Type-II B"acklund Transformations via Gauge Transformations

T yp e-I I B¨ acklund T ransfo rmations via Gauge T ransfo rmations A R Aguirre 1 , T R Araujo 2 , J F Gomes 3 and A H Zimerman 4 Instituto de F ´ ısica T e´ orica-UNESP Rua Dr. Ben to T eobaldo F erraz 271, Blo co I I, 01140-0 70, S˜ ao Paulo, Brazil Abstract The construction of type I I B¨ ac klund transform a tion for the sine-Gordon and the Tzitz ´ eica-Bullo u gh -Do dd mo dels are obtained from gauge transformation. An infinite n umb er of conserve d qu a ntitie s are constructed from the defect matrices. This guar- an tees that the introd uctio n of t yp e I I d efe cts for these mod e ls do es n o t sp oil their in tegrabilit y . In particular, mo d ified energy and m o mentum are derived and compared with th ose pr e sented in r ec ent literature. 1 In tro duct i o n In tegrable defects ha v e b ee n in tro duced some time ago [1] from the Lagrangian p oin t of view whic h, from the conserv ation of energy and momen tum, lead t o the deriv ation of B¨ ac klund transformations. Man y examples lik e the sine-Gordon, mKdV, nonlinear Schr¨ odinger, A (1) n - T o da equations, w ere explicitly dev elop ed [2], [3]. As a characteristic of these mo dels only ph ysical fields we re presen t within the formulation and the asso ciated B¨ ac klund transfor- mations we re called ty p e I [4]. The construction of t yp e-I B¨ ac klund transformatio n for the sine-Gordon and for the non-linear Sc hr¨ odinger mo dels thro ugh gauge transformatio ns we re also emplo y ed in [5], [6] resp ectiv ely . More recen tly in r ef. [4] an extension of the metho d emplo y ed in [2], [3] w as pro p osed by in tro ducing an auxiliary field and the asso ciated B¨ a cklund transformat ion w ere named type I I [4]. Examples of type II B¨ ack lund transformations w ere explicitly deriv ed in ref. [4] for the sine-Gordon and for the Tzitz ´ eica-Bullough-Do dd mo dels b y imposing conserv ation of the mo dified energy a nd momen tum by a defect. It is worth to note tha t for the sup ers ym- metric extensions of sine-Go rdon mo del [7],[8] and Thirring mo dels [9],[10] the in tro duction of auxiliary fields also app eared. The purp ose of the presen t in v estigation is to deriv e the ty p e I I B¨ acklund transformations via gauge tra ns format io ns . The c hoice of suc h ga ug e transforma t io ns naturally in tro duces the auxiliary fields. He re w e consider the sine-Gordon and Tzitz ´ eica-Bullough-D odd mo dels. 1 aleroa g u@ift.unesp.br 2 taraujo@ift.unesp.br 3 jfg@ift.unesp.br 4 zimerman@ift.unesp.br 1 In the sp ec ific case of the s ine-G o rdon mo de l, b y a particular limiting pro cedure, w e can reduce t he type I I B¨ ac klund transformation to the usual ty p e I. The problem w e address consists in relating tw o distinct solutions o f the linear problem Φ (1) and Φ (2) giv en b y ∂ + Φ ( k ) = − A ( k ) + Φ ( k ) , ∂ − Φ ( k ) = − A ( k ) − Φ ( k ) , k = 1 , 2 , (1.1) where A ( k ) ± are Lie alg e bra v alued functionals of the field ϕ k . In eq. (1.1) w e ha v e in tro duced the lig h t cone co ordinates x ± = 1 2 ( t ± x ) with deriv ativ es ∂ ± = ∂ t ± ∂ x , and ∂ + ∂ − = ∂ 2 t − ∂ 2 x . Under the gauge transformation ∂ ± K = K A (1) ± − A (2) ± K, (1.2) the linear problem (1.1) implies Φ (2) = K Φ (1) . This matrix K is commonly called defe c t matrix and induces relations b et we en tw o differen t field configurations ϕ 1 and ϕ 2 , whic h are exp ected to corresp ond t o the B¨ ac klund tra ns forma t io ns . Based up on the ga ug e tra ns for ma t io n ( 1.2), w e construct type I I B¨ ac klund transforma- tions connecting tw o field configurations for the explicit example s of the sine-Gordon and Tzitz ´ eica-Bullo ug h-Do dd mo dels. In section 2 w e construct gauge transformat io ns leading to ty p e I I B¨ ac klund tra nsfor ma - tions for the sine-Gordon mo del. Apart from rederiving the t yp e I I B¨ ac klund transformation of ref. [4] w e obta in a second solution and observ ed that, in fa c