New soliton solutions for Chen-Lee-Liu and Burgers hierarchies and its Bäcklund transformations

New soliton solutions for Chen-Lee-Liu and Burgers hierarc hies and its Bäc klund transformations 1 Y. F. A dans a,b , H. Arat yn c , C. P . Constan tinidis d , J. F. Gomes a , G. V. Lob o a , and T. C. San tiago a . a Univ ersidade Estadual Paulista (Unesp), Instituto de Física T eórica (IFT), São P aulo, Rua Dr. Ben to T eobaldo F erraz 271, 01140-070, São Paulo, SP , Brasil b Sc ho ol of Mathematics & Hamilton Mathematics Institute, T rinity College Dublin, Ireland c Departmen t of Physics, Univ ersit y of Illinois Chicago, 845 W. T a ylor St., 60607-7059, Chicago, IL, USA. d Univ ersidade F ederal do Espirito Santo, Depto. de Física, A v. F ernando F errari, 514., CEP 29075-900, Vitoria, ES, Brasil. ysla.franca@unesp.br , aratyn@uic.edu , clisthenis.constantinidis@ufes.br , francisco.gomes@unesp.br , gabriel.lobo@unesp.br , t.santiago@unesp.br . Abstract P ositive and negative flo ws of the Chen-Lee-Liu model and its v arious reductions, including Burgers hierarc hy , are form ulated within the framework of Riemann-Hilbert-Birkhoff decom- p osition with the constan t grade t wo generator. T wo classes of v acua, namely zero v acuum and constan t non-zero v acuum can b e realized within a centerless Heisenberg algebra. The tau functions for soliton solutions are obtained by a dressing method and v ertex op erators are constructed for both types of v acua. W e are able to select and classify the soliton solutions in terms of the type of vertices inv olv ed. A judicious c hoice of vertices yields in a closed form a particular set of multi soliton solutions for the Burgers hierarc hy . W e develop and analyze a class of gauge-Bäc klund transformations that generate further multi soliton solutions from those obtained by dressing metho d b y letting them interact with v arious integrable defects. 1 In tro duction In tegrable hierarc hies are often realized as t wo-dimensional field theories that allo w an infinite num- b er of conserv ation la ws whic h, in turn ensure stability of soliton solutions. A crucial ingredien t in 1 This pap er is dedicated to the memory of Abraham Hirsz Zimerman, 1928-2025, a dear friend, mentor and long- term collaborator. 1 constructing such in tegrable hierarc hies with an underlying affine algebraic structure is its grada- tion [1, 2]. The flo w equations are conv enien tly obtained in terms of a zero curv ature represen tation,  ∂ x + A x ( ϕ ) , ∂ t ± N + A t ± N ( ϕ )  = 0 , A x , A t ± N ∈ ˆ G , N ∈ N ∗ . (1.1) Notice that (1.1) generate a series of flows asso ciated to graded Heisen b erg algebra elements. The construction is well-kno wn for positive flo ws. More recently negativ e flows ha ve been incorp orated [3 – 6] and sho wn to generate new interesting symmetries [7 – 9]. There are many ansatzes for constructing the auxiliary , field dep enden t, t wo-dimensional gauge p oten tials (Lax op erators) A x ( ϕ ) and A t ± N ( ϕ ) . Many well-kno wn examples, as mKdV and AKNS, in volv e hierarchies classified according to the gr ading of the affine Lie algebra ˆ G = P i ∈ Z ˆ G i and a c hoice of gr ade one semi-simple generator E (1) . A systematic approac h for constructing the Lax op erators can b e form ulated in terms of the Riemann-Hilb ert-Birkhoff (RHB) decomposition (see for instance [10]). The underlying algebraic framew ork is very p o werful and allo ws for the systematic construction of soliton solutions from represen tation theory . The dressing metho d constructs soliton solutions emplo ying a gauge trans- formation to map the Lax op erators from a particular v acuum solution, A x ( ϕ v ac ) and A t ± N ( ϕ v ac ) in to a non-trivial configuration, A x ( ϕ ) and A t ± N ( ϕ ) . There are ho wev er examples in volving higher grade semi-simple elements, E ( a ) , a > 1 [11] and presen ting a v ariety of non-trivial b oundary conditions with different v acuum solutions [4, 12]. In this pap er w e follow a proposal [6] for the generalized Riemann-Hilb ert-Birkhoff (g-RHB) decomp osition formula that includes b oth, higher gr ading semi-simple elements and a variety of non-trivial vacuum c onfigur ations . The condition to encompass differen t v acuum solutions requires the existence of Heisenberg sub-algebras. In fact, Heisen b erg sub-algebras classify the p ossible b oundary conditions. Define the generalized Baker-Akhiezer function (g-BA), Ψ a = e − P N ( ϵ ( aN ) t N + ϵ ( − aN ) t − N ) . (1.2) Here, ϵ ( ± aN ) , a ∈ N ∗ are v acuum parameters dependent generators that satisfy a cen terless Heisen- b erg algebra [ ϵ ( aM ) , ϵ ( aN ) ] = 0 . Notice that Ψ a displa ys explicit space-time information (where t 1 ≡ x ). The simplest example corresp onds to the mKdV hierarch y with a = 1 . The Lax op erators acting on v acuum w ere constructed in [4, 6] and were shown to generate one-parameter deformed Heisen b erg algebras for p ositive o dd and ne gative even flows. In this pap er we engage the g-RHB decomp osition (2.3) and (1.2) with a = 2 to form ulate the Chen-Lee-Liu (CLL) hierarch y , and construct its soliton solutions in terms of differen t p ossible v acuum solutions and their reductions to Burgers hierarc h y . In section 2 w e discuss the construction of the positive and negativ e flows for the CLL hierarc h y in terms of v arious Heisen b erg sub-algebras, eac h describing different possible v acuum solutions. In section 3 the v arious reductions to he at and Bur gers equations are discussed. The systematic construction of soliton solutions is presen ted explicitly in Section 4. The algebraic structure provides an elegan t construction for soliton solutions. An important element introduced b y the Kyoto School approac h [13] is the associated vertex op erators which corresp ond to eigen vectors of the Heisenberg algebras. The asso ciated eigenv alues enco de the space-time dep endence for the soliton solutions. It is interesting to note that these v ertices may dep end up on v acuum parameters and henceforth pro vide a new class of soliton solutions. The dressing metho d emplo yed here follo ws directly from the g-RHB decomposition and implies gauge transforming the g-BA Ψ a with v acuum information to some non-trivial solution Φ = Θ + Ψ a = 2 Θ − Ψ a g . This is accomplished by the construction of a pair of v ertex op erators, namely V ± . The solutions are then classified in to class A , when p ow ers of only one of the v ertices, either V + or V − are considered and as a consequence, one of the fields remains constan t (non v anishing). Suc h structure unco vers the underlying Burgers hierarc hy asso ciated to class A solutions and the dressing metho d generates, in closed form, the n − soliton solution for the entire Burgers hierarch y . The second, class B , is obtained when p ow ers of the pro duct V + V − are considered and b oth fields are sho wn to be non trivial. In section 5 w e construct a gauge-Bäcklund transformation as a generalization of the dressing metho d, where t wo non-trivial solutions are connected by gauge transformation. The Bäcklund transformation is shown to describe in tegrable defects [14, 15] since it describes the connection b et ween t wo solutions at a sp ecific space p osition. W e then discuss explicit examples of p ossible in tegrable defects. The key ingredient is an ansatz in volving three consecutiv e graded terms with the virtue to accommo date t wo non-trivial soliton configurations. In section 6 we discuss in detail the tw o classes of Bäcklund solutions. Since class A con tains p o wers of a single vertex op erator and one of the fields, either r or s , remains constant for all flow equations. The CLL hierarc hy then reduces to the Burgers hierarc h y and so do es the corresp onding Bäc klund transformation. W e therefore discuss the scattering and transition of one-soliton and t wo-solitons solutions for Burgers hierarc h y . Next w e consider, in section 7, class B of Bäcklund solutions comp osed of p ow ers of mixed v ertices. W e discuss the scattering of one-soliton and the transition of one to tw o-soliton solutions for the CLL hierarc hy . 