Integrable couplings and matrix loop algebras

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1 Integrable coulings and matrix loo algebras Wen-Xiu Ma Citation: AIP Conference Proceedings 1562, 105 (2013); doi: / View online: htt://dx.doi.org/ / View Table of Contents: htt://scitation.ai.org/content/ai/roceeding/aic/1562?ver=dfcov Published by the AIP Publishing

2 Integrable coulings and matrix loo algebras Wen-Xiu Ma Deartment of Mathematics and Statistics, University of South Florida, Tama, FL , USA address: Abstract. We will discuss how to generate integrable coulings from zero curvature equations associated with matrix sectral roblems. The key elements are matrix loo algebras consisting of block matrices with blocks of the same size or different sizes. Hamiltonian structures are furnished by alying the variational identity defined over semi-direct sums of Lie algebras. Illustrative examles include integrable coulings of the AKNS hierarchy by using the irreducible reresentations V 2 and V 3 of the secial linear Lie algebra sl(2,r). Keywords: Zero curvature equation, Recursion oerator, Variational identity, Hamiltonian structure, Symmetry, Conservation law PACS: Ik, Xx, Fy, Yv 1. INTRODUCTION A system of differential equations or differential-difference equations is said to be integrable by quadratures if its solutions to well-ossed Cauchy roblems can be obtained after a finite number of stes involving algebraic oerations and integration of given functions [1]. Given a Hamiltonian system of ordinary differential equations (ODEs) on a symlectic sace, when is it integrable by quadratures? A first answer is the Liouville theorem using action-angle variables [2, 3]. The modern formulation of the Liouville theorem is called the Liouville-Arnold theorem. It states that for a Hamiltonian system of ODEs associated with a Hamiltonian function H defined on a symlectic manifold (M 2N,ω 2 ): t = {,H} = ω 2 (IdH,Id), q t = {q,h} = ω 2 (IdH,Idq), (1) the following conditions guarantee integrability by quadratures: (i) Existence of N integrals of motion {F i (,q)} N i=1 ; (ii) F 1,F 2,,F N commute with each other under the associated Poisson bracket: {F i,f j } = 0, 1 i, j N; (iii) F 1,F 2,,F N are functionally indeendent on the level surface {F i = a i 1 i N} containing the initial data. Those three conditions are nowadays called the Liouville conditions, and Liouville integrable systems of ODEs mean Hamiltonian systems ossessing the above Liouville conditions. For artial differential equations (PDEs) or differential-difference equations (DDEs), no similar Liouville conditions are found to guarantee their integrability by quadratures. There are, however, many different but related integrable criteria for PDEs or DDEs widely adoted in the soliton community [4], a few of which are listed as follows (see, e.g., [5, 6, 7] for details): (i) Existence of Lax air and solvable by the inverse scattering transform (S-integrable case); (ii) Transforming into linear equations (C-integrable case); (iii) Passing the Painlevé test and the singularity confinement; (iv) Existence of Darboux transformations; (v) Algebraic comlete integrability; (vi) Existence of three-soliton solutions; (vii) Existence of infinitely many symmetries; (viii) Existence of infinitely many conservation laws; (ix) Bi-Hamiltonian formulations (which often imly the existence of infinitely many symmetries and conservation laws); etc. In what follows, we say a system of PDEs or DDEs is integrable if it has infinitely many symmetries which are functionally indeendent on the jet saces, and Liouville integrable if it is Hamiltonian and ossesses infinitely many conserved densities being functionally indeendent on the jet saces. The Liouville integrability is stronger since Hamiltonian systems generate symmetries from conserved densities associated with conservation laws. The Nonlinear and Modern Mathematical Physics AIP Conf. Proc. 1562, (2013); doi: / AIP Publishing LLC /$