t it corresp onds to the sym- metry ϕ → − ϕ of the sine-Gordon equation. By a suitable limiting pro cedure , the system of t yp e I I B¨ ac klund equations can b e reduced to ty p e I. In section 3 we consider the Tzitz ´ eica- Bullough-Do dd mo del and sho w that the gauge transformatio n generates a set of equations that decouples into three subsets . One of whic h is solv ed and sho wn to correspond to the B¨ acklun d transformation deriv ed in [13]. A further c hange of v ariables sho ws that these equations reduce to those f ound in [4 ]. In subse ction 4 .1 w e emplo y the formalism of ref. [11] for t yp e I I defect matrix K to construct an infinite n umber of conserv ed quan tities for the sine-Gordon mo del in the presence of suc h defect. In particular, we giv e explicit expressions for the defec t energy and momentum. By analysing sp ec ific com binations of defect contri- butions for the conserv ed quantities w e made contact with the defect Lag rangian of ref. [4]. In subs ection 4 .2 w e discuss the conserv ation la ws for the Tzitz ´ eica-Bullough- Do dd mo del in v olving 3 × 3 matrices, a nd extend the formalism of ref. [11] to the Tzitz ´ eica-Bullough- Do dd mo del in the presence of type I I defect. W e also ha ve an infinite n umber of conserv ed quan tities f or this kind of defect. In particular, we obtain the defect energy and momen tum and compare with the defect Lagrangian of ref. [4]. 2 The sine-Gord on mo del F or the sine-Gordon mo del w e hav e A + = i 4 ∂ + ϕ m 2 λe iϕ 2 − m 2 λe − iϕ 2 − i 4 ∂ + ϕ ! , A − = − i 4 ∂ − ϕ m 2 λ − 1 e − iϕ 2 − m 2 λ − 1 e iϕ 2 i 4 ∂ − ϕ ! , (2.1) 2 and λ is t he sp ectral parameter. Cons ider a gauge tra ns forma t ion with the follo wing form for the matrix, K = α + λ − 1 β + λ − 2 γ . (2.2) Let ϕ 1 and ϕ 2 b e t wo distinct field configurat io n. By in tro ducing v a riables p = ϕ 1 + ϕ 2 2 , q = ϕ 1 − ϕ 2 2 , (2.3) w e find from (1 .2) and ( 2 .2) t ha t equations f o r the matrices α , β and γ can b e group ed in to t w o subsets: (i) The first one in v olve s α 11 , α 22 , β 12 , β 21 , γ 11 , γ 22 and leads to α 11 = a 11 e − i 2 q , α 22 = a 11 e i 2 q , γ 11 = c 11 e i 2 q , γ 22 = c 11 e − i 2 q . (2.4) P arametrizing β 21 in terms of an auxiliary field Λ, β 21 = b 21 e − i Λ e i 2 p , (2.5) w e find for the equations in volving ∂ + α 11 , ∂ − γ 11 , ∂ + β 21 and ∂ − β 21 resp e ctive ly , i∂ + q = m 2 a 11 ( b 21 e − i Λ e ip + e − i 2 p β 12 ) , i∂ − q = − m 2 c 11 ( e i 2 p β 12 + b 21 e − i Λ ) , i∂ + Λ = − mc 11 2 b 21 e − ip e i Λ ( e iq − e − iq ) , i∂ − (Λ − p ) = ma 11 2 b 21 e i Λ ( e iq − e − iq ) , (2.6) together with ∂ + β 12 = − i 2 ∂ + p β 12 + mc 11 2 e i 2 p ( e iq − e − iq ) , ∂ − β 12 = i 2 ∂ − p β 12 − ma 11 2 e − i 2 p ( e iq − e − iq ) . (2.7) A solution for (2.7) compatible with (2.6) is found to b e β 12 = − b 21 4 e − i 2 p + i Λ ( e iq + e − iq + η ) , (2.8) where η is an arbitrary constant. Therefore K =   e − iq 2 − 1 λ 2 c 11 e iq 2 − 1 4 λ b 21 e i Λ e − ip 2 ( e iq + e − iq + η ) 1 λ b 21 e − i Λ e ip 2 e iq 2 − 1 λ 2 c 11 e − iq 2   , (2.9) where we ha v e chosen a 11 = 1. 3 F or c 11 = 0 and in t he limiting case where Λ = i ˜ Λ, with ˜ Λ constant, for larg e η , ˜ Λ, a nd small b 21 limit we find b 21 e − i Λ = B , b 21 e i Λ → 0 , b 21 4 e i Λ η → B , (2.10) where B is a finite constan t. Then, we obtain K =   e − iq 2 − 1 λ B e − ip 2 1 λ B e ip 2 e iq 2   . (2.11) This has the structure of t yp e I B¨ ac klund tr a ns forma t ion [1]. In the limit sp ecified in ( 2 .10) eqs. (2.6) reduces to ∂ − p = − m B sin( q ) , ∂ + q = mB s in( p ) . (2.12) No w, b y in tro ducing σ = − 2 b 21 = b 21 2 c 11 and using eq. (2.10), the equation ( 2 .6) becomes (with a 11 = 1) , i∂ − ( p − Λ) = mσ 4 e i Λ ( e iq − e − iq ) , i∂ + Λ = − m 4 σ e − i ( p − Λ) ( e iq − e − iq ) , i∂ − q = mσ 4 ( e i Λ ( e iq + e − iq + η ) − 4 e − i Λ ) , i∂ + q = m 4 σ ( e − i ( p − Λ) ( e iq + e − iq + η ) − 4 e i ( p − Λ) ) , (2.13) and the expression for K in eq. (2.9) tak es the form K =   e − iq 2 − 1 ( σλ ) 2 e iq 2 1 2( σλ ) e i Λ e − ip 2 ( e iq + e − iq + η ) − 2 ( σλ ) e − i Λ e ip 2 e iq 2 − 1 ( σλ ) 2 e − iq 2   . (2.14) By cross-differen tiating the last tw o equations in (2.13 ), we find that if the field ϕ 1 satisfies the sine-Go r do n equation ∂ 2 t ϕ − ∂ 2 x ϕ = − m 2 sin ϕ, (2.15) then the field ϕ 2 also satisfies it. In additio n, differen tiating the second equation in (2.16) with resp ect to x − , w e obta in ∂ − ∂ + ( i Λ) = − m 2 16 e − ip  4 e 2 i Λ − ( e iq + e − iq )(4 − η e 2 i Λ )  . (2.16) Equations (2.13) and (2.16) w ere considered in [4]. 