2 The Generalized Riemann-Hilb ert-Birkhoff (g-RHB) Decomp o- sition Consider the generalized Baker-Akhiezer function (g-BA) (1.2). The connection with integrable hierarc hies is established with the identification of Heisenberg generators with v acuum configuration, A v ac t N = A t N ( ϕ v ac ) = E ( aN ) + D ( aN − 1) v ac + · · · + D (0) v ac ≡ ϵ ( aN ) , A v ac t − N = A t − N ( ϕ v ac ) = E ( − aN ) + D ( − aN − 1) v ac + · · · + D ( − 1) v ac ≡ ϵ ( − aN ) , D ( i ) v ac ∈ ˆ G i (2.1) where A t 1 ≡ A x . F or zero ( ϕ v ac = 0 ) or nonzero v acuum ( ϕ v ac = ϕ 0 ) configurations, the zero curv ature represen tation (1.1) yields an imp ortant (cen terless) Heisenberg algebra which ma y dep end up on complex parameters, namely ( ϕ 0 ) [6, 10], h A v ac x , A v ac t ± N i =  A x ( ϕ v ac ) , A t ± N ( ϕ v ac )  = 0 . (2.2) In order to derive a construction of the t wo dimensional Lax op erators A x and A t N consider the follo wing g-RHB decomp osition Θ( t ) = Ψ a ( t ) g Ψ − 1 a ( t ) = Θ − 1 − ( t )Θ + ( t ) (2.3) where g is an arbitrary constan t group element and Θ − ( t ) = ˜ B ∞ Y k =1 e − θ ( − k ) , Θ + ( t ) = ˜ B B ∞ Y k =1 e θ ( k ) , B = e θ (0) , ˜ B = e ˜ θ 0 , θ ( k ) ∈ ˆ G k . (2.4) 3 Notice that (2.3) do es not dep end up on ˜ B . Here ˜ B represen ts a gauge fr e e dom and can b e c hosen for con venience as ˜ B = B − c , 0 ≤ c ≤ 1 such that allo ws one to reshoufle the zero grade comp onen t to b e contained partially within the p ositive, Θ + → B − c Θ + or negativ e , Θ − → B − c Θ − graded subgroups as sho wn in (2.4). The flow structure ( t = t N ) of integrable hierarc hies is determined by a decomp osition of an affine algebra into graded subspaces, ˆ G = P i ∈ Z ˆ G i , and its corresp onding decomp osition of A x ( ϕ ) and A t N ( ϕ ) , as discussed in detail in the next sections. In particular, in [6] it w as shown that integrable hierarc hies depend upon t w o distinct structures, i) constant semisimple op erators of (higher) grade a ∈ N ∗ , E ( a ) , and ii) nonzero constant v acuum parameters defined from the Lax op erators in v acuum, A v ac x ( ϕ 0 ) and A v ac t ± N ( ϕ 0 ) . A x = Θ ± A v ac x Θ − 1 ± − ( ∂ x Θ ± ) Θ − 1 ± = (Θ − ϵ ( a ) Θ − 1 − ) ≥ −  ∂ x B − c  B c = E ( a ) + a − 1 X i =0 A i (2.5a) A t N = Θ ± A v ac t N Θ − 1 ± − ( ∂ t N Θ ± ) Θ − 1 ± = (Θ − ϵ ( aN ) Θ − 1 − ) ≥ −  ∂ t N B − c  B c = E ( aN ) + aN − 1 X i =0 D ( i ) (2.5b) A t − N = Θ ± A v ac t − N Θ − 1 ± −  ∂ t − N Θ ±  Θ − 1 ± = (Θ + ϵ ( − aN ) Θ − 1 + ) < −  ∂ t − N B − c  B c = E ( − aN ) + aN − 1 X i =0 D ( − i ) (2.5c) Notice that Θ ± are identified with the dressing matrices mapping the v acuum A v ac µ ( ϕ 0 ) to some non-trivial configuration, A µ ( ϕ ) . In fact the g-RHB decomposition (2.3) is the basis of the dressing metho d where non-trivial solutions are constructed from a sp ecific v acuum configuration [16 – 18]. An imp ortan t ingredient here is the construction of vertex op erators whic h corresp ond to eigenv alues and eigenstates of the Heisenberg algebra denoted by ϵ ( aN ) enco ded within the generalized Baker- Akhiezer function (1.2), and henceforth dep end up on the v acuum through the v acuum parameters ϕ 0 . On the other hand, equations (2.5a)-(2.5c) naturally generalizes to the idea of connecting tw o distinct configurations b y gauge transformation, i.e., A µ ( ϕ ) = U − 1 A µ ( ψ ) U − ∂ µ U U − 1 , ( µ = x or t ± N ) , (2.6) where U ( ϕ, ψ ) that dep ends of field configurations and eqn. (2.6) generate the gauge-Bäcklund tr ansformation . In fact, this is the key idea in constructing Bäc klund as gauge transformation acting on the tw o dimensional p otentials suc h that the zero curv ature and therefore, the equations of motion remain unc hanged. It is important to note that Bäcklund transformation connects t wo distinct solutions of the same equation. In particular, eqn. (2.5a)-(2.5c) represen t the case where ψ denotes the v acuum configuration. Suc h framew ork w as prop osed and emplo yed to describe integrable defects in the sense that the t wo solutions are in terp olated b y a defect [19 – 21]. 3 Lax pair for the Chen-Lee-Liu (CLL) flo ws Consider the loop-algebra L ( G ) = { h ( n ) , E ( n ) α , E ( n ) − α } , endo wed with the principal gradation, see A. The grading op erator Q p = 1 2 h (0) + 2 ˆ d decomp oses the algebra L ( G ) = P i ∈ Z G i in to graded 4 subspaces: G 2 m = n h ( m ) o , G 2 m +1 = n E ( m ) α , E ( m +1) − α o , n, m ∈ Z , of grade 2 m and 2 m + 1 , resp ectiv ely . A second decomp osition of L ( G ) into Kernel K and its complemen t M : K = n h ( n ) o , M = n E ( n ) α , E ( n ) − α o , is generated b y a constan t, grade tw o generator, E (2) = 1 2 h (1) ∈ G 2 . The kernel and its complemen t satisfy the follo wing relations: [ K , K ] ⊂ K , [ K , M ] ⊂ M , [ M , M ] ⊂ K . The abov e algebraic structure underlies the Chen-Lee-Liu (CLL) hierarch y , whic h can b e deriv ed from the spatial Lax op erator with a = 2 and c = 1 2 in (2.5a), [6, 22 – 24]: A x = E (2) + r E (0) α + sE (1) − α − 1 2 r sh (0) , (3.1) where r = r ( x, t ± N ) and s = s ( x, t ± N ) are fields of the theory asso ciated to p ositiv e (or negativ e) flo ws t N (or t − N ), with N ∈ N ∗ . The flo w equations asso ciated to the Lax op erator (3.1) are obtained by solving the zero curv ature equation 2  ∂ x + A x , ∂ t ± N + A t ± N  = ∂ x A t ± N − ∂ t ± N A x +  A x , A t ± N  = 0 , (3.2) where A t ± N is the temp oral Lax p otential asso ciated to a giv en t ± N . F or p ositiv e and negative sub-hierarc hies their structure is resp ectively giv en b y A t N = E (2 N ) + 2 N − 1 X i =0 D ( i ) , (3.3) and A t − N = E ( − 2 N ) + 2 N − 1 X i =0 D ( − i ) , (3.4) where E ( ± 2 N ) = 1 2 h ( ± N ) , D (2 j ) = a 2 j, ± N h ( j ) , D (2 j +1) = b 2 j +1 , ± N E ( j ) α + c 2 j +1 , ± N E ( j +1) − α and j ∈ Z , a 2 j, ± N , b 2 j +1 , ± N and c 2 j +1 , ± N are functions of x and t ± N , to b e determined. The flow equations are therefore obtained b y solving (3.2) for either (3.3) or (3.4). 3.1 P ositiv e flo ws F or the p ositiv e sub-hierarch y we find from (3.1) and (3.3) in the zero curv ature equation (3.2), h ∂ x + E (2) + A 1 + A 0 , ∂ t N + E (2 N ) + D (2 N − 1) + D (2 N − 2) + · · · + D (1) + D (0) i = 0 , (3.5) 2 In the case of flow t 1 , the solution is trivial: A t 1 = A x . 5 where E (2) = 1 2 h (1) , A 1 = rE (0) α + sE (1) − α and A 0 = − 1 2 r sh (0) . The general structure of L ( G ) decomp oses (3.2) in to graded subspaces, i.e., h E (2) , E (2 N ) i = 0 , h A 1 , E (2 N ) i + h E (2) , D (2 N − 1) i = 0 , h A 0 , E (2 N ) i + h A 1 , D (2 N − 1) i + h E (2) , D (2 N − 2) i = 0 , ∂ x D (2 N − 1) + h A 0 , D (2 N − 1) i + h A 1 , D (2 N − 2) i + h E (2) , D (2 N − 3) i = 0 , . . . ∂ x D (2) + h A 0 , D (2) i + h A 1 , D (1) i + h E (2) , D (0) i = 0 , ∂ t N A 1 − ∂ x D (1) − h A 0 , D (1) i − h A 1 , D (0) i = 0 , ∂ t N A 0 − ∂ x D (0) − h A 0 , D (0) i = 0 . (3.6) W e start solving from the highest grade equation, namely , 2 N + 1 , in order to determine the co efficien ts b 2 N − 1 ,N and c 2 N − 1 ,N in terms of fields r e s . W e solv e recursiv ely all equations un til the grade one comp onen t obtaining in this pro cess the equations of motion: ∂ t N r = ∂ x b 1 ,N − r (2 a 0 ,N + b 1 ,N s ) , ∂ t N s = ∂ x c 1 ,N + s (2 a 0 ,N + c 1 ,N r ) . (3.7) Solving (3.6) for the first few flo ws we find for N = 2 : A t 2 = 1 2 h (2) + r E (1) α + sE (2) − α − r sh (1) −  r 2 s + ∂ x r  E (0) α +  − r s 2 + ∂ x s  E (1) − α + + 1 2  r 2 s 2 − r ∂ x s + s∂ x r  h (0) . (3.8) Rep eating the pro cedure for N = 3 yields: A t 3 = 1 2 h (3) + r E (2) α + sE (3) − α − r sh (2) −  r 2 s + ∂ x r  E (1) α +  − r s 2 + ∂ x s  E (2) − α + +  r 2 s 2 − r ∂ x s + s∂ x r  h (1) +  r  r 2 s 2 + 3 s∂ x r − r ∂ x s  + ∂ 2 x r  E (0) α + +  s  r 2 s 2 − 3 r ∂ x s + s∂ x r  + ∂ 2 x s  E (1) − α + − 1 2  r 3 s 3 − ( ∂ x r ) ( ∂ x s ) + r  − 3 r s∂ x s + ∂ 2 x s  + s  3 r s∂ x r + ∂ 2 x r  h (0) . 6 and for N = 4 : A t 4 = 1 2 h (4) + r E (3) α + sE (4) − α − r sh (3) −  r 2 s + ∂ x r  E (2) α +  − r s 2 + ∂ x s  E (1) − α + +  r 2 s 2 − r ∂ x s + s∂ x r  h (2) +  r  r 2 s 2 + 3 s∂ x r − r ∂ x s  + ∂ 2 x r  E (1) α + +  s  r 2 s 2 − 3 r ∂ x s + s∂ x r  + ∂ 2 x s  E (2) − α + −  r 3 s 3 − ( ∂ x r ) ( ∂ x s ) + r  − 3 r s∂ x s + ∂ 2 x s  + s  3 r s∂ x r + ∂ 2 x r  h (1) + − n r h r 3 s 3 − ( ∂ x r ) ( ∂ x s ) + r  − 3 r s∂ x s + ∂ 2 x s  + 2 s  3 r s∂ x r + 2 ∂ 2 x r  i + 3 s ( ∂ x r ) 2 + ∂ 3 x r o E (0) α + + n − s h r 3 s 3 − ( ∂ x r ) ( ∂ x s ) + s  3 r s∂ x r + ∂ 2 x r  + 2 r  − 3 r s∂ x s + 2 ∂ 2 x s  i − 3 r ( ∂ x s ) 2 + ∂ 3 x s o E (1) − α + + 1 2 n r 4 s 4 − 4 r s ( ∂ x r ) ( ∂ x s ) + ( ∂ x r )  ∂ 2 x s  − ( ∂ x s )  ∂ 2 x r  + − r h 6 r 2 s 2 ∂ x s − 3 r ( ∂ x s ) 2 + 4 r s∂ 2 x s + ∂ 3 x s i + s h 6 r 2 s 2 ∂ x r + 3 s ( ∂ x r ) 2 − 4 r s∂ 2 x r + ∂ 3 x r io h (0) . leading resp ectiv ely to the following time ev olution equations, ∂ t 2 r = − ∂ 2 x r − 2 r s∂ x r , ∂ t 2 s = ∂ 2 x s − 2 r s∂ x s, (3.9) ∂ t 3 r = ∂ 3 x r + 3 r 2 s 2 ∂ x r + 3 s∂ x ( r ∂ x r ) , ∂ t 3 s = ∂ 3 x s + 3 r 2 s 2 ∂ x s − 3 r ∂ x ( s∂ x s ) , (3.10) ∂ t 4 r = − ∂ 4 x r − 4 r 3 s 3 ∂ x r − 6 s 2 ∂ x  r 2 ∂ x r  − 4 s∂ x  r ∂ 2 x r  − 6 s ( ∂ x r )  ∂ 2 x r  − 2 ∂ x [ r ( ∂ x r ) ( ∂ x s )] , ∂ t 4 s = ∂ 4 x s − 4 r 3 s 3 ∂ x s + 6 r 2 ∂ x  s 2 ∂ x s  − 4 r ∂ x  s∂ 2 x s  − 6 r ( ∂ x s )  ∂ 2 x s  − 2 ∂ x [ s ( ∂ x r ) ( ∂ x s )] . (3.11) The ab ov e equations of motion admit tw o classes of v acuum solutions, i ) zero v acuum, i.e. r = s = 0 and ii ) strictly nonzero constant v acuum, r = r 0  = 0 , s = s 0  = 0 solutions. Consider now the general v acuum configuration for the Lax operators A v ac t 1 = Σ (2) , A v ac t 2 = Σ (4) − br 0 s 0 Σ (2) , A v ac t 3 = Σ (6) − br 0 s 0 Σ (4) + br 2 0 s 2 0 Σ (2) , A v ac t 4 = Σ (8) − br 0 s 0 Σ (6) + br 2 0 s 2 0 Σ (4) − br 3 0 s 3 0 Σ (2) , (3.12) where Σ (2 N ) ≡ 1 2  h ( N ) − br 0 s 0 h ( N − 1)  + br 0 E ( N − 1) α + bs 0 E ( N ) − α . (3.13) and the parameter b is used to classify the t wo classes of v acua namely , b = 0 for zero v acuum and b = 1 for nonzero constan t v acuum solutions 3 . It therefore follows that  Σ (2) , Σ (2 N )  = 0 and henceforth, h Σ (2 M ) , Σ (2 N ) i = 0 , M , N = 1 , 2 , · · · for either b = 0 or b = 1 . 3 F or b=1, mixed v acuum configurations ( r, s ) = ( r 0 , 0) and ( r , s ) = (0 , s 0 ) can also b e considered. 