3 fundamental question in the integrability theory of PDEs and DDEs is how to exlore Hamiltonian structures and infinitely many commuting functionally indeendent conserved densities. A continuous (or discrete) Hamiltonian system of PDEs or DDEs reads u t = K(u)=K(u,u x, )[or K(u,Eu,E 1 u, )] = J H u, (2) where u denotes a column vector of deendent variables, u is the variational derivative with resect to u, J is a Hamiltonian oerator and H = Hdx [or H = n Z H] is called a Hamiltonian functional associated with a Hamiltonian function H. For a Hamiltonian system, there exists a relationshi chain from conserved densities through adjoint symmetries to symmetries: I u and a Lie algebra homomorhism J u : Idx J u Idx [or I u I J n Z u I] (3) n Z J u {I 1,I 2 } =[J I 1 u,j I 2 ], (4) u with the Poisson bracket of functionals being defined by ( I1 ) T I 2 {I 1,I 2 } = J u u dx [or {I 1,I 2 } = n Z and the Lie bracket of vector fields being defined by ( I1 u ) T I 2 J ], (5) u [K,S]=K (u)[s] S (u)[k]. (6) Here and in what follows, P (u)[v] denotes the Gateaux derivative of an object P = P(u,u x, ) with resect to u along a direction v: P (u)[v]= P(u + εv,u x + εv x, ). (7) ε ε=0 We say that two conserved densities of a Hamiltonian system (2) commute, if their corresonding functionals commute under the Poisson bracket (5) (see, e.g., [8] for details). Lax roosed an equivalent oerator formulation, now called the Lax air aroach, for studying the KdV equation [9]. A Lax air formulation is generally equivalent to a zero curvature equation formulation [10] and it also rovides a novel tye of searability [11]. An integrable system of PDEs or DDEs is said to ossess a zero curvature equation reresentation, if it is generated from a continuous zero curvature equation U t V x +[U,V ]=0 (8) or a discrete zero curvature equation U t +UV (EV)U = 0, (9) where the two square matrices, U and V, are called a Lax air, often belonging to simle matrix loo algebras [12]. One of our rimary tasks in the field of integrable systems is how to construct, from the continuous and discrete zero curvature equation formulations, a Hamiltonian structure with a recursion oerator [13] or a bi-hamiltonian structure [14]: u t = K(u)=J H 1 u = M H 2 u, (10) which naturally engenders infinitely many commuting conserved densities, for a given system of evolution equations. The existence of bi-hamiltonian structures often guarantee recursion oerators [14]-[17], which can be hereditary [18]. A Lax air of U = U(u,λ) and V = V (u,λ) involving a sectral arameter λ, from a matrix Lie algebra with a free arameter, called a matrix loo algebra, is a starting oint in formulating integrable systems which ossess bi- Hamiltonian structures. Simle matrix loo algebras yield tyical integrable systems (see, e.g., [19, 20] for examles 106