4 (ii) The second set of equations inv o lv es α 12 , α 21 , β 11 , β 22 , γ 12 and γ 21 whic h satisfy γ 12 = ¯ c 12 e − i 2 p , γ 21 = − ¯ c 12 e i 2 p , α 21 = − a 12 e − i 2 p , α 12 = a 12 e i 2 p , ∂ + α 12 = − i 2 ∂ + p α 12 + m 2 e i 2 ( p + q ) β 11 − m 2 e i 2 ( p − q ) β 22 , ∂ + α 21 = i 2 ∂ + p α 21 − m 2 e − i 2 ( p + q ) β 22 + m 2 e − i 2 ( p − q ) β 11 , ∂ + β 11 = i 2 ∂ + q β 11 − m 2 e − i 2 ( p + q ) γ 12 − m 2 e i 2 ( p − q ) γ 21 , ∂ + β 22 = − i 2 ∂ + q β 22 + m 2 e i 2 ( p + q ) γ 21 + m 2 e − i 2 ( p − q ) γ 12 , (2.17) and ∂ − β 11 = − i 2 ∂ − q β 11 − m 2 e i 2 ( p + q ) α 12 − m 2 e − i 2 ( p − q ) α 21 , ∂ − γ 12 = i 2 ∂ − p γ 12 + m 2 e − i 2 ( p + q ) β 11 − m 2 e − i 2 ( p − q ) β 22 , ∂ − γ 21 = − i 2 ∂ − p γ 21 − m 2 e i 2 ( p + q ) β 22 + m 2 e i 2 ( p − q ) β 11 , ∂ − β 22 = i 2 ∂ − q β 22 + m 2 e − i 2 ( p + q ) α 21 + m 2 e i 2 ( p − q ) α 12 . (2.18) W e now pro pose the change of v ariables (introducing the auxiliary field ¯ Λ) β 22 = b 22 e − i ¯ Λ − i 2 q . (2.19) The equations inv o lv ing ∂ − γ 12 , ∂ + β 22 , ∂ − β 22 and ∂ + α 12 yields resp ectiv ely , i∂ + p = − m 2 a 12 ( e − i ¯ Λ − iq b 22 − e i 2 q β 11 ) , (2.20) i∂ − ( ¯ Λ + q ) = − m 2 a 12 b 22 e i ¯ Λ ( e ip − e − ip ) , (2.21) i∂ + ¯ Λ = m 2 ¯ c 12 b 22 e i ¯ Λ+ iq ( e ip − e − ip ) , (2.22) i∂ − p = m 2 ¯ c 12 ( b 22 e − i ¯ Λ − e − i 2 q β 11 ) . (2.23) A solution of β 11 compatible with (2 .17) and (2.18) is β 11 = − a 12 ¯ c 12 b 22 e i ( ¯ Λ+ 1 2 q ) ( e ip + e − ip + ¯ η ) , (2.24) where ¯ η is an a rbitrary constant. K is then giv en in the following form K =   − 1 λ a 12 ¯ c 12 b 22 e i ( ¯ Λ+ 1 2 q ) ( e ip + e − ip + ¯ η ) a 12 e i 2 p + 1 λ 2 ¯ c 12 e − i 2 p − a 12 e − i 2 p − ¯ c 12 λ 2 e i 2 p b 22 λ e − i ( ¯ Λ+ 1 2 q )   . (2.25) 5 In the limit ¯ c 12 = 0 with large real constants i ¯ Λ , ¯ η and small b 22 satisfying b 22 e − i ¯ Λ = ¯ B , − a 12 ¯ c 12 b 22 e i ¯ Λ → 0 , − a 12 ¯ c 12 b 22 e i ¯ Λ ¯ η = ¯ B , (2.26) where ¯ B is a constan t, w e recov er the structure of t yp e I B¨ acklund t rans fo rmations, ∂ + p = m ¯ B ¯ a 12 sin( q ) , ∂ − q = − m ¯ a 12 ¯ B sin( p ) . (2.27) Notice that (2 .2 7) can b e obtained from ( 2.13) b y exc hanging ϕ 2 → − ϕ 2 . 3 The Tzitz ´ eica-Bullo ugh-Do dd mo de l The Tzitz ´ eica-Bulloug h- Do dd mo del is give n by the field equation ∂ + ∂ − φ = − e φ + e − 2 φ . (3.1) This can b e deriv ed from the zero curv a t ure condition o r La x -Z akharo v-Shabat equation, ∂ + A − − ∂ − A + + [ A + , A − ] = 0 , (3.2) where A ± are giv en b y [12]: A + =   0 − i λe φ 0 0 0 − i λe φ λe − 2 φ 0 0   , A − =   − ∂ − φ 0 − 1 λ − i λ 0 0 0 − i λ ∂ − φ   , ( 3 .3) with λ b eing the sp ectral parameter. Now , by redefining v = e φ [13], the field equation (3.1 ) b ecome s: ∂ + ∂ − v = ∂ + v ∂ − v v − v 2 + 1 v . (3.4) Let us now assume the f o rm for the matrix K (1.2), following ref. [13] a s follows, K = α + 1 λ β + 1 λ 2 δ + 1 λ 3 γ , (3.5) where α, β , δ and γ are 3 × 3 matrices. Equation (1.2) decomp oses in to three indep ende nt systems of equations. W e will consider the one in v olving v ariables { α 11 , α 22 , α 33 , β 13 , β 21 , β 32 , δ 12 , δ 23 , δ 31 , γ 11 , γ 22 , γ 33 } suc h that K =    α 11 + 1 λ 3 γ 11 1 λ 2 δ 12 1 λ β 13 1 λ β 21 α 22 + 1 λ 3 γ 22 1 λ 2 δ 23 1 λ 2 δ 31 1 λ β 32 α 33 + 1 λ 3 γ 33    . (3.6) 6 Equations (1.2) with K g iv en ab o ve are satisfied for γ 11 = γ 22 = γ 33 = ν = const, α 