D E − B ∗ − E ∗ D ∗ (77) A = 12 λ diag ( Q (1) , Q (2) , Q (3) ) , B = 1 λ + λ diag ( P (1) , P (2) , P (3) ) D = 12 λ diag ( T (1) , T (2) , T (3) ) , E = 1 λ ∗ , − λ diag ( K (1) , K (2) , K (3) ) Q ( j ) = h m T | Λ ( j )11 ( λ , λ ) | m i , P ( j ) = h m T | Λ ( j )12 ( λ , λ ) | m i T ( j ) = h m T | Λ ( j )22 ( λ , λ ) | m i , K ( j ) = h m T | Λ ( j )22 ( λ , − λ ∗ ) | m ∗ i (78)with Λ ( j ) lp = − λ l λ p E − j, j + λ l E − j, j + λ p E − j, j , j = 1 , , . (79)In order to obtain the 2-soliton solution of the Tzitzeica equation we take the limit λ → in the equations satisfied by the dressing factor u ( ξ, η, λ ) and integrate toget φ Ns ( ξ, η ) = −
12 ln (cid:12)(cid:12)(cid:12)(cid:12) − n , m , λ − n , m , λ − n ∗ , m ∗ , λ ∗ (cid:12)(cid:12)(cid:12)(cid:12) . (80)The above formulae can be easily generalized for any N and N .
5. Hirota Method for Building 1-soliton Solution of ¸T2 Equation
There are many methods for deriving the soliton solutions; we have demonstratedtwo of the most used: the dressing method and the Hirota method [3, 6, 17]. Bothmethods give the same results both for the kinks and for the breathers.We build the Hirota bilinear form of ¸T2 eq. using the substitution φ ( ξ, η ) = 12 ln g ( ξ, η ) f ( ξ, η ) . (81)Introducing it into the second equation in (1) and decoupling in the bilinear disper-sion relation and the soliton-phase constraint, we obtain the following system ∂ g∂ξ∂η g − ∂g∂ξ ∂g∂η − f + g = 0 ∂ f∂ξ∂η g − ∂f∂ξ ∂f∂η − f g + f = 0 . (82)We impose that: g ( ξ, η ) = 1 + az ( ξ, η ) + bz ( ξ, η ) f ( ξ, η ) = 1 + Az ( ξ, η ) + Bz ( ξ, η ) (83)where z ( ξ, η ) = e kξ − ωη , k - the wave number, ω - the angular frequency.Using a software for analytical computation like MATHEMATICA, we obtain that g ( ξ, η ) = 1 − A e kξ − k η + A kξ − k η f ( ξ, η ) = 1 + A e kξ − k η + A kξ − k η (84)where the dispersion relation is ω = k .Using the above results our 1-soliton solution for ¸T2 acquires the following form φ ( ξ, η ) = 12 ln (cid:20)
32 tanh (cid:18)
12 ( kξ − k η ) (cid:19) − (cid:21) . (85)This solution coincide with the one obtained by Mikhailov in [21] for k = √ ρ .In this very direct manner, Hirota method gives immediately the 1-soliton solutionof 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 offirst type each parametrized with real eigenvalue ρ k and a vector ( µ k, , µ k, , µ k, ) with µ k, = 0 . We believe, that using Hirota method one can derive also morecomplicated cases of one- and N -soliton solutions. To this end, however one needsa more complicated ansatz for the functions f ( ξ, η ) and g ( ξ, η ) which would solveequation (82) but could not be reduced to functions of z ( ξ, η ) only.Of course, the equation (82) can be solved in a more general case, but the onlyone 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 ( ξ, η ) and f ( ξ, η ) corresponding to the second type of soliton solutions is still an openproblem for us and it will be tackled in a next paper. A possible approach couldbe 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 ( ξ, η ) and f ( ξ, η ) 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 fundamentalanalytic solutions. We will denote them by χ ν ( ξ, η, λ ) where ν = 0 , . . . , denotesthe number of the sector Ω ν ≡ (2 ν +1) π ≤ arg λ ≤ (2 ν +3) π , i.e. those are thesectors closed by the rays ( l ν , l ν +1 ) . The dressing factor for solitons of first type(23) obviously has simple poles located at | λ | e π ß k/ , k = 0 , , . The inverse ofthis dressing factor has also simple poles located at | λ | e π ß(2 k +1) / , k = 0 , , .Each dressing factor for soliton of second type (59) has 6 simple poles located at | λ | e ß β +2 π ß k/ and | λ | e − ß β + π ß(2 k +1) / , k = 0 , , . The inverse of this dressingfactor has also 6 simple poles located at | λ | e ß β + π ß(2 k +1) / and | λ | e − ß β +2 π ß k/ , k = 0 , , .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 χ ν ( ξ, λ ) = u ( ξ, λ ) χ ν ( ξ, λ ) u − − ( λ ) , u − ( λ ) = lim ξ →−∞ u ( ξ, η, λ ) (86)where χ ν ( ξ, λ ) is a regular FAS and u ( ξ, λ ) is a dressing factor of general form(74). Let us introduce R ν ( ξ, ξ ′ , λ ) = 1ß χ ν ( ξ, λ )Θ ν ( ξ − ξ ′ ) ˆ χ ν ( ξ ′ , λ ) (87) Θ ν ( ξ − ξ ′ ) = diag (cid:16) η (1) ν θ ( η (1) ν ( ξ − ξ ′ )) , η (2) ν θ ( η (2) ν ( ξ − ξ ′ )) , η (3) ν θ ( η (3) ν ( ξ − ξ ′ )) (cid:17) (88) λ × ⊗× ⊗ ×⊗ + ⊕ + ⊕ + ⊕ + ⊕ + ⊕ + ⊕ l l l l l l b b b b b b Figure 1.