4 associated with sl(2, R) and so(3, R)). Semisimle matrix loo algebras lead to searated integrable systems, i.e., collections of tyical integrable systems, each of which corresonds to a simle matrix loo algebra. Non-semisimle matrix loo algebras generate integrable coulings [21, 22]. Secially, an integrable couling of an integrable system (2) is a triangular integrable system of the following form [23, 24]: { ut = K(u), (11) v t = S(u,v). If S is nonlinear with resect to the second sub-vector v of deendent variables, the integrable couling (11) is called nonlinear. An examle of integrable coulings is the first-order erturbation system [23]: { ut = K(u), v t = K (12) (u)[v], where K is the Gateaux derivative as defined earlier. A bi-integrable couling of a given integrable system (2) is an enlarged triangular integrable system of the following form [25]: u t = K(u), u 1,t = S 1 (u,u 1 ), u 2,t = S 2 (u,u 1,u 2 ); and similarly, by a tri-integrable couling, we mean an enlarged triangular integrable system of the following form [26, 27]: u t = K(u), u 1,t = S 1 (u,u 1 ), (14) u 2,t = S 2 (u,u 1,u 2 ), u 3,t = S 3 (u,u 1,u 2,u 3 ). Non-semisimle Lie algebras ossess semi-direct sum decomositions [28]: (13) ḡ = g g c, g - semisimle, g c - solvable, (15) and so, semi-direct sums of Lie algebras lay a foundation for constructing integrable coulings. The notion of semidirect sums ḡ = g g c (16) means that the two Lie subalgebras g and g c satisfy [g,g c ] g c, (17) where [g,g c ]={[A,B] A g, B g c }, with [, ] denoting the Lie bracket of ḡ. Obviously, g c is an ideal Lie subalgebra of ḡ. The subscrit c indicates a contribution to the construction of couling systems. We also require the closure roerty between g and g c under the matrix multilication gg c,g c g g c, (18) where g 1 g 2 = {AB A g 1, B g 2 }, while we use discrete zero curvature equations over semi-direct sums of matrix Lie algebras to generate discrete integrable coulings [22]. Integrable coulings rovide insiration and insights into classifying integrable systems with multi-comonents [23, 24]. Continuous and discrete zero curvature equation formulations over non-semisimle matrix loo algebras are the basis for generating integrable coulings, and the associated (classical and suer) variational identities offer fundamental tools to build their Hamiltonian structures [29]-[32]. In this reort, we would like to exlore non-semisimle loo algebras consisting of block matrices to generate integrable coulings. Hamiltonian structures of the resulting couling systems will be furnished through the variational identity on general matrix loo algebras. In articular, we will resent matrix loo algebras consisting of block matrices by using two irreducible reresentations V 2 and V 3 of the secial linear Lie algebra sl(2,r) and aly them to the construction of integrable coulings, based on enlarged zero curvature equations. As illustrative examles, two alications will be made for the AKNS soliton hierarchy, and Hamiltonian structures will be generated for one of the two resulting hierarchies of integrable coulings. The resented matrix loo algebras rovide key elements for constructing integrable coulings. 107

5 2. SOLITON HIERARCHIES 2.1. General rocedure Let us recall a standard rocedure for generating soliton hierarchies from Lax airs (see [33, 34, 35] for details). Assume that g is a given matrix Lie algebra, which can be either semisimle or non-semisimle, and its corresonding matrix loo algebra is formulated as follows: The beginning is to introduce a satial sectral roblem g = {M g entries of M - Laurent series of λ}. (19) φ x = Uφ, U = U(u,λ) g, (20) where u is a column vector of deendent variables and λ is a sectral arameter. We then take a solution of the form W = W(u,λ)= W 0,i λ i, W 0,i g, i 0, (21) i 0 to the stationary zero curvature equation Further, introduce the temoral sectral roblems with the Lax matrices being chosen as W x =[U,W] [or (EW)U UW = 0]. (22) φ tm = V [m] φ = V [m] (u,λ)φ, m 0, (23) V [m] =(λ m W) + + Δ m g, m 0, (24) where P + denotes the olynomial art of P in λ. The introduction of the modification terms Δ m g aims to guarantee that the zero curvature equations U tm V [m] x +[U,V [m] ]=0 [or U tm (EV [m] )U +UV [m] = 0], m 0, (25) engender a hierarchy of soliton equations with Hamiltonian structures: u tm = K m (u)=j H m, m 0. u (26) Those Hamiltonian functionals are usually generated by alying the trace identity: u tr( U γ W)dx = λ λ λ λ γ tr( U W), γ = λ d 2 dλ ln tr(w 2 ), (27) or u tr( U γ W)=λ n Z λ λ λ γ tr( U W), γ = λ d 2 dλ ln tr((wu)2 ), (28) and more generally, the variational identity: U γ,w dx = λ u λ λ λ γ U,W, γ = λ d ln W,W, 2 dλ (29) or u U γ,w = λ n Z λ λ λ γ U,W, γ = λ d ln WU,WU, 2 dλ (30) where, is a non-degenerate, symmetric and ad-invariant bilinear form on the matrix loo algebra g [31, 32]. 108