7 3.2 Negativ e flo ws In order to construct the negativ e flo ws w e insert the Lax pair from equations (3.1) and (3.4) in to the zero curv ature equation (3.2), h ∂ x + E (2) + A 1 + A 0 , ∂ t − N + E ( − 2 N ) + D ( − 2 N +1) + D ( − 2 N +2) + · · · + D ( − 1) + D (0) i = 0 (3.14) and decomp ose it in to the graded subspaces, h A 0 , E ( − 2 N ) i = 0 , ∂ x D ( − 2 N +1) + h A 1 , E ( − 2 N ) i + h A 0 , D ( − 2 N +1) i = 0 , ∂ x D ( − 2 N +2) + h E (2) , E ( − 2 N ) i + h A 1 , D ( − 2 N +1) i + h A 0 , D ( − 2 N +2) i = 0 , . . . ∂ x D ( − 1) + h E (2) , D ( − 3) i + h A 1 , D ( − 2) i + h A 0 , D ( − 1) i = 0 , ∂ t − N A 0 − ∂ x D (0) − h E (2) , D ( − 2) i − h A 1 , D ( − 1) i + h A 0 , D (0) i = 0 , ∂ t − N A 1 − h E (2) , D ( − 1) i − h A 1 , D (0) i = 0 , h E (2) , D (0) i = 0 . (3.15) The lo west grade comp onent − 2 N + 1 now determines the co efficients b − 2 N +1 ,N and c − 2 N +1 ,N in terms of fields r and s . The procedure follows recursively until w e reac h the time ev olution equation, ∂ t − N r = b − 1 , − N − 2 a 0 , − N r , ∂ t − N s = − c − 1 , − N + 2 a 0 , − N s. (3.16) Solving for the first few flo ws, we find in terms of new v ariables, R ≡ ∂ − 1 x  r e − J  , S ≡ ∂ − 1 x  se J  , J = ∂ − 1 x ( r s ) . (3.17) where w e employ ed more compact notation of ∂ − 1 x f = R x f ( y ) dy . W e therefore find, for N = 1 : A t − 1 = 1 2 h ( − 1) + e J RE ( − 1) α − e − J S E (0) − α + 1 2 RS h (0) . (3.18) F or N = 2 w e find: A t − 2 = 1 2 h ( − 2) + e J RE ( − 2) α − e − J S E ( − 1) − α + RS h ( − 1) − e J ∂ − 1 x ( R − 2 RS ∂ x R ) E ( − 1) α + − e − J ∂ − 1 x ( S + 2 R S ∂ x S ) E (0) − α − 1 2 h ( RS ) 2 − R∂ − 1 x ( S + 2 R S ∂ x s ) + S ∂ − 1 x ( R − 2 RS ∂ x R ) i h (0) , (3.19) 8 and for N = 3 : A t − 3 = 1 2 h ( − 3) + e J RE ( − 3) α − e − J S E ( − 2) − α + RS h ( − 2) − e J ∂ − 1 x ( R − 2 RS ∂ x R ) E ( − 2) α + − e − J ∂ − 1 x ( S + 2 R S ∂ x S ) E ( − 1) − α − h ( RS ) 2 − R∂ − 1 x ( S + 2 R S ∂ x S ) + S ∂ − 1 x ( R − 2 RS ∂ x R ) i h ( − 1) + + e J ∂ − 1 x n (1 − 2 S ∂ x R ) ∂ − 1 x ( R − 2 RS ∂ x R ) − 2 ∂ x R h ( RS ) 2 − R∂ − 1 x ( S + 2 R S ∂ x S ) io E ( − 1) α + − e − J ∂ − 1 x n (1 + 2 R∂ x S ) ∂ − 1 x ( S + 2 R S ∂ x S ) − 2 ∂ x S h ( RS ) 2 + S ∂ − 1 x ( R − 2 RS ∂ x R ) io E (0) − α + − 1 2 n − 2 ( RS ) 3 + ∂ − 1 x ( R − 2 RS ∂ x R ) ∂ − 1 x ( S + 2 R S ∂ x S ) + − 2 RS  R∂ − 1 x ( S + 2 R S ∂ x S ) + S ∂ − 1 x ( R − 2 RS ∂ x R )  + − R∂ − 1 x h (1 − 2 S ∂ x R ) ∂ − 1 x ( R − 2 RS ∂ x R ) − 2 ( RS ) 2 ∂ x R + 2 R ( ∂ x R ) ∂ − 1 x ( S + 2 R S ∂ x S ) i + − S ∂ − 1 x h (1 − 2 S ∂ x R ) ∂ − 1 x ( R − 2 RS ∂ x R ) − 2 ( RS ) 2 ∂ x R + 2 R ( ∂ x R ) ∂ − 1 x ( S + 2 R S ∂ x S ) io h (0) . yielding resp ectiv ely the following time ev olution equations, ∂ t − 1 r = Re J − r RS, (3.20) ∂ t − 1 s = S e − J + sRS, (3.21) ∂ t − 2 r = − e J ∂ − 1 x ( R − 2 RS ∂ x R ) + r h ( RS ) 2 − R∂ − 1 x ( S + 2 R S ∂ x S ) + S ∂ − 1 x ( R − 2 RS ∂ x R ) i , (3.22) ∂ t − 2 s = e − J ∂ − 1 x ( S + 2 R S ∂ x S ) − s h ( RS ) 2 − R∂ − 1 x ( S + 2 R S ∂ x S ) + S ∂ − 1 x ( R − 2 RS ∂ x R ) i , (3.23) ∂ t − 3 r = e J ∂ − 1 x n (1 − 2 S ∂ x R ) ∂ − 1 x ( R − 2 RS ∂ x R ) − 2 ∂ x R h ( RS ) 2 − R∂ − 1 x ( S + 2 R S ∂ x S ) io + (3.24) + r n − 2 ( RS ) 3 + ∂ − 1 x ( R − 2 RS ∂ x R ) ∂ − 1 x ( S + 2 R S ∂ x S ) + − 2 RS  R∂ − 1 x ( S + 2 R S ∂ x S ) + S ∂ − 1 x ( R − 2 RS ∂ x R )  + − R∂ − 1 x h (1 − 2 S ∂ x R ) ∂ − 1 x ( R − 2 RS ∂ x R ) − 2 ( RS ) 2 ∂ x R + 2 R ( ∂ x R ) ∂ − 1 x ( S + 2 R S ∂ x S ) i + − S ∂ − 1 x h (1 − 2 S ∂ x R ) ∂ − 1 x ( R − 2 RS ∂ x R ) − 2 ( RS ) 2 ∂ x R + 2 R ( ∂ x R ) ∂ − 1 x ( S + 2 R S ∂ x S ) io , ∂ t − 3 s = e − J ∂ − 1 x n (1 + 2 R∂ x S ) ∂ − 1 x ( S + 2 R S ∂ x S ) − 2 ∂ x S h ( RS ) 2 + S ∂ − 1 x ( R − 2 RS ∂ x R ) io + (3.25) − s n − 2 ( RS ) 3 + ∂ − 1 x ( R − 2 RS ∂ x R ) ∂ − 1 x ( S + 2 R S ∂ x S ) + − 2 RS  R∂ − 1 x ( S + 2 R S ∂ x S ) + S ∂ − 1 x ( R − 2 RS ∂ x R )  + − R∂ − 1 x h (1 − 2 S ∂ x R ) ∂ − 1 x ( R − 2 RS ∂ x R ) − 2 ( RS ) 2 ∂ x R + 2 R ( ∂ x R ) ∂ − 1 x ( S + 2 R S ∂ x S ) i + − S ∂ − 1 x h (1 − 2 S ∂ x R ) ∂ − 1 x ( R − 2 RS ∂ x R ) − 2 ( RS ) 2 ∂ x R + 2 R ( ∂ x R ) ∂ − 1 x ( S + 2 R S ∂ x S ) io . 9 Notice that all the ab ov e equations admit b oth zero v acuum ( r = 0 , s = 0 ) or nonzero constan t v acuum solutions, ( r = r 0 , s = s 0 ). F or b oth cases w e define the v acuum configuration Lax op erators for the negative sub-hierarc hy . Considering the limits ( r → 0 , s → 0 , R → 0 , S → 0 ) or ( r → r 0 , s → s 0 , R → − 1 s 0 e − r 0 s 0 x , S → 1 r 0 e r 0 s 0 x ) w e find 4 , A v ac t − 1 = Υ ( − 2) , A v ac t − 2 = Υ ( − 4) − b r 0 s 0 Υ ( − 2) , A v ac t − 3 = Υ ( − 6) − b r 0 s 0 Υ ( − 4) + b r 2 0 s 2 0 Υ ( − 2) , (3.26) where Υ ( − 2 N ) ≡ 1 2  h ( − N ) − b r 0 s 0 h ( − N +1)  − b s 0 E ( − N ) α − b r 0 E ( − N +1) − α . (3.27) satisfying the cen terless Heisenberg algebra h Υ ( − 2 M ) , Υ ( − 2 N ) i = 0 , M , N = 1 , 2 , . . . (3.28) for b = 0 and b = 1 . Notice that the zero v acuum limit is obtained by taking b = 0 in the relations (3.26) and (3.27). 4 CLL Reductions Sev eral in teresting reductions can b e obtained from CLL hierarc h y by making use of zero and constan t nonzero v acuum solutions (see T able 1). 4 W e should point out that h Σ (2) , Υ ( − 2 N ) i = 0 for either b = 0 or b = 1 . 10 Limit Field Flo ws r → 0 , s → ϕ ϕ = ϕ ( x, t ± N ) ∂ t N ϕ = ∂ N x ϕ ∂ t − N ϕ = ∂ − N x ϕ r → ψ , s → 0 ψ = ψ ( x, t ± N ) ∂ t N ψ = ( − 1) N +1 ∂ N x ψ ∂ t − N ψ = ( − 1) N +1 ∂ − N x ψ r → r 0 , s → w w = w ( x, t ± N ) ∂ t N w = − 1 r 0 ∂ x h e r 0 ∂ − 1 x w  ∂ N x e − r 0 ∂ − 1 x w i ∂ t − N w = 1 r 0 ∂ x h e r 0 ∂ − 1 x w  ∂ − N x e − r 0 ∂ − 1 x w i r → u, s → s 0 u = u ( x, t ± N ) ∂ t N u = 1 s 0 ( − 1) N +1 ∂ x h e − s 0 ∂ − 1 x u  ∂ N x e s 0 ∂ − 1 x u i ∂ t − N u = − 1 s 0 ( − 1) N +1 ∂ x h e − s 0 ∂ − 1 x u  ∂ − N x e s 0 ∂ − 1 x u i T able 1: Immediate reductions of the CLL hierarch y: the limits ( r → 0 , s = ϕ ) or ( r → ψ , s = 0) yield the heat equation for ϕ (or ψ ), while ( r → r 0 , s = w ) or ( r = u, s → s 0 ) with fixed nonzero constan ts r 0 and s 0 lead to the Burgers equation. The factor ( − 1) N +1 can b e absorb ed through t ± N → t ′ ± N = t ± N / ( − 1) N +1 , while r 0 and s 0 can b e remov ed by the rescaling w → r 0 w and u → s 0 u , showing that the mo dels for ϕ and ψ are equiv alen t, as are those for w and u when u = − w . 4.1 Burgers hierarch y Considering the CLL hierarc hy with one of the fields constrained to a constant (say r = r 0 , see T able 1) we obtain from (3.7) the p ositive Burgers hierarc hy (4.1) with the p ositiv e fluxes of the Burgers hierarc hy written in a compact closed form [25]: ∂ t ′ N w = α N ∂ x ( ∂ x − r 0 w ) N − 1 w , (4.1) where t ′ N = α N t N and α N is an arbitrary constan t. Explicitly , the first few flows can b e identified to the Burgers equation for t = t 2 , originally derived b y Bateman in 1915 [26] and later p opularized p or Burgers [27], ∂ t ′ 2 w = α 2  ∂ 2 x w − 2 r 0 w ∂ x w  , (4.2) and the Sharma–T asso–Olv er, derived in [28, 29], ∂ t ′ 3 w = α 3  ∂ 3 x w + 3 r 2 0 w 2 ∂ x w − 3 r 0 ∂ x ( w ∂ x w )  . (4.3) Moreo ver the same limiting pro cedure in (3.16) yields, in a closed form a new sub-hierarc hy whic h w e are dubbing ne gative Bur gers hier ar chy . The p ositiv e and negative Burgers sub-hierarc hies are giv en in the closed form as, ∂ t ′ N w = − α N r 0 ∂ x h e r 0 ∂ − 1 x w  ∂ N x e − r 0 ∂ − 1 x w i , (4.4) and ∂ t ′ − N w = α − N r 0 ∂ x h e r 0 ∂ − 1 x w  ∂ − N x e − r 0 ∂ − 1 x w i , (4.5) 11 where α N is an arbitrary constant. Both cases only admit nonzer o c onstant vacuum solutions, w = w 0  = 0 . Explicitly , the first tw o flo w equations for the negative sub-hierarc h y are: ∂ t ′ − 1 w = α − 1 r 0  1 + r 0 w e r 0 ∂ − 1 x w ∂ − 1 x e − r 0 ∂ − 1 x w  , (4.6) ∂ t ′ − 2 w = α − 2 r 0 e r 0 ∂ − 1 x w  ∂ − 1 x e − r 0 ∂ − 1 x w + r 0 w ∂ − 2 x e − r 0 ∂ − 1 x w  . (4.7) Eqn. (4.6) can b e re-written in a lo cal form as, ∂ t ′ − 1 ∂ x w = ∂ x w w  ∂ t ′ − 1 w − α − 1 r 0  + r 0 w ∂ t ′ − 1 w . (4.8) 5 The dressing metho d and tau functions for CLL In this section we emplo y the Dressing method [16 – 18] in order to generate systematically the soliton solutions for the en tire (p ositiv e and negative flows) CLL hierarc h y . The metho d relies up on a particular v acuum solution which could be c hosen to b e zero or constan t nonzero v acuum solution. The metho d in v olves the construction of v ertex operators from the Heisenberg operators describing the v arious v acuum configurations for the t wo dimensional gauge potentials (3.12)-(3.13) or (3.26)- (3.27). Their eigen v alues defines their space-time dependence. In fact we shall see that there will b e tw o types of vertices related to eigen v alues of opp osite signs. The class A is constructed out of pro ducts of the same v ertex and class B constructed out of products of opp osite sign vertices [6]. F or the CLL hierarc hy with zero v acuum solutions only class B allows non-trivial solutions. F or nonzero v acuum, b oth cases allow non-trivial soliton solutions and class A leads to the Burgers solutions. 5.1 Dressing transformation In order to employ the dressing metho d to generate soliton solutions we shall upgrade the affine algebra to include central terms. This is necessary to ensure highest w eigh t states. This implies the follo wing mo dification A x → A x − 1 2 ( ∂ x ν ) ˆ c, A t ± N → A t ± N − 1 2  ∂ t ± N ν  ˆ c, (5.1) where ν = ν ( x, t ± N ) is an extra field that v anishes in v acuum limit and ˆ c commutes with all generators of ˆ G . The Lax op erators for the CLL hierarc hy in v acuum, can b e written as A v ac t N = Σ (2 N ) + b N − 1 X i =1 ( − r 0 s 0 ) i Σ (2 N − 2 i ) , (5.2) A v ac t − N = Υ ( − 2 N ) + b N − 1 X i =1 ( − r 0 s 0 ) − i Υ ( − 2 N +2 i ) . (5.3) where b = 0 for zero v acuum and b = 1 for the constan t nonzero v acuum. W e consider the g-RHB decomp osition prop osed in [6] Θ − 1 − ( t ) Θ + ( t ) = Ψ a ( t ) g Ψ − 1 a ( t ) , (5.4) 12 where, Ψ is the generalized Bak er-Akhiezer function (1.2) with a = 2 , Ψ = exp " − ∞ X N =1  A v ac t N t N + A v ac t − N t − N  # , (5.5) and g = e Y , with Y ∈ ˆ G is arbitrary and constan t Lie algebra v alued ob ject. The left-hand side of (5.4) can b e in general written as 5 Θ + = e 1 2 θ (0) ∞ Y i =1 e θ ( i ) , Θ − = e − 1 2 θ (0) ∞ Y i =1 e − θ ( − i ) , where θ ( j ) ∈ ˆ G j . In particular, 6 θ (2 k ) = φ 2 k h ( k ) + δ k, 0 ν ˆ c, θ (2 k +1) = χ 2 k +1 E ( k ) α + ψ 2 k +1 E ( k +1) − α , k ∈ Z The co efficien ts ν , φ 2 k , χ 2 k +1 , e ψ 2 k +1 , kno wn as auxiliary fields are functionals of x and t ± N . The dressing operators Θ + and Θ − gauge transform the Lax op erators A v ac x = − ( ∂ x Ψ) Ψ − 1 and A v ac t ± N −  ∂ t ± N Ψ  Ψ − 1 in to its non-trivial configuration A x and A t ± N , i.e., A x = Θ ± A v ac x Θ − 1 ± − ( ∂ x Θ ± ) Θ − 1 ± = − [ ∂ x (Θ ± Ψ)] (Θ ± Ψ) − 1 , (5.6a) A t ± N = Θ ± A v ac t ± N Θ − 1 ± −  ∂ t ± N Θ ±  Θ − 1 ± = −  ∂ t ± N (Θ ± Ψ)  (Θ ± Ψ) − 1 . (5.6b) Solving eqns. (5.6a) and (5.6b) recursiv ely we determine the auxiliary fields θ ( ± i ) in terms of the ph ysical fields r ( x, t ± N ) and s ( x, t ± N ) defined in (3.1). Decomp osing (5.6a) using Θ + w e obtain from zero grade pro jection, ∂ x φ 0 = r s − br 0 s 0 . (5.7) Grade one pro jection yields, ∂ x χ 1 − br 0 s 0 χ 1 = − r e − φ 0 + br 0 , ∂ x ψ 1 + br 0 s 0 ψ 1 = − se φ 0 + bs 0 . and so on in order to determine higher order coefficients in Θ + . F or transformation Θ − , w e find from (5.6a), χ − 1 = r e φ 0 − br 0 , ψ − 1 = − se − φ 0 + bs 0 . (5.8) together with φ − 2 = − ν x − 1 2 ( r e φ 0 − br 0 )  se − φ 0 − bs 0  − bs 0 ( r e φ 0 − br 0 ) . (5.9) and so on un til Θ − is determined. Con versely , eqns. (5.8) allo w determining fields r and s in terms of φ 0 , χ − 1 e ψ − 1 , r = ( br 0 + χ − 1 ) e − φ 0 , s = ( bs 0 − ψ − 1 ) e φ 0 . (5.10) 5 In general w e may consider an asymmetric splitting of the zero grade comp onent θ (0) , i.e., Θ + = e (1 − c ) θ (0) Q ∞ i =1 e θ ( i ) , Θ − = e − cθ (0) Q ∞ i =1 e − θ ( − i ) . Here we consider c = 1 / 2 . 6 The term δ k, 0 denotes the Kronec ker delta. 13 5.2 T au functions In order to determine soliton solutions within the dressing metho d we introduce the τ − functions defined as τ kl ≡ ⟨ λ k | Θ − 1 − Θ + | λ l ⟩ = ⟨ λ k | Ψ g Ψ − 1 | λ l ⟩ , k , l = 0 , 1 , 2 , 3 , (5.11) where the states | λ k ⟩ and | λ l ⟩ are defined as | λ 0 ⟩ = | µ 0 ⟩ , | λ 1 ⟩ = | µ 1 ⟩ , | λ 2 ⟩ = E ( − 1) α | µ 0 ⟩ , | λ 3 ⟩ = E (0) − α | µ 1 ⟩ , with | µ 0 ⟩ and | µ 1 ⟩ b eing the highest w eight states of ˆ A 1 . F rom the left-hand-side of (5.11), w e can define, τ 00 = ⟨ µ 0 | · · · e θ ( − 1) e θ (0) e θ (1) · · · | µ 0 ⟩ = ⟨ µ 0 | e ν ˆ c | µ 0 ⟩ = e ν , τ 11 = ⟨ µ 1 | · · · e θ ( − 1) e θ (0) e θ (1) · · · | µ 1 ⟩ = ⟨ µ 1 | e φ 0 h (0) + ν ˆ c | µ 1 ⟩ = e φ 0 + ν , τ 20 = ⟨ µ 0 | E 1 − α · · · e θ ( − 1) e θ (0) e θ (1) · · · | µ 0 ⟩ = − e ν ⟨ µ 0 | h θ ( − 1) , E (1) − α i | µ 0 ⟩ = χ − 1 e ν , τ 31 = ⟨ µ 0 | E 0 α · · · e θ ( − 1) e θ (0) · · · | µ 1 ⟩ = − e φ 0 + ν ⟨ µ 1 | h θ ( − 1) , E (0) α i | µ 1 ⟩ = ψ − 1 e φ 0 + ν , The follo wing relations follow straigh tforw ardly , e ν = τ 00 , e φ 0 + ν = τ 11 , ψ − 1 = τ 31 τ 11 χ − 1 = τ 20 τ 00 . (5.12) Substituting these v alues in to (5.10), we find fields r and s in terms of the τ -functions τ 00 , τ 11 , τ 20 , and τ 31 , r = br 0 τ 00 + τ 20 τ 11 , s = bs 0 τ 11 − τ 31 τ 00 . (5.13) 5.3 V ertex op erators An imp ortant ingredien t in constructing and classifying solutions are the v ertex op erators. These are the eigenstates of the Heisen b erg sub-algebras whose eigenv alues lead to the space-time dep endence of the solitons for the en tire hierarch y , i.e.,  V ± i , A v ac x  = ± κ x V ± i , h V ± i , A v ac t ± N i = ± ω ± N V ± i . (5.14) It can b e c heck ed that 7 V + i ≡ V + ( k i ) = − br 0 ˆ c + ∞ X j = −∞  br 0 k − j i h ( j ) + br 2 0 k − j i E ( j − 1) α − k − j +1 i E ( j ) − α  , (5.15) V − i ≡ V − ( k i ) = ∞ X j = −∞  bs 0 k − j i h ( j ) − k − j +1 i E ( j − 1) α + bs 2 0 k − j i E ( j ) − α  , (5.16) where k i is a complex parameter, with i ∈ Z , that satisfy (5.14) with κ x = k i + br 0 s 0 , ω N = k N i − b ( − r 0 s 0 ) N , ω − N = (1 − 2 b ) k − N i + b ( − r 0 s 0 ) − N . (5.17) 7 Notice that for b = 1 these corresp ond to deformed v ertex op erators depending up on parameters r 0 and s 0 . 14 It therefore follo ws that Ψ V ± i Ψ − 1 = V ± i + h V ± i , A v ac x x + A v ac t ± N t ± N i + 1 2 hh V ± i , A v ac x x + A v ac t ± N t ± N i , A v ac x x + A v ac t ± N t ± N i + · · · , (5.18) and Ψ V ± i Ψ − 1 = ρ i V ± i , ρ i = ρ i ( x, t ± N ) = e κ x x + ω ± N t ± N . (5.19) Using the iden tity Ψ − 1 Ψ = 1 , it follo ws that Ψ  V ± i  n Ψ − 1 =  Ψ V ± i Ψ − 1  n (5.20) and hence, Ψ e V ± i Ψ − 1 = exp  ρ i V ± i  . (5.21) The τ functions (5.11) can b e exactly ev aluated b y choosing g = n Q i =1 e V ± i . 5.4 Class A and solitons for Burgers hierarch y Assuming g = n Q i =1 e V ± i , we obtain a class of solutions in volving pro ducts of single vertices, either V + i or V − i , τ kl = ⟨ λ k | n Y i =1  1 + ρ i V + i + ρ 2 i  V + i  2 + · · ·  | λ l ⟩ . (5.22) Ev aluating the τ − functions τ 00 , τ 11 , τ 20 , and τ 31 , τ 00 = 1 − br 0 n X i =1 ρ i , τ 11 = 1 , τ 20 = br 2 0 n X i =1 ρ i , τ 31 = − n X i =1 k i ρ i . (5.23) Substituting in (5.13) w e find for general v alues of n, r = br 0 , s = bs 0 + n X i =1 k i ρ i 1 − br 0 n X i =1 ρ i . (5.24) F or the particular case where b = 0 , w e find the trivial wa ve solution for the asso ciated heat equation for field ϕ (see table 1), r → 0 , s → ϕ = n X i =1 k i exp n k i x + ( k i ) ± N t ± N o , (5.25) F or b = 1 r → r 0 , s → w = s 0 + n X i =1 k i exp n ( k i + r 0 s 0 ) x ± h ( k i ) ± N − ( − r 0 s 0 ) ± N i t ± N o 1 − r 0 n X i =1 exp n ( k i + r 0 s 0 ) x ± h ( k i ) ± N − ( − r 0 s 0 ) ± N i t ± N o , (5.26) 15 w e find w in (5.26) to solve the Burgers hierarch y . Re-writing the n -solitons solution as w = − ( r 0 ) − 1 ∂ x Φ Φ , Φ = exp n − r 0 s 0 x ± α ± N ( − r 0 s 0 ) ± N t ′ ± N o − r 0 n X i =1 exp n k i x ± α ± N ( k i ) ± N t ′ ± N o , (5.27) w e can express w in terms of v ariable Φ = Φ ( x, t ± N ) satisfying ∂ t N Φ = α N ∂ N x Φ , ∂ t − N Φ = − α − N ∂ − N x Φ , (5.28) via the Cole-Hopf transformation. W e should point out that the Cole-Hopf transformation [30, 31], w as emplo yed to all p ositiv e flo ws of the Burgers hierarch y by Kudry ashov [25]. Later in [6] it w as extended to all negative sub-hierarc h y . In fact this was sho wn to be realized as a gauge transformation of Miura t yp e b etw een CLL and AKNS hierarchies. Exc hanging V + i for V − i in (5.22) w e find a similar result after exc hanging r → s and ρ i → − ρ − 1 i . 5.5 Class B and solitons for CLL hierarch y Let us no w consider pro ducts of mixed vertices, g = n Q i =1 e V + i e V − i +1 suc h that, τ kl = ⟨ λ k | n Y i =1  1 + ρ i V + i + ρ − 1 i +1 V − i +1 + ρ i ρ − 1 i +1 V + i V − i +1 + · · ·  | λ l ⟩ . (5.29) Ev aluating τ 00 , τ 11 , τ 20 , and τ 31 w e find for n = 1 , τ 00 = 1 − br 0 ρ 1 + k 2 ( k 1 + br 0 s 0 ) 2 ( k 2 − k 1 ) 2 ρ 1 ρ − 1 2 , (5.30a) τ 11 = 1 + bs 0 ρ − 1 2 + k 1 ( k 2 + br 0 s 0 ) 2 ( k 2 − k 1 ) 2 ρ 1 ρ − 1 2 , (5.30b) τ 20 = br 2 0 ρ 1 − k 2 ρ − 1 2 + br 0 k 2 ( k 1 + k 2 + 2 br 0 s 0 ) k 2 − k 1 ρ 1 ρ − 1 2 , (5.30c) τ 31 = − k 1 ρ 1 + bs 2 0 ρ − 1 2 + bs 0 k 1 ( k 1 + k 2 + 2 br 0 s 0 ) k 2 − k 1 ρ 1 ρ − 1 2 . (5.30d) Substituting these relations in (5.13), w e obtain the 2-soliton solution for the CLL hierarch y r = br 0 − k 2 ρ − 1 2 + br 0 k 2 ( k 2 + br 0 s 0 ) 2 ( k 2 − k 1 ) 2 ρ 1 ρ − 1 2 1 + bs 0 ρ − 1 2 + k 1 ( k 2 + br 0 s 0 ) 2 ( k 2 − k 1 ) 2 ρ 1 ρ − 1 2 , s = bs 0 + k 1 ρ 1 + bs 0 k 1 ( k 1 + br 0 s 0 ) 2 ( k 2 − k 1 ) 2 ρ 1 ρ − 1 2 1 − br 0 ρ 1 + k 2 ( k 1 + br 0 s 0 ) 2 ( k 2 − k 1 ) 2 ρ 1 ρ − 1 2 . (5.31) for b = 0 or b = 1 . Con versely , exchanging e V + 1 e V − 2 with e V − 1 e V + 2 , w e find another pair of solutions, r = br 0 − k 1 ρ − 1 1 + br 0 k 1 ( k 1 + br 0 s 0 ) 2 ( k 2 − k 1 ) 2 ρ − 1 1 ρ 2 1 + bs 0 ρ − 1 1 + k 2 ( k 1 + br 0 s 0 ) 2 ( k 2 − k 1 ) 2 ρ − 1 1 ρ 2 , s = bs 0 + k 2 ρ 2 + bs 0 k 2 ( k 2 + br 0 s 0 ) 2 ( k 2 − k 1 ) 2 ρ − 1 1 ρ 2 1 − br 0 ρ 2 + k 1 ( k 2 + br 0 s 0 ) 2 ( k 2 − k 1 ) 2 ρ − 1 1 ρ 2 . (5.32) 16 6 Gauge-Bäc klund transformation Bäc klund transformations play an important role in the construction and c haracterization of solu- tions in integrable systems. These transformations may b e obtained through sev eral formulations and techniques [32]. In this work, w e form ulate the Bäcklund transformations as gauge transforma- tions that preserve the zero curv ature as this prop erty ensures that the resulting relations extend to al l flows . This univ ersality arises from the fact that all flo ws of a given in tegrable hierarch y share the same underlying algebraic structure and p ossess a common Lax pair, whose spatial comp onent w e denote by A x . The sp ecial case in whic h it connects t w o configurations within the same equation of motion is referred to as auto-Bäc klund transformations and w e shall explore it for the CLL hierarc h y case in the present section. Within the algebraic framework, the Bäcklund transformation can b e represen ted b y a gauge transformation, since the zero-curv ature condition is gauge in v arian t and therefore flo ws equations are unchanged. Consider then the gauge-transformed Lax pair giv en by: A µ ( ψ ) = U ( ϕ, ψ , λ ) A µ ( ϕ ) U − 1 ( ϕ, ψ , λ ) + U ( ϕ, ψ , λ ) ∂ µ U − 1 ( ϕ, ψ , λ ) ( µ = x or t ± N ) . (6.1) A µ ( ϕ ) and A µ ( ψ ) are Lax pairs in different field configurations and U is a group element (expanded in terms of algebra elemen ts), that dep end on the field configurations and the sp ectral parameter λ . In a series of works, w e developed an approac