11 = ξ v 2 v 1 = α, α 22 = ξ = const, α 33 = ξ v 1 v 2 = ξ 2 α . (3.7) In tro ducing new v ariables β 21 = Y v 1 = αY ξ v 2 , (3.8) the matr ix K giv en in (3.6) leads to K =     α + ν λ − 3 2 ξ v 2 ν αY ( α + ξ ) λ − 2 2 ξ 2 v 2 2 ν α 2 Y 2 ( α + ξ ) 2 λ − 1 αY ξ v 2 λ − 1 ξ + ν λ − 3 2 ξ v 2 ν αY ( α + ξ ) λ − 2 α Y 2 2 v 2 2 ξ 2 λ − 2 Y v 2 λ − 1 ξ 2 1 α + ν λ − 3     , (3.9) The fields α and Y satisfy the follo wing equations ∂ + α − i α Y ξ − 2 ν Y 2 ( α + ξ ) 2 = 0 , (3.10) 1 v 2 ∂ + Y − Y v 2 2 ∂ + v 2 + 2 ν ξ α v 2 Y ( α + ξ ) + i Y 2 2 v 2 ξ = 0 , (3.11) ∂ − α − i 2 ν v 2 ξ Y ( α + ξ ) + α 2 Y 2 2 v 2 2 ξ 2 = 0 , (3.12) ∂ − Y v 2 + i ξ  ξ α − 1  = 0 , (3.13) whic h w ere written in [1 3]. W e no w define the functions p = φ 1 + φ 2 2 , q = φ 1 − φ 2 2 , α = ξ exp ( − 2 q ) . (3.14) It t hen follows that equations (3.1 0 ) to (3.13) are giv en no w by ∂ + q = − 1 2  i ξ e Λ + 2 ν ξ e − 2Λ  e q + e − q  2  , ∂ − q = − 1 2  2 i ν ξ e p − Λ  e q + e − q  − e 2Λ − 2 p 2 ξ  , ∂ + (Λ − p ) = − ν ξ e − 2Λ ( e 2 q − e − 2 q ) , ∂ − Λ = iξ e − Λ+ p ( e − q − e q ) , (3.15) where Y ≡ e Λ . These equations correspond to those given in [4] with Λ → − λ, ∂ ± → ∂ ∓ , and φ → − φ . 7 4 Conser v ation la ws for ty p e I I defect matrice s 4.1 The sine-Gordon mo del The conserv ation laws for the sine-Gor do n mo del with defects can b e derive d [11] b y making use of the equations of motion as a compatibility condition for the asso ciated linear problem, ∂ x Φ( x, t ; λ ) = U ( x, t ; λ ) Φ( x, t ; λ ) , ∂ t Φ( x, t ; λ ) = V ( x, t ; λ ) Φ( x, t ; λ ) , (4.1) where t he Lax pair is taken in the follo wing form U = " − i 4 ( ∂ t ϕ ) q ( λ ) r ( λ ) i 4 ( ∂ t ϕ ) # , V = " − i 4 ( ∂ x ϕ ) A ( λ ) B ( λ ) i 4 ( ∂ x ϕ ) # , (4.2) and w e hav e defined the following fields, q ( λ ) = − m 4 ( λe iϕ 2 − λ − 1 e − iϕ 2 ) , r ( λ ) = m 4 ( λe − iϕ 2 − λ − 1 e iϕ 2 ) , (4.3) A ( λ ) = − m 4 ( λe iϕ 2 + λ − 1 e − iϕ 2 ) , B ( λ ) = m 4 ( λe − iϕ 2 + λ − 1 e iϕ 2 ) . (4.4) F rom the linear system (4.1), we can derive t wo conserv ation equations, ∂ t  q Γ − i 4 ( ∂ t ϕ )  = ∂ x  A Γ − i 4 ( ∂ x ϕ )  , (4.5) ∂ t  r Ψ + i 4 ( ∂ t ϕ )  = ∂ x  B Ψ + i 4 ( ∂ x ϕ )  , (4.6) where w e hav e in tro duced the auxiliary functions Γ = Φ 2 Φ − 1 1 and Ψ = Φ 1 Φ − 1 2 . These functions satisfy a set of R ic att i equations, ∂ x Γ = r + i 2 ( ∂ t ϕ )Γ − q Γ 2 , (4.7) ∂ x Ψ = q − i 2 ( ∂ t ϕ )Ψ − r Ψ 2 , (4.8) Firstly , let us consider the equation (4.7 ) to solv e Γ. Hence, expanding Γ as λ → ∞ Γ = ∞ X n =0 Γ n λ n , (4.9) w e get, Γ 0 = ie − iϕ 2 , (4.10) Γ 1 = − i m ( ∂ t ϕ + ∂ x ϕ ) e − iϕ 2 , (4.11) 8 Γ 2 = e − iϕ 2  − 2 m 2 ∂ x ( ∂ t ϕ + ∂ x ϕ ) + i 2 m 2 ( ∂ t ϕ + ∂ x ϕ ) 2 + sin ϕ  , (4.12) Γ 3 = 2 im e − i 2 ϕ  − 2 m 2 ∂ 2 x ( ∂ t ϕ + ∂ x ϕ ) + i m 2 ( ∂ t ϕ + ∂ x ϕ )( ∂ x ∂ t ϕ + ∂ 2 x ϕ ) + cos ϕ ( ∂ x ϕ ) + 1 2 ( ∂ t ϕ + ∂ x ϕ ) e − iϕ  . (4.13) Th us, w e hav e a first infinite set of conserv ed quantitie s g enerated f rom I = Z ∞ −∞ dx  q Γ − i 4 ( ∂ t ϕ )  . (4.14) No w, if w e consider the expansion of Γ as λ → 0 , Γ = ∞ X n =0 ˆ Γ n λ n , (4.15) w e obtain the following co efficien ts, ˆ Γ 0 = ie iϕ 2 , ˆ Γ 1 = i m ( ∂ t ϕ − ∂ x ϕ ) e iϕ 2 , (4.16) ˆ Γ 2 = e iϕ 2  − 2 m 2 ∂ x ( ∂ t ϕ − ∂ x ϕ ) + i 2 m 2 ( ∂ t ϕ − ∂ x ϕ ) 2 − sin ϕ  , (4.17) ˆ Γ 3 = 2 im e i 2 ϕ  − 2 m 2 ∂ 2 x ( ∂ t ϕ − ∂ x ϕ ) + i m 2 ( ∂ t ϕ − ∂ x ϕ )( ∂ x ∂ t ϕ − ∂ 2 x ϕ ) − cos ϕ ( ∂ x ϕ ) + 1 2 ( ∂ t ϕ − ∂ x ϕ ) e − iϕ  . (4.18) No w, let us consider the Ricatti equation (4.8) to solv e for Ψ. Clearly , using the same sc heme we can obtain the first few co efficien ts fo r the auxiliary function. The results are listed b elo w, Ψ 0 = ie iϕ 2 , Ψ 1 = − i m ( ∂ t ϕ + ∂ x ϕ ) e iϕ 2 , (4.19) Ψ 2 = e iϕ 2  2 m 2 ∂ x ( ∂ t ϕ + ∂ x ϕ ) + i 2 m 2 ( ∂ t ϕ + ∂ x ϕ ) 2 − sin ϕ  , (4.20) Ψ 3 = − 2 im e i 2 ϕ  2 m 2 ∂ 2 x ( ∂ t ϕ + ∂ x ϕ ) + i m 2 ( ∂ t ϕ + ∂ x ϕ )( ∂ x ∂ t ϕ + ∂ 2 x ϕ ) − cos ϕ ( ∂ x ϕ ) − 1 2 ( ∂ t ϕ + ∂ x ϕ ) e iϕ  , (4.21) and ˆ Ψ 0 = ie − iϕ 2 , ˆ Ψ 1 = i m ( ∂ t ϕ − ∂ x ϕ ) e − iϕ 2 , (4.22) ˆ Ψ 2 = e − iϕ 2  2 m 2 ∂ x ( ∂ t ϕ − ∂ x ϕ ) + i 2 m 2 ( ∂ t ϕ − ∂ x ϕ ) 2 + sin ϕ  , (4.23) 9 ˆ Ψ 3 = 2 im e − i 2 ϕ  2 m 2 ∂ 2 x ( ∂ t ϕ − ∂ x ϕ ) + i m 2 ( ∂ t ϕ − ∂ x ϕ )( ∂ x ∂ t ϕ − ∂ 2 x ϕ ) + cos ϕ ( ∂ x ϕ ) − 1 2 ( ∂ t ϕ − ∂ x ϕ ) e − iϕ  . (4.24) Therefore, I = Z ∞ −∞ dx  i 4 ( ∂ t ϕ ) + r Ψ  , (4.25) generates an infinite num ber o f conserv ation laws. No w, w e will discus s how the presence of in ternal b oundary conditions, or more common- ly called j ump defe cts , mo dify the conserv ed c harges of the sine-Gordon mo del using the Lax pair a pp ro a c h. First, let us place a defect at x = 0, and consider the g e nerating functional of infinite charges giv en by (4.14) in the presence of a defect as follow s, I = Z 0 −∞ dx  q 1 Γ[ ϕ 1 ] − i 4 ( ∂ t ϕ 1 )  + Z ∞ 0 dx  q 2 Γ[ ϕ 2 ] − i 4 ( ∂ t ϕ 2 )  . (4.26) and taking the time-deriv a tiv e, w e hav e dI dt =  A 1 Γ[ ϕ 1 ] − i 4 ( ∂ x ϕ 1 )     x =0 −  A 2 Γ[ ϕ 2 ] − i 4 ( ∂ x ϕ 2 )     x =0 = − dI D dt , (4.27) where [1 1 ], I D = ln h K 11 + K 12 Γ[ ϕ 1 ] i    x =0 . (4.28) No w, follow ing the same pro cedure for the generating function (4.25 ) , one gets d I dt =  B 1 Ψ[ ϕ 1 ] + i 4 ( ∂ x ϕ 1 )     x =0 −  B 2 Ψ[ ϕ 2 ] + i 4 ( ∂ x ϕ 2 )     x =0 = − d I D dt , ( 4 .29) where I D = ln h K 21 Ψ[ ϕ 1 ] + K 22 i    x =0 , (4.30) No w, ta king into accoun t the type I I defect matrix K in (2.14) and using the formulas (4.28) and (4.30) to compute the resp ec tive defect contributions to the mo dified conserv ed quan tities for this case, w e o btain the following results, ( I D ) − 1 = i 2 σ e − i ( p − Λ) ( e iq + e − iq + η ) , ( I D ) +1 = − iσ 2 e i Λ ( e iq + e − iq + η ) , (4.31) ( I D ) − 1 = − 2 i σ e i ( p − Λ) , ( ˆ I D ) +1 = 2 iσ e − i Λ . (4.32) 10 The corresp onding t yp e I I defect energy and momen tum for the sine-Gordon mo del can b e written as, E D = im  ( I D ) − 1 − ( I D ) − 1 − ( ˆ I D ) +1 + ( ˆ I D ) +1  = − m 2 σ (4 e i ( p − Λ) + e − i ( p − Λ) ( e iq + e − iq + η )) − mσ 2 (4 e − i Λ + e i Λ ( e iq + e − iq + η )) , P D = im  ( I D ) − 1 − ( I D ) − 1 + ( ˆ I D ) +1 − ( ˆ I D ) +1  = − m 2 σ (4 e i ( p − Λ) + e − i ( p − Λ) ( e iq + e − iq + η )) + mσ 2 (4 e − i Λ + e i Λ ( e iq + e − iq + η )) . (4.33) Notice that ( I D ) − 1 − ( I D ) − 1 and ( I D ) +1 − ( ˆ I D ) +1 corresp onds resp ectiv ely to the quan tities f and − g of ref. [4]. T aking in to accoun t the type I I defect matrix K in (2.25) and using (4.30) a nd (4.32) we obtain the following results for the mo dified conserv ed quan tities ( I D ) − 1 = i ¯ c 12 b 22 e i ( q + ¯ Λ) ( e ip + e − ip + ¯ η ) − 1 m ( ∂ t ϕ 1 + ∂ x ϕ 1 ) , ( ˆ I D ) +1 = i a 12 b 22 e i ¯ Λ ( e ip + e − ip + ¯ η ) + 1 m ( ∂ t ϕ 1 − ∂ x ϕ 1 ) , ( I D ) − 1 = i b 22 a 12 e − i ( q + ¯ Λ) − 1 m ( ∂ t ϕ 1 + ∂ x ϕ 1 ) , ( ˆ I D ) +1 = i b 22 ¯ c 12 e − i ¯ Λ + 1 m ( ∂ t ϕ 1 − ∂ x ϕ 1 ) , (4.34) yielding defect energy and momen tum E D = im  ( I D ) − 1 − ( I D ) − 1 − ( ˆ I D ) +1 + ( ˆ I D ) +1  = m b 22 a 12 e − i ( ¯ Λ+ q ) − m ¯ c 12 b 22 e i ( ¯ Λ+ q )  e ip + e − ip + ¯ η  − m b 22 ¯ c 12 e − i ¯ Λ + m a 12 b 22 e i ¯ Λ  e ip + e − ip + ¯ η  , P D = im  ( I D ) − 1 − ( I D ) − 1 + ( ˆ I D ) +1 − ( ˆ I D ) +1 )  = m b 22 a 12 e − i ( ¯ Λ+ q ) − m ¯ c 12 b 22 e i ( ¯ Λ+ q )  e ip + e − ip + ¯ η  + m b 22 ¯ c 12 e − i ¯ Λ − m a 12 b 22 e i ¯ Λ  e ip + e − ip + ¯ η  . (4.35) Notice that (4.33) and (4.35) reflect the symmetry ϕ 2 → − ϕ 2 already p oin ted out with resp e ct t o eqs. (2 .2 7) a nd (2.12 ). 