6 The non-degenerate roerty means that if A,B = 0 for all A g, then B = 0; and if A,B = 0 for all B g, then A = 0. The symmetric and ad-invariant roerties are A,B = B,A and A,[B,C] = [A,B],C, where [, ] denotes the Lie bracket of g, resectively. When the underlying Lie algebra g is semisimle, the trace identity will most likely work. When g is nonsemisimle, we have to use the variational identity since the trace identity is not valid, and the bilinear form, in the variational identity (29) must not be of the Killing tye. If A, A are ositive for non-zero matrices A g, then the Lie algebra ( g,, ) becomes quadratic. Matrix loo algebras rovide key elements to study soliton hierarchies, including hierarchies of integrable coulings, and both the trace identity and the variational identity are basic tools for generating Hamiltonian structures which usually guarantee the Liouville integrability The AKNS soliton hierarchy Let us recall the AKNS soliton hierarchy associated with g = sl(2,r) [19]. The traditional satial sectral roblem for the AKNS hierarchy is given by [ ] [ ] [ ] λ φ1 φ x = Uφ, U = U(u,λ)=, u =, φ =. (31) q λ q φ 2 The stationary continuous zero curvature equation in (22), i.e., W x =[U,W], is equivalent to a x = c qb, b x = 2λ b 2 a, c x = 2qa + 2λ c, (32) if we assume that W is of the form [ ] a b W = c a [ ] = W 0,i λ i ai b i, W 0,i =, i 0. (33) i 0 c i a i Equivalently, the system (32) leads to the recursion relations: b i+1 = 1 2 b i,x a i, c i+1 = 1 2 c i,x qa i, a i+1,x = c i+1 qb i+1, i 0, (34) uon taking the initial values a 0 = 1, b 0 = c 0 = 0. (35) We imose the conditions on constants of integration: a i u=0 = b i u=0 = c i u=0 = 0, i 1, (36) such that the recursion relations in (34) will uniquely determine the sequence of {a i,b i,c i i 1}. This way, the first three sets can be worked out as follows: b 1 =, c 1 = q, a 1 = 0; b 2 = 1 2 x, c 2 = 1 2 q x, a 2 = 1 2 q; b 3 = 1 4 xx q, c 3 = 1 4 q xx 1 2 q2, a 3 = 1 4 (q x x q). 109

7 Now, the zero curvature equations U tm V [m] x +[U,V [m] ]=0 with V [m] =(λ m W) +, m 0, (37) where all modification terms, Δ m, m 0, are taken as zero, generate the AKNS soliton hierarchy: [ ] [ ] 2bm+1 2 u tm = K m = = Φ m = J H m, m 0, (38) 2c m+1 2q u where the Hamiltonian oerator, the hereditary recursion oerator and the Hamiltonian functionals are given by [ ] [ ] J =, Φ = q 1 2am q 1 1 q 2 q 1, H m = dx, m 0, (39) m + 1 resectively. The above Hamiltonian structures are generated by alying the trace identity [33, 35]. 3. MATRIX LOOP ALGEBRAS To generate integrable coulings, one needs to create non-semisimle Lie algebras, also called enlarged Lie algebras (see, e.g., [36]-[45] for tyical examles). We will resent a few classes of matrix Lie algebras consisting of block matrices with blocks of the same size or different sizes. Larger numbers of blocks show diversity of integrable coulings, though they bring comlexity in theoretical verification Lie algebras consisting of block matrices with blocks of the same size In this subsection, we would articularly like to show matrix Lie algebras consisting of 2 2, 3 3or4 4 block matrices, which can be used to generate dark equations and bi- and tri-integrable coulings [26]. Class 1 - Matrix Lie algebras consisting of 2 2 block matrices: All 2 2 block matrices of the tye: [ A1 A 2 M 1 (A 1,A 2 )= 0 A 1 ], (40) where A 1 and A 2 are square matrices of the same size, define an enlarged matrix Lie algebra with the following semi-direct sum decomosition: The corresonding matrix roduct reads ḡ = g g c, where g = {M(A 1,0)} and g c = {M(0,A 2 )}. M(A 1,A 2 )M(B 1,B 2 )=M(C 1,C 2 ), (41) where C 1 and C 2 are given by { C1 = A 1 B 1, (42) C 2 = A 1 B 2 + A 2 B 1. This kind of enlarged matrix Lie algebras can be used to generate dark equations [32, 46, 47], and the variational identity in this case reduces to the bi-trace identity [48]. Class 2 - Matrix Lie algebras consisting of 3 3 block matrices: Let α and β be two arbitrarily given constants, which can be zero. All 3 3 block matrices of the tye: M(A 1,A 2,A 3 )= A 1 A 2 A 3 0 A 1 + αa 2 βa 2 + αa A 1 + αa 2, (43) 110