h that uses the affine structure of the algebra to prop ose different graded ansatzes for the Bäc klund transformation. In [33, 34] we hav e shown that differen t graded ansatz are related to Type I and Type I I Bäcklund transformations for the sinh-Gordon hierarc hy [14, 20] and generalized this result to A r -mKdV hierarc hy . More recen tly , we ha ve extended this approac h to the negativ e sector of b oth mKdV [5]. Let us denote the differen t CLL configurations as follows: A µ ( ϕ ) ≡ A CLL µ ( r 1 , s 1 ) and A µ ( ψ ) ≡ A CLL µ ( r 2 , s 2 ) . (6.2) suc h that (6.1) b ecome A CLL µ ( r 2 , s 2 ) U − U A CLL µ ( r 1 , s 1 ) + ∂ µ U = 0 , with µ = x or t ± N , (6.3) Next, we propose an 2 × 2 matrix ansatz in order to implemen t the gauge–Bäc klund transformation b y using the graded structure present in the ˆ sl (2) affine algebra, as in [33]. T o accomplish this, we consider the follo wing 2 × 2 graded matrices U (2 n ) =    λ n a (2 n ) 1 , 1 0 0 λ n a (2 n ) 2 , 2    , U (2 n +1) =    0 λ n a (2 n +1) 1 , 2 λ n +1 a (2 n +1) 1 , 2 0    (6.4) where λ is the, previously introduced, sp ectral parameter, u i,j are functional of the fields r i , s i and the upp er index indicates the grade of the matrix. Then, for each Bäcklund transformation, we consider a differen t expansion given b y: Ansatz Bäc klund T ransformation U 0 = U (2 n ) B 0 U I = U (2 n ) + U (2 n +1) B I U I I = U (2 n ) + U (2 n +1) + U (2 n +2) B II 17 to enable us to solv e (6.3) for each U i , determining b oth a ij and the Bäcklund transformation. In the following section, we present this pro cedure for ansatz I I. W e shall see, therefore, that the most relev an t information in the gauge ansatz is the sum of successiv e graded subspaces, since differen t sums lead to differen t Bäcklund transformations. 6.1 Determining the transformation W e no w prop ose the ansatz I I for the gauge–Bäcklund transformation, which is the most general one and co vers the previous ansatz as w e tak e appropriated limits. F or particular choice n = − 1 , it tak es the form U II = U (0) + U ( − 1) + U ( − 2) =    a 1 , 1 + 1 λ b 1 , 1 1 λ a 1 , 2 a 2 , 1 a 2 , 2 + 1 λ b 2 , 2    , (6.5) where w e in tro duced a (0) ij = a i,j , a ( − 1) ij = a i,j and a ( − 2) ij = b i,j to simplify the notation. Substituting this ansatz in to (6.3) yields the following system of equations − a 1 , 1 r 1 + a 2 , 2 r 2 + a 1 , 2 = 0 , (6.6a) a 1 , 1 s 2 − a 2 , 2 s 1 − a 2 , 1 = 0 , (6.6b) ∂ x b 1 , 1 + 1 2 b 1 , 1 ∂ x ( J 1 − J 2 ) = 0 , (6.6c) ∂ x b 2 , 2 − 1 2 b 2 , 2 ∂ x ( J 1 − J 2 ) = 0 , (6.6d) ∂ x a 1 , 1 + 1 2 a 1 , 1 ∂ x ( J 1 − J 2 ) + a 2 , 1 r 2 − a 1 , 2 s 1 = 0 , (6.6e) ∂ x a 2 , 2 − 1 2 a 2 , 2 ∂ x ( J 1 − J 2 ) − a 2 , 1 r 1 + a 1 , 2 s 2 = 0 , (6.6f ) ∂ x a 1 , 2 − 1 2 a 1 , 2 ∂ x ( J 1 + J 2 ) − b 1 , 1 r 1 + b 2 , 2 r 2 = 0 , (6.6g) ∂ x a 2 , 1 + 1 2 a 2 , 1 ∂ x ( J 1 + J 2 ) − b 2 , 2 s 1 + a 1 , 1 s 2 = 0 . (6.6h) where ∂ x J i = r i s i with i = 1 , 2 . By direct in tegration of equations (6.6c)– (6.6d) we obtain b 1 , 1 = γ 1 e − 1 2 ( J 1 − J 2 ) and b 2 , 2 = γ 2 e 1 2 ( J 1 − J 2 ) . (6.7) F rom (6.6a)– (6.6b) w e can isolate a 1 , 2 = a 1 , 1 r 1 − a 2 , 2 r 2 , a 2 , 1 = a 1 , 1 s 2 − a 2 , 2 s 1 , (6.8) and after substituting (6.8) into (6.6e) and (6.6f), we obtain ∂ x a 1 , 1 − 1 2 ∂ x ( J 1 − J 2 ) = 0 , ⇒ a 1 , 1 = α 1 e 1 2 ( J 1 − J 2 ) , (6.9a) ∂ x a 2 , 2 + 1 2 ∂ x ( J 1 − J 2 ) = 0 , ⇒ a 2 , 2 = α 2 e − 1 2 ( J 1 − J 2 ) . (6.9b) Hence the functions a 1 , 2 and a 2 , 1 b ecome a 1 , 2 = α 1 e 1 2 ( J 1 − J 2 ) r 1 − α 2 e − 1 2 ( J 1 − J 2 ) r 2 , (6.10a) a 2 , 1 = α 1 e 1 2 ( J 1 − J 2 ) s 2 − α 2 e − 1 2 ( J 1 − J 2 ) s 1 . (6.10b) 18 Ha ving determined all en tries of the matrix asso ciated with ansatz II, the resulting gauge–Bäc klund transformation is tak es the form U II =     α 1 e 1 2 ( J 1 − J 2 ) + γ 1 e 1 2 ( J 2 − J 1 ) λ e − 1 2 ( J 1 + J 2 ) ( α 1 e J 1 r 1 − α 2 e J 2 r 2 ) λ e − 1 2 ( J 1 + J 2 )  α 1 e J 1 s 2 − α 2 e J 2 s 1  α 2 e 1 2 ( J 2 − J 1 ) + γ 2 e 1 2 ( J 1 − J 2 ) λ     , (6.11) whic h satisfies (6.3) provided that the follo wing differen tial relations hold: α 2 ∂ x  r 2 e − J 1  + γ 1 e − J 1 r 1 = α 1 ∂ x  r 1 e − J 2  + γ 2 e − J 2 r 2 , (6.12a) α 1 ∂ x  s 2 e J 1  − γ 2 e J 1 s 1 = α 2 ∂ x  s 1 e J 2  − γ 1 e J 2 s 2 . (6.12b) These relations constitute the type I I Bäcklund transformation or simply denoted as B II . They con tain spatial deriv ativ es as commonly occurs in similar transformations for other mo dels. The appropriate limits reduce the B II to the simplest cases, t yp e 0 ( B 0 ) and t yp e I ( B I ). 6.2 Reductions The equations (6.12) p ossesses t wo pairs of Bäc klund parameters, ( α 1 , α 2 ) and ( γ 1 , γ 2 ) . W e now analyze ho w Bäcklund transformations reduces under certain limits for these parameters. I In the limit α i → 0 , the transformations reduce to the simplest case, namely B 0 , suc h that (6.12) b ecomes γ 1 e − J 1 r 1 = γ 2 e − J 2 r 2 , ⇒ r 2 = γ 1 γ 2 e − J 1 + J 2 r 1 , (6.13a) γ 2 e J 1 s 1 = γ 1 e J 2 s 2 , ⇒ s 2 = γ 2 γ 1 e J 1 − J 2 s 1 . (6.13b) Hence, r 2 s 2 = r 1 s 1 ⇒ ∂ x J 1 = ∂ x J 2 ⇒ J 2 = J 1 + δ, (6.14) where δ is a constant. Thus the Bäcklund transformations on the fields configurations ma y b e written as r 2 = γ 1 γ 2 e δ r 1 , s 2 = γ 2 γ 1 e − δ s 1 , (6.15) that can b e in terpreted as a trivial scaling transformation. Indeed, previous w ork has shown that a zero order expansion alw ays leads to this trivial transformation. A ccordingly , the corresp onding gauge–Bäc klund transformation (6.11) reduces to U I I − → α i → 0 U 0 =  γ 1 λ e δ / 2 0 0 γ 2 λ e − δ / 2  . (6.16) I I On the other hand, the limit γ i → 0 leads to B I , and (6.12) b ecomes α 2 ∂ x  r 2 e − J 1  = α 1 ∂ x  r 1 e − J 2  , ⇒ α 2 r 2 e − J 1 = α 1 r 1 e − J 2 + β 1 (6.17a) α 1 ∂ x  s 2 e J 1  = α 2 ∂ x  s 1 e J 2  , ⇒ α 1 s 2 e J 1 = α 2 s 1 e J 2 + β 2 , (6.17b) 19 where β 1 , β 2 are constan ts of integration (Bäc klund parameters). The T yp e I Bäc klund transforma- tion indeed gives a non-trivial transformation. It cannot b e reduced to a scaling transformation as in 6.19 and do es not dep end on deriv atives of fields as one usually exp ects for suc h kind of expansion. The corresp onding gauge–Bäc klund transformation (6.11) turns out to b e: U I I − → γ i → 0 U I =    α 1 e 1 2 ( J 1 − J 2 ) β 1 e 1 2 ( J 1 + J 2 ) λ β 2 e − 1 2 ( J 1 + J 2 ) α 2 e 1 2 ( J 2 − J 1 )    , (6.18) taking the limit β i → 0 we recov er the type I Bäcklund transformation from t yp e I I Bäcklund transformation as we discussed b efore. In this wa y , the type I I Bäcklund transformation B II con tains the previous cases as limits. 6.3 Bäc klund T ransformations Ha ving obtained the gauge transformations ab ov e, we summarize our results. Eac h gauge-Bäc klund transformation U i , (6.16), (6.18) and (6.11) has a differen t Bäcklund transformation asso ciated to it in order to satisfy (6.1). These transformations are listed b elo w: (i) T yp e 0 B 0 : γ 2 r 2 = γ 1 e δ r 1 , (6.19a) γ 1 s 2 = γ 2 e − δ s 1 , (6.19b) (ii) T yp e I B I : α 2 r 2 e − J 1 = α 1 r 1 e − J 2 + β 1 , (6.20a) α 1 s 2 e J 1 = α 2 s 1 e J 2 + β 2 . (6.20b) (iii) T yp e I I B II : α 2 ∂ x  r 2 e − J 1  + γ 1 e − J 1 r 1 = α 1 ∂ x  r 1 e − J 2  + γ 2 e − J 2 r 2 (6.21a) α 1 ∂ x  s 2 e J 1  − γ 2 e J 1 s 1 = α 2 ∂ x  s 1 e J 2  − γ 1 e J 2 s 2 . (6.21b) Finally , using any of the U i , the gauge-transformation can b e applied to the temp oral Lax op erators: A CLL t N ( r 2 , s 2 ) U i − U i A CLL t N ( r 1 , s 1 ) + ∂ t N U i = 0 . (6.22) This equation is satisfied using only Bäcklund transformation and the equation of motion corre- sp onding for eac h U i and flo w t N . This r einfor c es the universality of the Bäcklund tr ansformation within the hier ar chy . 7 Bäc klund transformations and in tegrable defects In tegrable defects, or “jump-defects”, can b e understoo d as lo calized discon tin uities that connect the fields of the model on b oth sides of the defect while preserving in tegrability . It is well established that jump-defects in integrable systems are typically related to Bäc klund transformations frozen at 20 the defect position. This framew ork of Bäcklund transformations is particularly relev an t, since our main interest lies in the in teraction b etw een solitons and such defects. A large b o dy of work has in vestigated integrable defects in several mo dels, including KdV, mKdV, sinh-Gordon, sine-Gordon, T zitzéica, Boussinesq, nonlinear Sc hrö dinger equation (NLS), among others [14, 15, 19 – 21, 35 – 37]. In our case, we assume that Bäc klund transformations introduced in the previous section and form ulated as gauge transformations describ e a “jump-defect” lo cated at a fixed p osition in the CLL hierarch y . In order to analyze the in teraction b etw een solitons and defects, w e reformulate the Bäc klund transformations in terms of tau functions, which provide a more efficien t computational framew ork. W e then presen t the soliton solutions b efore and after the defect, and inv estigate the conditions under whic h the Bäcklund transformations are satisfied. As previously stated, B II con tains the remaining classes of Bäcklund transformations, so w e restrict our analysis to this t yp e, since the other cases can b e recov ered as reductions of this one 7.1 Bäc klund transformations and τ -functions The structure for the CLL fields written in terms of tau functions w as previously determined via the dressing metho d in the previous section 5. F or instance, we recall that (5.7), (5.12) and (5.13) determine r = br 