The contour of the RHP with Z -symmetry fills up the rays l , . . . , l . The symbols × and ⊗ (resp. + and ⊕ ) denote the locations of the discreteeigenvalues corresponding to a soliton of first (resp. second) type. where θ ( ξ − ξ ′ ) is the step-function and η ( k ) ν = ± , see the table 2.Then the following theorem holds true [4]: Theorem 2
Let Q ( ξ ) be a Schwartz-type function and let λ ± j be the simple zeroesof the dressing factor u ( ξ, λ ) (74). Then1. The functions R ν ( ξ, ξ ′ , λ ) are analytic for λ ∈ Υ ν where b ν : arg λ = π ( ν + 1)3 , Υ ν : π ( ν + 1)3 ≤ arg λ ≤ π ( ν + 2)3 (89) 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 kernelof an unbounded integral operator; Υ Υ Υ Υ Υ Υ η (1) ν − − − + + + η (2) ν + + − − − + η (3) ν − + + + − − Table 2.
The set of signs η ( k ) ν for each of the sectors Υ ν (89). R ν ( ξ, ξ ′ , λ ) satisfy the equation L ( λ ) R ν ( ξ, ξ ′ , λ ) = δ ( ξ − ξ ′ ) . (90) Remark 3
The dressing factor u ( ξ, λ ) has N + 6 N simple poles located at λ l q p , λ r q p and λ ∗ r q p where l = 1 , . . . , N , r = 1 , . . . , N and p = 0 , , . Itsinverse u − ( ξ, λ ) has also N + 6 N poles located − λ l q p , − λ r q p and − λ ∗ r q p . Inwhat follows for brevity we will denote them by λ j , − λ j for j = 1 , . . . , N + 6 N . It remains to show that the poles of R ν ( ξ, ξ ′ , λ ) coincide with the poles of thedressing factors u ( ξ, λ ) and its inverse u − ( ξ, λ ) .The proof follows immediately from the definition of R ν ( ξ, ξ ′ , λ ) and from eq.(86), taking into account that the limiting value u − ( λ ) commutes with the corre-sponding matrix Θ ν ( ξ − ξ ′ ) .Thus we have established that dressing by the factor u ( ξ, λ ) , we in fact add tothe discrete spectrum of the Lax operator N + 12 N discrete eigenvalues; for N = N = 1 they are shown on Figure 1.
7. Conclusions
Shortly before finishing this paper we became aware of the fact, that appropriatecombination of changes of variables, considered in Section 2 can take each mem-ber of Tzitzeica family (2) into one of its 4 versions that we introduced. Let usdemonstrate how this can be done of the equation ∂ φ∂ξ∂η = − c e φ + c e − φ (91)where c and c are real positive constants. Now we shall use somewhat moregeneral change of variables. First we apply the transformation (8) with s = 0 and φ ′ = φ + ln( c /c ) . Then we change ξ → ξ ′ /k , η → η ′ /k where k is also realpositive constant taken to be k = p c c . Easy calculation shows that as a result eq. (91) goes into ¸T2 for φ ′ ( ξ, η ) . Using in addition Table 1 we can transform eachmember of Tzitzeica family into ¸T2 and then solve it using the results above.Let us consider the soliton solutions Tzitzeica eq. in a small neighborhood aroundthe singularities, where φ as ( ξ, η ) tends to ∞ . Then the second term in the ¸T2equation can be neglected and the asymptotically we get ∂φ as ∂ξ∂η = e φ as . Similarly if in a small neighborhood around the singularity φ ′ as ( ξ, η ) tends to −∞ we have ∂φ ′ as ∂ξ∂η = − e − φ ′ as . In both cases we find equations, equivalent to the Liouville equation. Thus theasymptotical behavior of the solutions of Tzitzeica equation around the singulari-ties 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 jointproject between the Bulgarian academy of sciences and the Romanian academyof sciences. One of the authors (CNB) acknowledges the support of the strategicgrant POSDRU/159/1.5/S/133255, Project ID 133255 (2014), co-financed by theEuropean Social Fund within the Sectorial Operational Program Human ResourcesDevelopment 2007-2013, and also the support of the project IDEI, PN-II-ID-PCE-2011-3-0083 (MECTS).
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E-mail address : [email protected] Vladimir S. GerdjikovInstitute for Nuclear Research and Nuclear EnergyBulgarian Academy of Sciences1784 Sofia, Bulgaria
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