8 where A 1, A 2 and A 3 be square matrices of the same size, define an enlarged matrix Lie algebra with the following semi-direct sum decomosition: The corresonding matrix roduct reads ḡ = g g c, where g = {M(A 1,0,0)} and g c = {M(0,A 2,A 3 )}. M(A 1,A 2,A 3 )M(B 1,B 2,B 3 )=M(C 1,C 2,C 3 ), (44) where C 1,C 2 and C 3 are given by C 1 = A 1 B 1, C 2 = A 1 B 2 + A 2 B 1 + αa 2 B 2, C 3 = A 1 B 3 + A 3 B 1 + βa 2 B 2 + αa 2 B 3 + αa 3 B 2. (45) This kind of enlarged matrix Lie algebras can be used to generate bi-integrable coulings [25]. Class 3 - Matrix Lie algebras consisting of 4 4 block matrices: Let α,β,μ and ν be four arbitrarily given constants, which can be zero. All following 4 4 block matrices: M 3 (A 1,A 2,A 3,A 4 )= A 1 A 2 A 3 A 4 0 A 1 + αa 2 αa 3 βa 2 + αa A 1 + αa 2 + μa 3 νa A 1 + αa 2, (46) where A i,1 i 4, be square matrices of the same size, define an enlarged matrix Lie algebra with the semi-direct sum decomosition: The corresonding matrix roduct reads ḡ = g g c, where g = {M(A 1,0,0,0)} and g c = {M(0,A 2,A 3,A 4 )}. M(A 1,A 2,A 3,A 4 )M(B 1,B 2,B 3,B 4 )=M(C 1,C 2,C 3,C 4 ), (47) where C 1,C 2,C 3 and C 4 are given by C 1 = A 1 B 1, C 2 = A 1 B 2 + A 2 B 1 + αa 2 B 2, C 3 = A 1 B 3 + A 3 B 1 + αa 2 B 3 + αa 3 B 2 + μa 3 B 3, C 4 = A 1 B 4 + A 4 B 1 + αa 2 B 4 + αa 4 B 2 + βa 2 B 2 + νa 3 B 3. (48) This kind of enlarged matrix Lie algebras can be used to generate tri-integrable coulings [27] Lie algebras consisting of block matrices with blocks of different sizes We will use the finite-dimensional irreducible reresentations V d of the secial linear Lie algebra sl(2,r) (see, e.g., [49]), to create enlarged matrix Lie algebras. Class 1 - Matrix Lie algebra by using the irreducible reresentation V 2 : Let M(V 2,A a )=ḡ be a Lie algebra of enlarged matrices of the tye: 2a b 0 2c 0 2b A a f 0 M(a,b,c, f,g)= 0 c 2a with A a = g f. (49) a b 0 g c a 111