0 τ 0 , 0 + τ 2 , 0 τ 1 , 1 , s = bs 0 τ 1 , 1 − τ 3 , 1 τ 0 , 0 , J = ln  τ 1 , 1 τ 0 , 0  − br 0 s 0 x, (7.1) where τ i,j = τ i,j ( x, t m ) and ( r 0 , s 0 ) denotes the v acuum pair. Since the Bäcklund transformations relate a pair of solution ( r 1 , s 1 ) to another pair of solution ( r 2 , s 2 ) , w e will adopt the following notation for eac h configuration Solution T au F unction ( r 1 , s 1 ) ( τ i,j , r 0 , s 0 ) ( r 2 , s 2 ) ( ¯ τ i,j , ¯ r 0 , ¯ s 0 ) so that w e can reformulate the Bäc klund transformation as τ -functions. Accordingly , the Bäcklund transformations originally expressed as functional of the fields r , s and the Bäc klund parameters, are here rewritten in terms of tau functions as: 21 B II : Γ 1 ( α 1 br 0 ¯ r 0 ¯ s 0 τ 0 , 0 τ 1 , 1 ¯ τ 0 , 0 ¯ τ 1 , 1 + α 1 b ¯ r 0 ¯ s 0 τ 1 , 1 τ 2 , 0 ¯ τ 0 , 0 ¯ τ 1 , 1 − α 1 br 0 τ 1 , 1 ∂ x τ 0 , 0 ¯ τ 0 , 0 ¯ τ 1 , 1 + α 1 br 0 τ 0 , 0 ∂ x τ 1 , 1 ¯ τ 0 , 0 ¯ τ 1 , 1 − α 1 br 0 τ 0 , 0 τ 1 , 1 ¯ τ 1 , 1 ∂ x ¯ τ 0 , 0 + α 1 br 0 τ 0 , 0 τ 1 , 1 ¯ τ 0 , 0 ∂ x ¯ τ 1 , 1 − bγ 2 ¯ r 0 τ 2 1 , 1 ¯ τ 2 0 , 0 + α 1 τ 2 , 0 ∂ x τ 1 , 1 ¯ τ 0 , 0 ¯ τ 1 , 1 − α 1 τ 1 , 1 ∂ x τ 2 , 0 ¯ τ 0 , 0 ¯ τ 1 , 1 − α 1 τ 1 , 1 τ 2 , 0 ¯ τ 1 , 1 ∂ x ¯ τ 0 , 0 + α 1 τ 1 , 1 τ 2 , 0 ¯ τ 0 , 0 ∂ x ¯ τ 1 , 1 − γ 2 τ 2 1 , 1 ¯ τ 0 , 0 ¯ τ 2 , 0  + Γ 2 ( α 2 b ¯ r 0 τ 1 , 1 ∂ x τ 0 , 0 ¯ τ 0 , 0 ¯ τ 1 , 1 − α 2 br 0 s 0 ¯ r 0 τ 0 , 0 τ 1 , 1 ¯ τ 0 , 0 ¯ τ 1 , 1 − α 2 br 0 s 0 τ 0 , 0 τ 1 , 1 ¯ τ 1 , 1 ¯ τ 2 , 0 − α 2 b ¯ r 0 τ 0 , 0 ∂ x τ 1 , 1 ¯ τ 0 , 0 ¯ τ 1 , 1 + α 2 b ¯ r 0 τ 0 , 0 τ 1 , 1 ¯ τ 1 , 1 ∂ x ¯ τ 0 , 0 − α 2 b ¯ r 0 τ 0 , 0 τ 1 , 1 ¯ τ 0 , 0 ∂ x ¯ τ 1 , 1 + bγ 1 r 0 τ 2 0 , 0 ¯ τ 2 1 , 1 + α 2 τ 1 , 1 ∂ x τ 0 , 0 ¯ τ 1 , 1 ¯ τ 2 , 0 − α 2 τ 0 , 0 ∂ x τ 1 , 1 ¯ τ 1 , 1 ¯ τ 2 , 0 − α 2 τ 0 , 0 τ 1 , 1 ¯ τ 2 , 0 ∂ x ¯ τ 1 , 1 + α 2 τ 0 , 0 τ 1 , 1 ¯ τ 1 , 1 ∂ x ¯ τ 2 , 0 + γ 1 τ 0 , 0 τ 2 , 0 ¯ τ 2 1 , 1  = 0 , (7.2a) Γ 1 ( α 1 br 0 s 0 ¯ s 0 τ 0 , 0 τ 1 , 1 ¯ τ 0 , 0 ¯ τ 1 , 1 − α 1 br 0 s 0 τ 0 , 0 τ 1 , 1 ¯ τ 0 , 0 ¯ τ 3 , 1 − α 1 b ¯ s 0 τ 1 , 1 ∂ x τ 0 , 0 ¯ τ 0 , 0 ¯ τ 1 , 1 + α 1 b ¯ s 0 τ 0 , 0 ∂ x τ 1 , 1 ¯ τ 0 , 0 ¯ τ 1 , 1 − α 1 b ¯ s 0 τ 0 , 0 τ 1 , 1 ¯ τ 1 , 1 ∂ x ¯ τ 0 , 0 + α 1 b ¯ s 0 τ 0 , 0 τ 1 , 1 ¯ τ 0 , 0 ∂ x ¯ τ 1 , 1 − bγ 2 s 0 τ 2 1 , 1 ¯ τ 2 0 , 0 + α 1 τ 1 , 1 ∂ x τ 0 , 0 ¯ τ 0 , 0 ¯ τ 3 , 1 − α 1 τ 0 , 0 ∂ x τ 1 , 1 ¯ τ 0 , 0 ¯ τ 3 , 1 + α 1 τ 0 , 0 τ 1 , 1 ¯ τ 3 , 1 ∂ x ¯ τ 0 , 0 − α 1 τ 0 , 0 τ 1 , 1 ¯ τ 0 , 0 ∂ x ¯ τ 3 , 1 + γ 2 τ 1 , 1 τ 3 , 1 ¯ τ 2 0 , 0  + Γ 2 ( α 2 b ¯ r 0 ¯ s 0 τ 0 , 0 τ 3 , 1 ¯ τ 0 , 0 ¯ τ 1 , 1 + α 2 bs 0 τ 1 , 1 ∂ x τ 0 , 0 ¯ τ 0 , 0 ¯ τ 1 , 1 − α 2 bs 0 ¯ r 0 ¯ s 0 τ 0 , 0 τ 1 , 1 ¯ τ 0 , 0 ¯ τ 1 , 1 − α 2 bs 0 τ 0 , 0 ∂ x τ 1 , 1 ¯ τ 0 , 0 ¯ τ 1 , 1 + α 2 bs 0 τ 0 , 0 τ 1 , 1 ¯ τ 1 , 1 ∂ x ¯ τ 0 , 0 − α 2 bs 0 τ 0 , 0 τ 1 , 1 ¯ τ 0 , 0 ∂ x ¯ τ 1 , 1 + bγ 1 ¯ s 0 τ 2 0 , 0 ¯ τ 2 1 , 1 − α 2 τ 3 , 1 ∂ x τ 0 , 0 ¯ τ 0 , 0 ¯ τ 1 , 1 + α 2 τ 0 , 0 ∂ x τ 3 , 1 ¯ τ 0 , 0 ¯ τ 1 , 1 − α 2 τ 0 , 0 τ 3 , 1 ¯ τ 1 , 1 ∂ x ¯ τ 0 , 0 + α 2 τ 0 , 0 τ 3 , 1 ¯ τ 0 , 0 ∂ x ¯ τ 1 , 1 − γ 1 τ 2 0 , 0 ¯ τ 1 , 1 ¯ τ 3 , 1  = 0 , (7.2b) where w e defined the auxiliary fields Γ i that enco des v acuum information Γ 1 = exp( br 0 s 0 x ) and Γ 2 = exp( b ¯ r 0 ¯ s 0 x ) . (7.3) Finally , we recall that we must alwa ys satisfy ∂ x J i − r i s i = 0 for i = 1 , 2 , whic h is equiv alen t to imp ose J 1 : (1 − b ) br 0 s 0 τ 0 , 0 τ 1 , 1 + br 0 τ 0 , 0 τ 3 , 1 − bs 0 τ 1 , 1 τ 2 , 0 + τ 3 , 1 τ 2 , 0 − τ 1 , 1 ∂ x τ 0 , 0 + τ 0 , 0 ∂ x τ 1 , 1 = 0 , (7.4a) J 2 : (1 − b ) b ¯ r 0 ¯ s 0 ¯ τ 0 , 0 ¯ τ 1 , 1 + b ¯ r 0 ¯ τ 0 , 0 ¯ τ 3 , 1 − b ¯ s 0 ¯ τ 1 , 1 ¯ τ 2 , 0 + ¯ τ 3 , 1 ¯ τ 2 , 0 − ¯ τ 1 , 1 ∂ x ¯ τ 0 , 0 + ¯ τ 0 , 0 ∂ x ¯ τ 1 , 1 = 0 . (7.4b) In the next section, we will combine this set of equations with the previous solutions obtained in section 5. As the tau are written as linear combinations of ρ i , B II this will lead to a set of polynomial equations for ρ i that are easier to solv e than the original set of differential equations, allowing us to implemen t an efficient routine to study each case. Another adv an tage of such approach is that as w e use the spacial part of the Lax pair to determine the Bäcklund transformation, the results here are v alid for the en tire hierarc hy . 7.2 Solitons and in tegrable defects W e hav e established the necessary framework to study different soliton solutions interacting with v arious in tegrable defects. W e now pro vide different soliton solutions b efore and after the defect and require the Bäc klund transformations B II to b e satisfied. This will allow us to determine the final configuration of the soliton after in teracting with the defect. The results are presented in the follo wing order. First, w e will consider the soliton solutions obtained via v ertex op erators, referred to as class A. W e will p erform this analysis for the follo wing 22 cases: one-soliton to one-soliton, one-soliton to tw o-soliton and tw o-soliton to t wo-soliton. Sub- sequen tly , we will apply the same pro cedure to the solitons obtained via vertex op erators of the referred to as class B. Due to its complex structure, w e only presen t the case of tw o-soliton to t wo-soliton transformation. F or simplicity , in all cases we assume the same v acuum structure, i.e, ¯ r 0 = r 0 and ¯ s 0 = s 0 . F or all the cases, the solutions are written in terms of ρ i , such for each integer flo w N, we ha ve: ρ i = e ( k i + br 0 s 0 ) x + ω ± N i t with ω N i = k N i − b ( − r 0 s 0 ) N or ω − N i = (1 − 2 b ) k − N i + b ( − r 0 s 0 ) − N . 7.2.1 One-soliton → one-soliton Consider the one-soliton solution from class A (5.23), passing through the defect and emerging as another one-soliton class A shifted b y a delay factor R : τ 0 , 0 = 1 − br 0 ρ 1 , τ 1 , 1 = 1 , τ 2 , 0 = br 2 0 ρ 1 , τ 3 , 1 = − k 1 ρ 1 , ¯ τ 0 , 0 = 1 − br 0 Rρ 1 , ¯ τ 1 , 1 = 1 , ¯ τ 2 , 0 = br 2 0 Rρ 1 , ¯ τ 3 , 1 = − k 1 Rρ 1 . (7.5) W e recall that class A solitons represent a natural reduction to the Burgers’ equation (4.1) as discussed b efore. The type II Bäc klund transformations are satisfied with the follo wing conditions imp osed on the dela y factor ( R ) and Bäc klund parameters ( α i , γ i ): • Scattering Conditions: – b = 1 R = k 1 α 2 + γ 1 k 1 α 1 + γ 2 suc h γ 1 = γ 2 + ( α 2 − α 1 ) r 0 s 0 . (7.6) – b = 0 R = k 1 α 2 + γ 1 k 1 α 2 + γ 2 . (7.7) • Solutions: ( r 1 , s 1 ) =  br 0 , bs 0 + k 1 ρ 1 1 − br 0 ρ 1  − → B II ( r 2 , s 2 ) =  br 0 , bs 0 + k 1 R ρ 1 1 − br 0 R ρ 1  . (7.8) Suitable limits as ( r 0 , 0) or (0 , s 0 ) can b e considered for p ositive flo ws. After in teracting with the defect, the one-soliton acquires a dela y factor R . 7.2.2 One-soliton → tw o-soliton No w, w e prop ose a one-soliton solution from the class A vertex op erators, passing through the defect and emerging as another t wo-soliton class A. In terms of τ -functions, w e hav e τ 0 , 0 = 1 − br 0 ρ 1 , τ 1 , 1 = 1 , τ 2 , 0 = br 2 0 ρ 1 , τ 3 , 1 = − k 1 ρ 1 , ¯ τ 0 , 0 = 1 − br 0 ( Rρ 1 + ρ 2 ) , ¯ τ 1 , 1 = 1 , ¯ τ 2 , 0 = br 2 0 ( Rρ 1 + ρ 2 ) , ¯ τ 3 , 1 = − k 1 R ρ 1 − k 2 ρ 2 , (7.9) 23 In this case, the defect con verts a one-soliton configuration in to a t wo-soliton configuration. W e will ha ve the follo wing conditions up on the dela y factor R and Bäc klund parameters: • Scattering Conditions: – b = 1 R = γ 1 + k 1 α 2 γ 2 + k 1 α 1 suc h γ 1 = γ 2 − ( α 1 − α 2 ) r 0 s 0 and γ 2 = − α 1 k 2 . (7.10) – b = 0 R = k 1 α 2 + γ 2 k 1 α 1 + γ 1 and γ 1 = − α 1 k 2 . (7.11) • Solutions: ( r 1 , s 1 ) =  br 0 , bs 0 + k 1 ρ 1 1 − br 0 ρ 2  − → B II ( r 2 , s 2 ) =  br 0 , bs 0 + k 1 Rρ 1 + k 2 ρ 2 1 − br 0 ( Rρ 1 + ρ 2 )  . (7.12) The resulting t wo-soliton emerges from the in teractions of the one-soliton solution with the defect. It inherits the w av e n umber k 1 and acquires a second w av e n umber k 2 . 7.2.3 T w o-soliton → tw o-soliton No w, w e prop ose a t w o-soliton solution from the class A vertex op erators, passing through the defect and emerging as another t wo-soliton class A, differing from the original b y phases R 1 and R 2 . In terms of τ -functions, we hav e τ 0 , 0 = 1 − br 0 ( ρ 1 + ρ 2 ) , τ 1 , 1 = 1 , τ 2 , 0 = br 2 0 ( ρ 1 + ρ 2 ) , τ 3 , 1 = − k 1 ρ 1 − k 2 ρ 2 , ¯ τ 0 , 0 = 1 − br 0 ( R 1 ρ 1 + R 2 ρ 2 ) , ¯ τ 1 , 1 = 1 , ¯ τ 2 , 0 = br 2 0 ( R 1 ρ 1 + R 2 ρ 2 ) , ¯ τ 3 , 1 = − k 1 R 1 ρ 1 − k 2 R 2 ρ 2 . (7.13) The final result is giv en by: • Scattering Conditions: – b = 1 R i = γ 1 + k i α 2 γ 2 + k i α 1 suc h γ 1 = γ 2 − ( α 1 − α 2 ) r 0 s 0 . (7.14) – b = 0 R i = γ 2 + k i α 2 γ 1 + k i α 1 . (7.15) • Solutions: ( r 1 , s 1 ) =  br 0 , bs 0 + k 1 ρ 1 + k 2 ρ 2 1 − br 0 ( ρ 1 + ρ 2 )  − → B II ( r 2 , s 2 ) =  br 0 , bs 0 + k 1 R 1 ρ 1 + k 2 R 2 ρ 2 1 − br 0 ( R 1 ρ 1 + R 2 ρ 2 )  . (7.16) In this case t wo-soliton solution of class A interacting with defect, preserv es the original w av es n umber, k 1 and k 2 and acquires dela y factors, R 1 and R 2 . 