4.2 The Tzitz ´ eica-Bullough-Do dd Mo del F or this mo del, the matrices U and V , are conv enien tly written as, U =     − ( ∂ − v ) 2 v iλv 2 − 1 2 λ − i 2 λ 0 iλv 2 − λ 2 v − 2 − i 2 λ ( ∂ − v ) 2 v     , V =     ( ∂ − v ) 2 v iλv 2 1 2 λ i 2 λ 0 iλv 2 − λ 2 v − 2 i 2 λ − ( ∂ − v ) 2 v     . (4.36) 11 In terms of these, we can write down the set of linear differen tial equations (4.1) a s , ∂ t Φ 1 =  ∂ − v 2 v  Φ 1 +  iλv 2  Φ 2 +  1 2 λ  Φ 3 , (4.37) ∂ t Φ 2 =  i 2 λ  Φ 1 +  iλv 2  Φ 3 , (4.38) ∂ t Φ 3 = −  λ 2 v 2  Φ 1 +  i 2 λ  Φ 2 −  ∂ − v 2 v  Φ 3 , (4.39) and ∂ x Φ 1 = −  ∂ − v 2 v  Φ 1 +  iλv 2  Φ 2 −  1 2 λ  Φ 3 , (4.40) ∂ x Φ 2 = −  i 2 λ  Φ 1 +  iλv 2  Φ 3 , (4.41) ∂ x Φ 3 = −  λ 2 v 2  Φ 1 −  i 2 λ  Φ 2 +  ∂ − v 2 v  Φ 3 . (4.42) No w, b y defining the a ux iliary functions Γ 12 = Φ 2 Φ − 1 1 and Γ 13 = Φ 3 Φ − 1 1 , we can construct an infinite set of conserv ation law s fro m the equations (4.37) and (4.40) as follo ws, ∂ t  − ( ∂ − v ) 2 v +  iλv 2  Γ 12 − 1 2 λ Γ 13  = ∂ x  ( ∂ − v ) 2 v +  iλv 2  Γ 12 + 1 2 λ Γ 13  , (4.43) where t he auxiliary functions satisfy the following coupled Ricatti equations for the x -part, ∂ x Γ 12 =  ( ∂ − v ) 2 v  Γ 12 +  iλv 2   Γ 13 − (Γ 12 ) 2  − 1 2 λ ( i − Γ 12 Γ 13 ) , (4.44) ∂ x Γ 13 =  ( ∂ − v ) v  Γ 13 − λ 2  1 v 2 + iv Γ 12 Γ 13  − 1 2 λ  i Γ 12 − (Γ 13 ) 2  , ( 4 .45) and for the t -part, ∂ t Γ 12 = − ( ∂ − v ) 2 v Γ 12 + iλv 2 (Γ 13 − (Γ 12 ) 2 ) + 1 2 λ ( i − Γ 12 Γ 13 ) , (4.46) ∂ t Γ 13 = − ( ∂ − v ) v Γ 13 − λ 2  1 v 2 + iv Γ 12 Γ 13  + 1 2 λ ( i Γ 12 − (Γ 13 ) 2 ) . (4.47) No w, a s usual these differen tial equations can b e recursiv ely solved b y considering an expan- sion in non-p ositiv e p o w ers of λ , Γ 12 = ∞ X n =0 Γ ( n ) 12 λ n , Γ 13 = ∞ X n =0 Γ ( n ) 13 λ n . (4.48) 12 The lo w est co efficien ts are simply given b y , Γ (0) 12 = i ( µv ) − 1 , Γ (0) 13 = µv − 2 , (4.49) Γ (1) 12 = − i ( ∂ + v ) v − 2 , Γ (1) 13 = µ − 1 ( ∂ + v ) v − 3 , (4.50) Γ (2) 12 = − 4 iµ 3 ∂ x  ( ∂ + v ) v − 1  v − 1 + iµ 3  ( ∂ + v ) v − 1  2 v − 1 + 2 iµ 3  1 − v − 3  , (4.51) Γ (2) 13 = 2 3 ∂ x  ( ∂ + v ) v − 1  v − 2 − 2 3  ( ∂ + v ) v − 1  2 v − 2 − 1 3  v − 1 − v − 4  , (4.52) where µ is an ar bitrary constan t satisfying µ 3 = − 1. Assumin g sufficien tly smo oth deca ying fields a s | x | → ±∞ , the corresp onding conserv ed quantities reads I 1 = Z ∞ −∞ dx  − ( ∂ − v ) 2 v + iλv 2 Γ 12 − 1 2 λ Γ 13  . (4.53) By substituting the expansion of the auxiliary functions into ab o ve definition, we get an infi- nite num b e r of conserv ed c harges I ( k ) 1 . It is very easy to c hec k that the conserv ed quantities corresp onding to k = 1 is trivial, a nd for k = 0 w e obtain a top ological term. The first non-v anishing conserv ed charge is explicitly giv en b y , I ( − 1) 1 = 1 3 Z ∞ −∞ dx  1 2 (( ∂ + v ) v − 1 ) 2 +  v + 1 2 v − 2  , (4.54) where without loss of generality , w e hav e c hosen µ = − 1. Then, rep eating this pro cedure w e can construct a nother set of conserv ed quantities corresp onding to the expansion of the auxiliary functions in non- nega t iv e p o w ers of λ , namely , Γ 12 = ∞ X n =0 ˆ Γ ( n ) 12 λ n , Γ 13 = ∞ X n =0 ˆ Γ ( n ) 13 λ n . (4.55) F rom the Ricatti equations we get, ˆ Γ (0) 12 = iµ, ˆ Γ (0) 13 = µ − 1 , (4.56) ˆ Γ (1) 12 = 0 , ˆ Γ (1) 13 = − ( ∂ − v ) v − 1 , (4.57) ˆ Γ (2) 12 = − 2 i 3 ∂ x  ( ∂ − v ) v − 1  + i 3  ( ∂ − v ) v − 1  2 − i 3  v − v − 2  , (4.58) ˆ Γ (2) 13 = − 2 µ 3 ∂ x  ( ∂ − v ) v − 1  + µ 3  ( ∂ − v ) v − 1  2 − µ 3  v − v − 2  . (4.59) F rom these results and c hosing µ = − 1, the first no n-v anishing conserv ed charge is give n b y ˆ I (+1) 1 = 1 3 Z ∞ −∞ dx  1 2 (( ∂ − v ) v − 1 ) 2 +  v + 1 2 v − 2  . (4.60) Then, we clearly can combine I ( − 1) 1 and ˆ I (+1) 1 in order to obtain t he usual energy and momen- tum quan tities. Ho w ev er, it is not enough b ecause w e are not considering all the information 13 coming from the Lax pair. So, it is also p ossible to construct another infinite sets o f con- serv ed quantities b y considering tw o more conserv ation equations that can b e deriv ed from the equations (4 .3 8), (4 .39), (4.41) and (4.42), namely , ∂ t  −  i 2 λ  Γ 21 +  iλv 2  Γ 23  = ∂ x  i 2 λ  Γ 21 +  iλv 2  Γ 23  , (4.61) ∂ t  ∂ − v 2 v  −  λ 2 v 2  Γ 31 −  i 2 λ  Γ 32  = ∂ x  −  ∂ − v 2 v  −  λ 2 v 2  Γ 31 +  i 2 λ  Γ 32  , (4.62) where w e ha ve in tro duced some other auxiliary functions Γ 21 = Φ 1 Φ − 1 2 , Γ 23 = Φ 3 Φ − 1 2 , Γ 31 = Φ 1 Φ − 1 3 , a nd Γ 32 = Φ 2 Φ − 1 3 . It is quite straigh tfo r ward that these functions satisfy a