9 Here the (1,1)-block is generated from the irreducible reresentation V 2 of sl(2,r). This Lie algebra has the semi-direct sum decomosition: ḡ = g g c, where g = {M(a,b,c,0,0)} and g c = {M(0,0,0, f,g}. The corresonding matrix commutator reads where [M(a,b,c, f,g),m(a,b,c, f,g )] = M(a,b,c, f,g ), (50) a = bc cb, b = 2ab 2ba, c = 2ca 2ac, f = af fa + bg gb, g = cf fc + ga ag. This enlarged matrix Lie algebra will be used to generate a class of integrable coulings of different tye from the revious ones. Class 2 - Matrix Lie algebra by using the irreducible reresentation V 3 : Let M(V 3,A a )=ḡ be a Lie algebra of enlarged matrices of the tye: 3a b 0 0 3c a 2b 0 A f 0 a 0 2c a 3b e f M(a,b,c,e, f,g)= with A 0 0 c 3a a = g e. (52) a b 0 g c a Here the (1,1)-block is generated from the irreducible reresentation V 3 of sl(2,r). This enlarged matrix Lie algebra has the semi-direct sum decomosition: The corresonding matrix commutator reads where ḡ = g g c,where g = {M(a,b,c,0,0,0)} and g c = {M(0,0,0,e, f,g}. [M(a,b,c,e, f,g),m(a,b,c,e, f,g )] = M(a,b,c,e, f,g ), (53) a = bc cb, b = 2ab 2ba, c = 2ca 2ac, e = 2cf 2 fc + 2bg 2gb, f = 2af 2 fa + be eb, g = ce ec 2ag + 2ga. This enlarge matrix Lie algebra will be used to generate another class of integrable coulings of different tye from the revious ones. (51) (54) 112

10 4. INTEGRABLE COUPLINGS BY USING IRREDUCIBLE REPRESENTATIONS 4.1. Using the irreducible reresentation V 2 of sl(2,r) Based on M(V 2,A a ), let us introduce a satial sectral roblem φ x = Ūφ = Ū(ū,λ)φ, (55) where λ is a sectral arameter, φ =(φ 1,,φ 5 ) T is an eigenfunction, and the enlarged satial matrix Ū is given by 2λ 0 r 0 2q 0 2 s r Ū = Ū(ū,λ)= 0 q 2λ 0 s, ū = q λ r. (56) s q λ With W = M(V 2,W a ) being chosen from the enlarged matrix loo algebra M(V 2,A a ): 2a b 0 f 0 2c 0 2b g f f 0 W = M(V 2,W a )= 0 c 2a 0 g, W a = g f, (57) a b 0 g c a the enlarged stationary zero curvature equation leads equivalently to both (32) and Assume that W a is of the following form W x =[Ū, W] (58) { fx = λ f + g ra sb, g x = qf + sa rc + λg. W a = W a,i λ i, W a,i = i 0 (59) f i 0 g i f i, i 0. (60) 0 g i Then, uon taking the initial values f 0 = g 0 = 0, (61) the system (59) equivalently yields the recursion relations: { fi+1 = f i,x + g i ra i sb i, g i+1 = g i,x qf i sa i + rc i, i 0. (62) The first three sets then can be comuted as follows: Further, taking b 1 = r, g 1 = s; f 2 = r x, g 2 = s x ; f 3 = r xx + s x 1 2 qr xs, g 3 = s xx + qr x 1 2 qs q xr. V [m] =(λ m W) + = m i=0 W i λ m i, m 0, (63) 113

11 where W i M(V 2,A a ), i 0, we see that the zero curvature equations Ū tm V [m] x +[Ū, V [m] ]=0, m 0, (64) generate a hierarchy of integrable coulings for the AKNS hierarchy (38): 2b m+1 2 ū tm = q 2c = K m = m+1 r f m+1 = Φ m 2q r s s t m g m+1, m 0, (65) where the oerator Φ is determined by using the recursion relations in (34) and (62): [ ] [ 12 Φ 0 r 1 q 1 2 Φ =, Φ 1 = s 1 2 r 1 ] [ Φ 1 Φ 2 2 1s 1 q 2 1s r, Φ 2 = q ]. (66) This recursion oerator is of different tye from the revious ones (see, e.g., [26, 31, 32] for revious examles). We oint out that any symmetric and ad-invariant bilinear form on the enlarged matrix loo algebra M(V 2,A a ) is degenerate. Therefore, the variational identity does not work for M(V 2,A a ), and we do not know how to exlore Hamiltonian structures, if any, for the hierarchy of integrable coulings (65) Using the irreducible reresentation V 3 of sl(2,r) Soliton hierarchy of integrable coulings Based on M(V 3,A a ), let us introduce a satial sectral roblem φ x = Ūφ = Ū(ū,λ)φ, (67) where λ is a sectral arameter, φ =(φ 1,,φ 6 ) T is an eigenfunction, and the enlarged satial matrix Ū is given by 3λ 0 0 r 0 3q λ r 0 2q λ 3 s 0 Ū = Ū(ū,λ)= 0 0 q 3λ 0 s, ū = q r λ s q λ With W = M(V 3,W a ) being chosen from the enlarge matrix loo algebra M(V 3,A a ): 3a b 0 0 f 0 3c a 2b 0 e f 0 2c a 3b g e W = M(V 3,W a )= 0 0 c 3a 0 g, W a = a b c a f 0 e f g e 0 g, (68) the enlarged stationary zero curvature equation W x =[Ū, W] (69) 114