24 7.2.4 T w o-soliton → tw o-soliton No w, w e prop ose a tw o-soliton solution from the class B vertex op erators, passing through the defect and emerging as another t wo-soliton class B, differing from the original by phase shifts R 1 and R 2 : τ 0 , 0 = 1 − br 0 ρ 1 + k 2 ( br 0 s 0 + k 1 ) 2 ( k 1 − k 2 ) 2 ρ 1 ρ − 1 2 , τ 1 , 1 = 1 + bs 0 ρ − 1 2 + k 1 ( br 0 s 0 + k 2 ) 2 ( k 1 − k 2 ) 2 ρ 1 ρ − 1 2 , τ 2 , 0 = br 2 0 ρ 1 − k 2 ρ − 1 2 − br 0 k 2 ( k 1 + k 2 + 2 br 0 s 0 ) ( k 1 − k 2 ) ρ 1 ρ − 1 2 , τ 3 , 1 = bs 2 0 ρ − 1 2 − k 1 ρ 1 − bs 0 k 1 ( k 1 + k 2 + 2 br 0 s 0 ) ( k 1 − k 2 ) ρ 1 ρ − 1 2 , (7.17) and ¯ τ 0 , 0 = 1 − br 0 R 1 ρ 1 + k 2 ( br 0 s 0 + k 1 ) 2 ( k 1 − k 2 ) 2 R 1 R 2 ρ 1 ρ − 1 2 , ¯ τ 1 , 1 = 1 + bs 0 R 2 ρ − 1 2 + k 1 ( br 0 s 0 + k 2 ) 2 ( k 1 − k 2 ) 2 R 1 R 2 ρ 1 ρ − 1 2 , ¯ τ 2 , 0 = br 2 0 R 1 ρ 1 − k 2 R 2 ρ − 1 2 − br 0 k 2 ( k 1 + k 2 + 2 br 0 s 0 ) ( k 1 − k 2 ) R 1 R 2 ρ 1 ρ − 1 2 , ¯ τ 3 , 1 = bs 2 0 R 2 ρ − 1 2 − k 1 R 1 ρ 1 − bs 0 k 1 ( k 1 + k 2 + 2 br 0 s 0 ) ( k 1 − k 2 ) R 1 R 2 ρ 1 ρ − 1 2 . (7.18) This leads to the follo wing results • Scattering Conditions: – b = 1 γ 1 = γ 2 − ( α 1 − α 2 ) r 0 s 0 , R 1 = k 1 α 2 + γ 1 k 1 α 1 + γ 2 and R 2 = k 2 α 1 + γ 2 k 2 α 2 + γ 1 . (7.19) – b = 0 R 1 = k 1 α 2 + γ 2 k 1 α 1 + γ 1 and R 2 = k 2 α 1 + γ 1 k 2 α 2 + γ 2 . (7.20) • Solutions: r 1 = ( k 1 − k 2 ) 2  br 0 − k 2 ρ − 1 2  + k 2 ρ − 1 2 ρ 1 r 0 ( k 2 + br 0 s 0 ) 2 k 1 ρ − 1 2 ρ 1 ( k 2 + br 0 s 0 ) 2 + ( k 1 − k 2 ) 2  ρ − 1 2 bs 0 + 1  , (7.21a) s 1 = k 1 ρ 1  ρ − 1 2 bs 0 ( k 1 + br 0 s 0 ) 2 + ( k 1 − k 2 ) 2  + ( k 1 − k 2 ) 2 bs 0 ρ 1  k 2 ρ − 1 2 ( k 1 + br 0 s 0 ) 2 − ( k 1 − k 2 ) 2 br 0  + ( k 1 − k 2 ) 2 , (7.21b) 25 and r 2 = ( k 1 − k 2 ) 2  br 0 − k 2 R 2 ρ − 1 2  + k 2 R 2 R 1 ρ − 1 2 ρ 1 br 0 ( k 2 + br 0 s 0 ) 2 k 1 R 2 R 1 ρ − 1 2 ρ 1 ( k 2 + br 0 s 0 ) 2 + ( k 1 − k 2 ) 2  R 2 ρ − 1 2 bs 0 + 1  , (7.22a) s 2 = k 1 R 1 ρ 1  R 2 ρ − 1 2 bs 0 ( k 1 + br 0 s 0 ) 2 + ( k 1 − k 2 ) 2  + ( k 1 − k 2 ) 2 bs 0 R 1 ρ 1  k 2 R 2 ρ − 1 2 ( k 1 + br 0 s 0 ) 2 − ( k 1 − k 2 ) 2 br 0  + ( k 1 − k 2 ) 2 . (7.22b) In this case, the t wo-soliton in teracting with the defect acquires dela y factors R 1 and R 2 . 8 Discussion and further dev elopmen ts In this pap er w e prop ose a universal framework to deal with generalized in tegrable hierarc hies in the sense that higher grading semi-simple elemen ts (of grade a ≥ 1 ) can b e incorporated within the Riemann-Hilb ert-Birkhoff decomp osition. The framew ork extends the usual class of soliton solutions asso ciated to zero v acuum solutions. These, in turn define a centerless Heisenberg sub-algebra that include different t yp es of b oundary conditions. In fact, the construction of differen t Heisen b erg sub-algebras classify the p ossible v acua of the soliton solutions. In particular, w e ha ve shown the existence of a no v el class of constant non-zero v acuum solutions whic h are constructed from a one parameter dep endent (deformed) Heisen b erg sub-algebra. Explicit examples w ere found and discussed for the mKdV ( a = 1 ) [12] and CLL ( a = 2 ) hierarc hies, [6]. Other in teresting new examples, for a > 2 , follo w the general pattern and are under in vestigation. In particular for the CLL hierarch y , b y a judicious choice of vertex op erators, we ha ve constructed a class of solutions in which one of the field remains constant. This pro vides, in a closed form, a systematic construction of soliton solutions for the en tire underlying Burgers hierarch y . Moreo ver the grading structure of the affine algebra pro vide a systematic construction of Bäc k- lund transformation in terms of graded group elemen ts. The main idea is to consider graded gauge transformations to map differen t solutions of the same flow equation. Examples of suc h construction were successfully emplo y ed for the generalized A r -mKdV hierar- c hies [33, 34]. Here we ha ve determined the Bäcklund transformation for the entire CLL hierarc hy . The reduction pro cedure yields the Bäc klund transformation for the Burgers hierarch y . Sev eral Bäcklund solutions w ere explicitly w orked out for the CLL and its underlined Burgers hierarc hies. So far we hav e employ ed the principal gradation in our examples. An in teresting construction will b e to consider higher grading with mixed gradations, e.g., higher grading Y a jima-Oik aw a hierarch y (deriv ativ e Y a jima-Oik a wa). An interesting pattern emerging naturally from our construction is the parameter dependence in the Lax op erators in v acuum, (3.12) and (3.26). Notice that the grading added to the p ow er of r 0 or s 0 in eac h term in (3.12) and (3.26) is a constant. This suggests a second lo op described b y ζ is a dimension of either r 0 or s 0 and an effectiv e grading can b e defined to b e ˜ Q = Q + ζ d dζ , The idea of affine Lie algebra with t wo loops was in tro duced in [38]. 26 A c kno wledgmen ts JF G thank CNPq and F APESP for supp ort. YF A thanks F APESP for financial supp ort under gran t #2022/13584-0. GVL thanks F APESP for financial supp ort under gran t #2024/16787-4. TCS thanks F APESP for financial supp ort under gran t #2026/02077-0. A The sl (2) affine algebra The affine Kac-Mo o dy algebra ˆ G is an infinite extension of a Lie algebra G , ˆ G = L ( G ) ⊕ C ˆ c ⊕ C ˆ d, where L ( G ) = G ⊗ C  ζ , ζ − 1  = { X ⊗ ζ n | X ∈ G , n ∈ Z } is the lo op algebra of G , where ζ ∈ C is its sp ectral parameter [39]. The cen tral term ˆ c commutes with all other generators and the sp ectral deriv ative ˆ d = ζ d dζ measures the p ow er of the parameter ζ . Considering G = A 1 ∼ sl (2) , the generators { h, E α , E − α } ob ey the follo wing comm utation relations [ h, E ± α ] = ± 2 E ± α and [ E α , E − α ] = h . The corresp onding affine algebra ˆ G = ˆ A 1 is obtained considering L ( G ) = { h ( n ) = ζ n h, E ( n ) α = ζ n E α , E ( n ) − α = ζ n E − α } with the normalization α 2 = 2 . Th us, the affine Kac-Mo o dy ˆ G = ˆ A 1 generators read { h ( n ) , E ( m ) α , E ( m ) − α , ˆ c, ˆ d } , and the comm utation relations in the Chev alley basis are giv en by h h ( n ) , h ( m ) i = 2 nδ n + m, 0 ˆ c, h h ( n ) , E ( m ) ± α i = ± 2 E ( n + m ) ± α , h E ( n ) α , E ( m ) − α i = h ( n + m ) + nδ n + m, 0 ˆ c, h E ( n ) ± α , E ( m ) ± α i = 0 , h ˆ c, T ( n ) i = 0 , h ˆ d, T ( n ) i = nT ( n ) , (A.1) with n, m ∈ Z , where T ( n ) ∈ { h ( n ) = ζ n h, E ( n ) α = ζ n E α , E ( n ) − α = ζ n E − α } . The sl (2) lo op algebra is obtained considering ˆ c = 0 in to the commutation relations ab o ve. An algebra ˆ G can be decomp osed into graded subspaces as follo ws: ˆ G = M n ∈ Z ˆ G n , h ˆ G n , ˆ G m i ⊂ ˆ G n + m , n, m ∈ Z , where ˆ G n is a subspace of degree n according to a grading operator Q such that h Q, ˆ G n i = n ˆ G n . F or ˆ G = ˆ A 1 it is p ossible to define tw o gradations, the homogeneous and the principal, whose grading op erators are Q h = ˆ d, Q p = 1 2 h (0) + 2 ˆ d. Observ e that in order to construct the in tegrable hierarc hies b y an algebraic method it is enough to consider only the loop algebra, but once w e w ant to obtain the solutions of the equations we need to tak e into accoun t the complete Kac-Mo o dy algebra and its represen tation theory . 27 The highest w eight states of ˆ A 1 , namely | µ 0 ⟩ and | µ 1 ⟩ , are annihilated by all generators T ( n ) with n > 0 and satisfy also: h (0) | µ 0 ⟩ = 0 , E (0) α | µ 0 ⟩ = 0 , ˆ c | µ 0 ⟩ = | µ 0 ⟩ , h (0) | µ 1 ⟩ = | µ 1 ⟩ , E (0) α | µ 1 ⟩ = 0 , ˆ c | µ 1 ⟩ = | µ 1 ⟩ . The adjoin t relations read  h ( n )  † = h ( − n ) ,  E ( n ) ± α  † = E ( − n ) ∓ α , (ˆ c ) † = ˆ c, (A.2) th us, ⟨ µ 0 | and ⟨ µ 1 | are annihilated b y all generators T ( n ) ( n < 0 ). A 2 × 2 matrix represen tation of the sl (2) lo op algebra is given by: h ( n ) =  ζ n 0 0 − ζ n  , E ( n ) α =  0 ζ n 0 0  , E ( n ) − α =  0 0 ζ n 0  . B Matrix Elemen ts In this section w e presen t the matrix elemen ts used in order to obtain the solutions of Section 5. In all cases w e consider the vertices (5.15) and (5.16) with their resp ective parameters k i  = 0 . F or those matrix elements whic h inv olv e only one vertex ( V + i or V − i ) with its resp ective parameter k i , w e obtain: ⟨ λ 0 | V + i | λ 0 ⟩ = − br 0 , ⟨ λ 1 | V + i | λ 1 ⟩ = 0 , ⟨ λ 2 | V + i | λ 0 ⟩ = br 2 0 , ⟨ λ 3 | V + i | λ 1 ⟩ = − k i , ⟨ λ 0 | V − i | λ 0 ⟩ = 0 , ⟨ λ 1 | V − i | λ 1 ⟩ = bs 0 , ⟨ λ 2 | V − i | λ 0 ⟩ = − k i , ⟨ λ 3 | V − i | λ 1 ⟩ = bs 2 0 . All matrix elemen ts ⟨ λ k |  V ± i  n | λ l ⟩ with n ⩾ 2 are zero for vertices (5.15) and (5.16). The same holds for matrix elements whic h in volv e the pro duct of t wo or more vertices of the same kind ( V + i or V − i ), ev en when related to distinct parameters k i as: ⟨ λ k | V ± i V ± j | λ l ⟩ = 0 , ⟨ λ k | V ± i V ± j V ± m | λ l ⟩ = 0 . Finally , for the pro duct V + i with V − j (or V − i with V + j ), the matrix elemen ts are: ⟨ λ 0 | V + i V − j | λ 0 ⟩ = k j ( k i + br 0 s 0 ) 2 ( k i − k j ) 2 , ⟨ λ 1 | V + i V − j | λ 1 ⟩ = k i ( k j + br 0 s 0 ) 2 ( k i − k j ) 2 , ⟨ λ 2 | V + i V − j | λ 0 ⟩ = − br 0 k j ( k i + k j + 2 br 0 s 0 ) k i − k j , ⟨ λ 3 | V + i V − j | λ 1 ⟩ = − bs 0 k i ( k i + k j + 2 br 0 s 0 ) k i − k j , ⟨ λ 0 | V − i V + j | λ 0 ⟩ = k i ( k j + br 0 s 0 ) 2 ( k i − k j ) 2 , ⟨ λ 1 | V − i V + j | λ 1 ⟩ = k j ( k i + br 0 s 0 ) 2 ( k i − k j ) 2 , ⟨ λ 2 | V − i V + j | λ 0 ⟩ = br 0 k i ( k i + k j + 2 br 0 s 0 ) k i − k j , ⟨ λ 3 | V − i V + j | λ 1 ⟩ = bs 0 k j ( k i + k j + 2 br 0 s 0 ) k i − k j . 