set of Ricatti equations that can b e written for the x -par t as follow s, ∂ x Γ 21 =  iλv 2  −  ∂ − v 2 v  Γ 21 −  1 2 λ  Γ 23 −  iλv 2  (Γ 21 Γ 23 ) +  i 2 λ  (Γ 21 ) 2 , (4.63) ∂ x Γ 23 = −  i 2 λ  −  λ 2 v 2  Γ 21 +  ∂ − v 2 v  Γ 23 +  i 2 λ  (Γ 21 Γ 23 ) −  iλv 2  (Γ 23 ) 2 , (4.64) ∂ x Γ 31 = −  1 2 λ  −  ∂ − v 2 v  Γ 31 +  iλv 2  Γ 32 +  i 2 λ  (Γ 31 Γ 32 ) +  λ 2 v 2  (Γ 31 ) 2 , (4.6 5) ∂ x Γ 32 =  iλv 2  −  i 2 λ  Γ 31 −  ∂ − v 2 v  Γ 32 +  λ 2 v 2  (Γ 31 Γ 32 ) +  i 2 λ  (Γ 32 ) 2 , (4.66) and for the t -part, ∂ t Γ 21 =  iλv 2  +  ∂ − v 2 v  Γ 21 +  1 2 λ  Γ 23 −  iλv 2  (Γ 21 Γ 23 ) −  i 2 λ  (Γ 21 ) 2 , (4.67) ∂ t Γ 23 =  i 2 λ  −  λ 2 v 2  Γ 21 −  ∂ − v 2 v  Γ 23 −  i 2 λ  (Γ 21 Γ 23 ) −  iλv 2  (Γ 23 ) 2 , (4.68) ∂ t Γ 31 =  1 2 λ  +  ∂ − v 2 v  Γ 31 +  iλv 2  Γ 32 −  i 2 λ  (Γ 31 Γ 32 ) +  λ 2 v 2  (Γ 31 ) 2 , (4.69) ∂ t Γ 32 =  iλv 2  +  i 2 λ  Γ 31 +  ∂ − v 2 v  Γ 32 +  λ 2 v 2  (Γ 31 Γ 32 ) −  i 2 λ  (Γ 32 ) 2 . (4.70) As w as already sho wn, these equations can be r e cursiv ely solv ed b y in tro ducing a n expan- sion of the resp ectiv e auxiliary functions in p ositiv e and/or negative p o w ers o f the sp ectral parameter λ . Doing so, after a lengthy calculation the first few co efficien ts for these auxiliary functions can b e determined, a nd the results are sho wn in tables 1 and 2. No w, from equations (4.6 1) and (4.62) w e obtain directly the follo wing tw o generating functions of the conserv ed quantities, I 2 ( λ ) = Z ∞ −∞ dx  −  i 2 λ  Γ 21 +  iv λ 2  Γ 23  , (4.71) I 3 ( λ ) = Z ∞ −∞ dx  ∂ − v 2 v  −  λ 2 v 2  Γ 31 −  i 2 λ  Γ 32  . (4.72) 14 Γ (0) 21 = iv Γ (0) 23 = − iv − 1 ˆ Γ (0) 21 = i ˆ Γ (0) 23 = − i Γ (0) 31 = − v 2 Γ (0) 32 = iv ˆ Γ (0) 31 = − 1 ˆ Γ (0) 32 = i Γ (1) 21 = − i ( ∂ + v ) Γ (1) 23 = 0 ˆ Γ (1) 21 = 0 ˆ Γ (1) 23 = − i ( ∂ − v ) v − 1 Γ (1) 31 = ( ∂ + v ) v Γ (1) 32 = 0 ˆ Γ (1) 31 = ( ∂ − v ) v − 1 ˆ Γ (1) 32 = − i ( ∂ − v ) v − 1 T able 1: The zero-th a nd first or der co efficien ts. Γ (2) 21 = − 4 3  ∂ x ( ∂ + v ) + 1 2 v ( ∂ − v ) ( ∂ + v ) + 1 2 ( v − 1 − v 2 )  Γ (2) 23 = 2 3 i  ∂ x ( ∂ + v ) v − 2 + 1 2 v 3 ( ∂ − v ) ( ∂ + v ) + 1 2 ( v − 3 − 1)  ˆ Γ (2) 21 = 2 3 i  ∂ x ( ∂ − v ) v − 1 − 1 2 v 2 ( ∂ − v ) ( ∂ + v ) − 1 2 ( v − 2 − v )  ˆ Γ (2) 23 = 4 3 i  − ∂ x ( ∂ − v ) v − 1 + 1 2 v 2 ( ∂ − v ) ( ∂ + v ) + 1 2 ( v − 2 − v )  Γ (2) 31 = − 2 3  ∂ x ( ∂ + v ) v + 1 2 ( ∂ − v ) ( ∂ + v ) + 1 2 (1 − v 3 )  Γ (2) 32 = 2 3 iv  ∂ x ( ∂ + v ) v + 1 2 ( ∂ − v ) ( ∂ + v ) + 1 2 (1 − v 3 )  T able 2: Second-or de r co efficien ts. 15 Then, by substituting t he respectiv e expansion o f eac h a ux iliary function and using the co e- fficien ts sho w ed in tables 1 and 2, w e immediately g et the first few non-v anishing conserv ed quan tities, whic h are explicitly give n b y , I ( − 1) 2 = I ( − 1) 3 = 1 3 Z ∞ −∞ dx  1 2 (( ∂ + v ) v − 1 ) 2 +  v + 1 2 v − 2  , (4.73) ˆ I (+1) 2 = ˆ I (+1) 3 = 1 3 Z ∞ −∞ dx  1 2 (( ∂ − v ) v − 1 ) 2 +  v + 1 2 v − 2  . (4.74) F rom the a bov e results, w e can notice that there is a simple com bination o f all these con tri- butions giving us the usual energy and momen tum conserv ed quan tities. In fact, if w e define the fo llo wing consev ed quantities , I ( − 1) = I ( − 1) 1 + I ( − 1) 2 + I ( − 1) 3 = Z ∞ −∞ dx  1 2 (( ∂ + v ) v − 1 ) 2 +  v + 1 2 v − 2  , (4.75) ˆ I (+1) = ˆ I (+1) 1 + ˆ I (+1) 2 + ˆ I (+1) 3 = Z ∞ −∞ dx  1 2 (( ∂ − v ) v − 1 ) 2 +  v + 1 2 v − 2  , (4.76) the conserv ed energy and momen tum can b e written as E = ( I ( − 1) + ˆ I (+1) ) 2 = Z ∞ −∞ dx  1 2  ( ∂ x φ ) 2 + ( ∂ t φ ) 2  +  e φ + 1 2 e − 2 φ  , (4.77) P = ( I ( − 1) − ˆ I (+1) ) 2 = Z ∞ −∞ dx ( ∂ x v ) ( ∂ t v ) v 2 = Z ∞ −∞ dx ( ∂ x φ ) ( ∂ t φ ) . (4.78) Then, it sho ws that giv en a Lax pair for the Tzitz ´ eica-Bullough-Do dd mo del we can construct an infinite set of conserv ed c harges b y using all the info rmation coming from t he asso ciated linear pro ble m. No w, w e will compute the mo dified conserv ed c harges coming fro m the defect con tribu- tions f or the Tzitz ´ eica-Bullough- D odd mo del using the Lax pair approac h. The n, considering the defect placed at x = 0, the set of infinite c harges giv en b y (4.53 ) in the presence o f a defect reads, I 1 ( λ ) = Z 0 −∞ dx  − ( ∂ − v 1 ) 2 v 1 + iλv 1 2 Γ 12 ( v 1 ) − 1 2 λ Γ 13 ( v 1 )  + Z ∞ 0 dx  − ( ∂ − v 2 ) 2 v 2 + iλv 2 2 Γ 12 ( v 2 ) − 1 2 λ Γ 13 ( v 2 )  , (4.79) differen tiating with resp ect to time, w e get d I 1 ( λ ) dt =  ( ∂ − v 1 ) 2 v 1 + iλv 1 2 Γ 12 ( v 1 ) + 1 2 λ Γ 13 ( v 1 )     x =0 −  ( ∂ − v 2 ) 2 v 2 + iλv 2 2 Γ 12 ( v 2 ) + 1 2 λ Γ 13 ( v 2 )     x =0 . (4.80) 16 Then, using the asso ciated linear system w e o btain the follow ing relation b et w een the a ux - iliary f un ctions o f eac h side, Γ 12 ( v 2 ) = K 21 + K 22 Γ 12 ( v 1 ) + K 23 Γ 13 ( v 1 ) K 11 + K 12 Γ 12 ( v 1 ) + K 13 Γ 13 ( v 1 ) , Γ 13 ( v 2 ) = K 31 + K 32 Γ 12 ( v 1 ) + K 33 Γ 13 ( v 1 ) K 11 + K 12 Γ 12 ( v 1 ) + K 13 Γ 13 ( v 1 ) . (4.81) No w, from the partial diff erential equations satisfied by K and the t w o Ricatti equations (4.46) and (4.47), w e finally get that d I 1 ( λ ) dt = − d dt h ln ( K 11 + K 12 Γ 12 ( v 1 ) + K 13 Γ 13 ( v 1 )) i    x =0 . (4.82) Then, w e hav e that the mo dified conserv ed quan tities derive d from this conserv ation equation (4.43) is I 1 ( λ ) + I D 1 ( λ ), where I D 1 ( λ ) = ln h K 11 + K 12 Γ 12 ( v 1 ) + K 13 Γ 13 ( v 1 ) i    x =0 . (4.83) F rom the ab o v e form ula w e can deriv e t w o differen t sets of defect con tribution b y considering the expansion o f the auxiliary functions in p ositiv e and negativ e p o w ers of λ . In particular for the λ ± 1 -terms, we obtain ˆ I (+1) D 1 = − 2 iξ e ( p − Λ)  e q + e − q  , I ( − 1) D 1 = − 2 ξ e − 2Λ  e q + e − q  2 . (4.84) As w e no w know, to obtain the exact f orm of the corresp onding defect con tributions t o t he energy and momen tum, w e need to consider the others conserv ation equations and conse- quen tly the other charges give n in (4 .71) and ( 4 .72). Applyin g the same steps to derive the defect con tributions, one g e ts I D 2 ( λ ) = ln h K 21 Γ 21 ( v 1 ) + K 22 + K 23 Γ 23 ( v 1 ) i    x =0 , (4.85) I D 3 ( λ ) = ln h K 31 Γ 31 ( v 1 ) + K 32 Γ 32 ( v 1 ) + K 33 i    x =0 . (4.86) F rom these, w e find that t he first non-v anishing terms are giv en explicitly by , ˆ I (+1) D 2 = − 2 iξ e ( p − Λ)  e q + e − q  , I ( − 1) D 2 = i ξ e Λ , (4.87) ˆ I (+1) D 3 = − 1 2 e 2(Λ − p ) , I ( − 1) D 3 = i ξ e Λ . (4.88) No w, defining b y analogy the follo wing tw o conserv ed quan tities, I ( − 1) D = I ( − 1) D 1 + I ( − 1) D 2 + I ( − 1) D 3 = − 2 ξ e − 2Λ  e q + e − q  2 + 2 i ξ e Λ , (4.89) ˆ I (+1) D = ˆ I ( − 1) D 1 + ˆ I ( − 1) D 2 + ˆ I ( − 1) D 3 = − 4 iξ e ( p − Λ)  e q + e − q  − 1 2 e 2(Λ − p ) , (4.90) 17 w e can write down the defect energy and momen tum as follows , E D = 1 2  I ( − 1) D + ˆ I (+1) D  = − ξ e − 2Λ  e q + e − q  2 + i ξ e Λ − 2 iξ e ( p − Λ)  e q + e − q  − 1 4 e 2(Λ − p ) , (4.91) P D = 1 2  I ( − 1) D − ˆ I (+1) D  = − ξ e − 2Λ  e q + e − q  2 + i ξ e Λ + 2 i ξ e ( p − Λ)  e q + e − q  + 1 4 e 2(Λ − p ) . (4.92) These are exactly the defect energy and mo mentum whic h are o bt a ine d b y using the La- grangian formalism. In particular, w e can note that E D corresp onds t o f + g of ref. [4], which w as exp ecte d (with the observ atio ns g iven at the end of section 2). It is in teresting to em- phasize that for obtaining the most general form o f the defect energy and momen tum for the Tzitz ´ eica-Bullo ug h-Do dd mo del is necess ary to consider all the con tributions coming from all conserv a tion equations and, fo r b oth expansions of ev ery eac h of the auxiliary functions in p ositiv e and negativ e p o w ers of the sp ectral parameter λ . 5 Conclus ions In conclusion, w e ha ve constructe d the t yp e I I B¨ ac klund transforma t ions via gauge trans- formations and the corresp onding conserv ed quan tities for t he sine-Gordo n and Tzitz ´ eica- Bullough-Do dd mo dels . 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