28 References [1] V. G. Drinfel’d, V. V. Sok olo v, Lie algebras and equations of K orteweg-de V ries type, Journal of So viet Mathematics 30 (2) (1985) 1975–2036. doi:10.1007/BF02105860 . URL h ttp://link.springer.com/10.1007/BF02105860 [2] M. F. De Gro ot, T. J. Hollow o o d, J. L. Miramontes, Generalized Drinfel’d-Sok olo v hierarchies, Comm unications in Mathematical Physics 145 (1) (1992) 57–84. doi:10.1007/BF02099281 . URL h ttp://link.springer.com/10.1007/BF02099281 [3] H. Arat yn, L. A. F erreira, J. F. Gomes, A. H. Zimerman, The complex sine-Gordon equation as a symmetry flow of the AKNS hierarc hy, Journal of Ph ysics A: Mathematical and General 33 (35) (2000) L331–L337. doi:10.1088/0305- 4470/33/35/101 . URL h ttps://iopscience.iop.org /article/10.1088/0305- 4470/33/35/101 [4] J. F. Gomes, G. S. F ranca, G. R. de Melo, A. H. Zimerman, Negative Ev en Grade mKdV Hierarc hy and its Soliton Solutions, Journal of Physics A: Mathematical and Theoretical 42 (44) (2009) 445204, arXiv:0906.5579 [hep-th, physics:nlin]. d oi :10 .1 088 /1 751 - 8 1 13 /4 2 /4 4/ 4 45 2 04 . URL h [5] Y. F. Adan s, G. F rança, J. F. Gomes, G. V. Lob o, A. H. Zimerman, Negative flows of generalized KdV and mKdV hierarc hies and their gauge-Miura transformations, Journal of High Energy Ph ysics 2023 (8) (2023) 160, arXiv:2304.01749 [hep-th, ph ysics:math-ph, ph ysics:nlin]. d o i : 10.1007/JHEP08(2023)160 . URL h [6] H. Aratyn, C. Constantinidis, J. Gomes, T. Santiago, A. Zimerman, Generalized Riemann- Hilb ert-Birkhoff decomp osition and a new class of higher grading in tegrable hierarc hies, Nuclear Ph ysics B 1018 (2025) 117080. doi:10.1016/j.nuclphysb.2025.117080 . URL h ttps://linkinghub.elsev ier.com/retriev e/pii/S0550321325002895 [7] V. E. Adler, Negative flows for several in tegrable mo dels, Journal of Mathematical Physics 65 (2) (2024) 023502. doi:10.1063/5.0181692 . URL h t tps :// pub s.a ip. org / jmp /ar tic le/ 65/ 2/0 235 02/ 326 699 6/N eg ati v e- f lo ws- f or - se ve ral - int egrable- mo dels [8] V. E. Adler, 3D consistency of negativ e flows, Theoretical and Mathematical Ph ysics 221 (2) (2024) 1836–1851. doi:10.1134/S0040577924110047 . URL h ttps://link.springer.com/10.1134/S0040577924110047 [9] M. P . Kolesnik o v, The negativ e symmetry classification problem, Theoretical and Mathematical Ph ysics 224 (2) (2025) 1398–1413. doi:10.1134/S0040577925080057 . URL h ttps://link.springer.com/10.1134/S0040577925080057 [10] H. Arat yn, J. Gomes, A. Zimerman, In tegrable hierarc hy for multidimensional T o da equations and top ological–an ti-top ological fusion, Journal of Geometry and Ph ysics 46 (1) (2003) 21–47. doi:10.1016/S0393- 0440(02)00126- 2 . URL h ttps://linkinghub.elsev ier.com/retriev e/pii/S0393044002001262 [11] L. A. F erreira, J.-L. Gerv ais, J. Sánc hez Guillén, M. V. Sav elie, Affine T o da systems coupled to matter fields, Nuclear Physics B 470 (1-2) (1996) 236–288. do i:10.1 016/05 50- 321 3(9 6)0 29 0146- 0 . URL h ttps://linkinghub.elsev ier.com/retriev e/pii/0550321396001460 [12] J. F. Gomes, G. S. F rança, A. H. Zimerman, Non v anishing b oundary condition for the mKdV hierarc hy and the Gardner equation, Journal of Physics A: Mathematical and Theoretical 45 (1) (2012) 015207, arXiv:1110.3247 [math-ph, ph ysics:nlin]. d oi :1 0. 10 88 / 17 51 - 8 11 3/ 45 /1 /0 15 207 . URL h [13] M. Jim b o, T. Miwa, Solitons and Infinite Dimensional Lie Algebras, Publications of the Re- searc h Institute for Mathematical Sciences 19 (3) (1983) 943–1001. doi:10.2977/prims/1195 182017 . URL h ttps://ems.press/doi/10.2977/prims/1195182017 [14] E. Corrigan, C. Zambon, A new class of in tegrable defects, Journal of Physics A: Mathematical and Theoretical 42 (47) (2009) 475203. doi:10.1088/1751- 8113/42/47/475203 . URL h ttps://iopscience.iop.org /article/10.1088/1751- 8113/42/47/475203 [15] E. Corrigan, C. Zambon, T yp e I I defects revisited, Journal of High Energy Ph ysics 2018 (9) (2018) 19. doi:10.1007/JHEP09(2018)019 . URL h ttps://link.springer.com/10.1007/JHEP09(2018)019 [16] O. Bab elon, D. Bernard, Dressing symmetries, Communications in Mathematical Ph ysics 149 (2) (1992) 279–306. doi:10.1007/BF02097626 . URL h ttp://link.springer.com/10.1007/BF02097626 [17] O. Bab elon, D. Bernard, Affine Solitons: A Relation Betw een T au F unctions, Dressing and B\"ac klund T ransformations, In ternational Journal of Mo dern Ph ysics A 08 (03) (1993) 507– 543, arXiv:hep-th/9206002. doi:10.1142/S0217751X93000199 . URL h ttp://arxiv.org/abs/hep- th/9206002 [18] L. A. F erreira, J. L. Miramon tes, J. S. Guillen, T au-functions and Dressing T ransformations for Zero-Curv ature Affine In tegrable Equations, Journal of Mathematical Ph ysics 38 (2) (1997) 882–901, arXiv:hep-th/9606066. doi:10.1063/1.531895 . URL h ttp://arxiv.org/abs/hep- th/9606066 [19] P . Bow co c k, E. Corrigan, C. Zambon, Classically integrable field theories with defects, In- ternational Journal of Mo dern Ph ysics A 19 (supp02) (2004) 82–91, arXiv:hep-th/0305022. doi:10.1142/S0217751X04020324 . URL h ttp://arxiv.org/abs/hep- th/0305022 [20] E. Corrigan, C. Zam b on, Jump-defects in the nonlinear Schrödinger mo del and other non- relativistic field theories, Nonlinearity 19 (6) (2006) 1447–1469. doi :10 .10 88/ 0 951 - 77 15/ 19 /6/012 . URL h ttps://iopscience.iop.org /article/10.1088/0951- 7715/19/6/012 [21] V. Caudrelier, On a systematic approach to defects in classical integrable field theories, In- ternational Journal of Geometric Metho ds in Mo dern Ph ysics 05 (07) (2008) 1085–1108, arXiv:0704.2326 [hep-th, ph ysics:math-ph, physics:nlin]. doi:10.1142/S0219887808003223 . URL h 30 [22] D.-y . Chen, k -constraint for the mo dified Kadomtsev–P etviashvili system, Journal of Mathe- matical Ph ysics 43 (4) (2002) 1956–1965. doi:10.1063/1.1446665 . URL https://pubs.aip.org /jmp/ar ticle/43/4/1 956/830177/k- co nstraint- f or- the- mo dif ied- Kad omtsev [23] H. H. Chen, Y. C. Lee, C. S. Liu, In tegrability of Nonlinear Hamiltonian Systems b y Inv erse Scattering Method, Ph ysica Scripta 20 (3-4) (1979) 490–492. do i:1 0.1 088 /00 31- 894 9/ 20/ 3 - 4/026 . URL h ttps://iopscience.iop.org /article/10.1088/0031- 8949/20/3- 4/026 [24] D.-j. Zhang, The discrete Burgers equation, Partial Differential Equations in Applied Mathe- matics 5 (2022) 100362. doi:10.1016/j.padiff.2022.100362 . URL h ttps://linkinghub.elsev ier.com/retriev e/pii/S2666818122000535 [25] N. A. Kudryasho v, D. I. Sinelshc hiko v, Exact solutions of equations for the Burgers hierarch y, Applied Mathematics and Computation 215 (3) (2009) 1293–1300. doi:10.1016 /j.amc.2009 .06.010 . URL h ttps://linkinghub.elsev ier.com/retriev e/pii/S0096300309005761 [26] H. Bateman, SOME RECENT RESEAR CHES ON THE MOTION OF FLUIDS, Mon thly W eather Review 43 (4) (1915) 163–170. doi:1 0 .1175 /1520 - 0493(19 15)43< 16 3:SRR OTM> 2. 0 .CO;2 . URL h ttp://journals.ametso c.org /doi/10.1175/1520- 0493(1915)43< 163:SRROTM> 2.0.CO;2 [27] J. Burgers, A Mathematical Mo del Illustrating the Theory of T urbulence, in: A dv ances in Applied Mechanics, V ol. 1, Elsevier, 1948, pp. 171–199. doi : 10.10 16/S0 065- 215 6(08)70 1 00 - 5 . URL h ttps://linkinghub.elsev ier.com/retriev e/pii/S0065215608701005 [28] A. S. Sharma, H. T asso, Connection b et ween wa ve env elop e and explicit solution of a nonlinear disp ersiv e wa ve equation, T ech. rep. (1977). URL h ttps://hdl.handle.net/11858/00- 001M- 0000- 0027- 6CD5- C [29] P . J. Olver, Evolution equations p ossessing infinitely many symmetries, Journal of Mathemat- ical Ph ysics 18 (6) (1977) 1212–1215. doi:10.1063/1.523393 . URL https://pubs.aip.org /jmp/article /18/6/1212/2248 65/Evol ution- equations- poss essing- inf initely- many [30] E. Hopf, The P artial Differential Equation u_t + uu_x = \mu_{xx}, Comm unications on Pure and Applied Mathematics 3 (3) (1950) 201–230. doi:10.1002/cpa.3160030302 . URL h ttps://onlinelibrary .wiley .com/doi/10.1002/cpa.3160030302 [31] J. D. Cole, On a quasi-linear parab olic equation o ccurring in aero dynamics, Quarterly of Ap- plied Mathematics 9 (3) (1951) 225–236. doi:10.1090/qam/42889 . URL h ttps://ww w.ams.org/qam/1951- 09- 03/S0033- 569X- 1951- 42889- X/ [32] C. Rogers, W. F. Shadwic k, Bäcklund transformations and their applications, no. v. 161 in Mathematics in science and engineering, A cademic Press, New Y ork, 1982. [33] J. M. De Carv alho F erreira, J. F. Gomes, G. V. Lob o, A. H. Zimerman, Gauge Miura and Bäc klund transformations for generalized A n -KdV hierarc hies, Journal of Physics A: Mathe- matical and Theoretical 54 (43) (2021) 435201. doi:10.1088/1751- 8121/ac2718 . URL h ttps://iopscience.iop.org /article/10.1088/1751- 8121/ac2718 31 [34] J. M. De Carv alho F erreira, J. F. Gomes, G. V. Lobo, A. H. Zimerman, Generalized Bäc klund transformations for affine T o da hierarchies, Journal of Physics A: Mathematical and Theoretical 54 (6) (2021) 065202. doi:10.1088/1751- 8121/abd8b2 . URL h ttps://iopscience.iop.org /article/10.1088/1751- 8121/abd8b2 [35] P . Bow co ck, E. Corrigan, C. Zambon, Affine T o da field theories with defects, Journal of High Energy Ph ysics 2004 (01) (2004) 056–056. doi:10.1088/1126- 6708/2004/01/056 . URL h t tp:/ /st ack s.i op. org /1 126- 6 708 /20 04/ i=0 1/a =05 6?key= cro ssre f .0 e24 433 20a 30c 301 30e 9eab30846dd2b [36] V. Caudrelier, Multisymplectic approac h to in tegrable defects in the sine-Gordon model, Jour- nal of Physics A: Mathematical and Theoretical 48 (19) (2015) 195203. doi:10.1088/ 1 7 51- 8 113/48/19/195203 . URL h ttps://iopscience.iop.org /article/10.1088/1751- 8113/48/19/195203 [37] E. Corrigan, C. Zambon, A dding in tegrable defects to the Boussinesq equation, Journal of Ph ysics A: Mathematical and Theoretical 56 (38) (2023) 385701. doi:10.1088/1751- 8121/ac eec9 . URL h ttps://iopscience.iop.org /article/10.1088/1751- 8121/aceec9 [38] H. Aratyn, L. F erreira, J. Gomes, A. Zimerman, Kac-Mo o dy construction of T o da t yp e field theories, Ph ysics Letters B 254 (3-4) (1991) 372–380. doi:10.1016/0370- 2693(91)91171- Q . URL h ttps://linkinghub.elsev ier.com/retriev e/pii/037026939191171Q [39] V. G. Kac, Infinite-Dimensional Lie Algebras, 3rd Edition, Cam bridge Universit y Press, 1990. doi:10.1017/CBO9780511626234 . URL h ttps://ww w.cam bridg e.org/core/product/identif ier/9780511626234/t y p e/bo ok 32