12 leads equivalently to both (32) and Assume that W a is of the following form f x = 2λ f + e 2ra, g x = qe + 2sa + 2λg, e x = 2qf + 2g 2rc 2sb. W a = W a,i λ i, W a,i = i 0 (70) f i 0 e i f i, i 0. (71) g i e i 0 g i Then, uon taking the initial values e 0 = 1, f 0 = g 0 = 0, (72) the system (70) equivalently yields the recursion relations: f i+1 = 1 2 f i,x e i ra i, g i+1 = 1 2 g i,x 1 2 qe i sa i, i 0. (73) e i+1,x = 2qe i+1 + 2g i+1 2rc i+1 2sb i+1, We imose the conditions on constants of integration e i ū=0 = f i ū=0 = g i ū=0 = 0, i 1, (74) to determine the sequence of {e i, f i,g i i 1} uniquely. The first three sets then can be comuted as follows: Further, taking f 1 = r, g 1 = 1 2 q + s, e 1 = 0; f 2 = 1 4 x 1 2 r x, g 2 = 1 4 q x s x, e 2 = 1 2 q + s qr; f 3 = 1 8 xx r xx q s qr, g 3 = 1 8 q xx s xx 1 4 q2 qs 1 2 q2 r, e 3 = 1 4 ( xq q x ) 1 2 ( xs s x ) 1 2 (q xr qr x ). V [m] =(λ m W) + = m i=0 where W i M(V 3,A a ), i 0, we see that the zero curvature equations W i λ m i, m 0, (75) Ū tm V [m] x +[Ū, V [m] ]=0, m 0, (76) generate a hierarchy of integrable coulings for the AKNS hierarchy (38): ū tm = q r s t m 2b m+1 2c = K m = m+1 2 f m+1 2g m+1 = Φ m 2 2q 2r q + 2s, m 0, (77) where the oerator Φ is determined by using the recursion relations in (34) and (73): [ ] Φ 0 Φ =, (78) Φ 1 Φ 2 115

13 with the sub-matrix oerators Φ 1 and Φ 2 being given by [ 1 s + r 1 q 1 r + r 1 ] [ q 1 ] Φ 1 =, Φ 2 =. (79) q 1 s s 1 q q 1 r s 1 q 1 1 q 2 q 1 This recursion oerator is also of different tye from the revious ones (see, e.g, [26, 31, 32] for revious examles) Hamiltonian structures It can be directly shown that any symmetric and ad-invariant bilinear form on the enlarged matrix loo algebra M(V 3,A a ) is given by M(a,b,c,e, f,g),m(a,b,c,e, f,g ) =(a,b,c,e, f,g)f(a,b,c,e, f,g ) T = η 1 (aa bc cb )+η 2 (ae + bg cf + ea fc + gb ), (80) where the square matrix F is defined by η η η η F = 2 η η 2 0, (81) η η η with η 1 and η 2 being arbitrary constants. Note that the determinant of F is η2 6. Thus, the bilinear form, is non-degenerate iff the constant η 2 is non-zero. Now a simle alication of the variational identity [29, 31, 32]: where γ is a constant, engenders ū ū W, Ū γ λ dx = λ λ λ λ γ W, Ū, (82) ū ( η 1 a η 2 e)dx = λ γ λ λ γ 1 2 η 1c + η 2 g 1 2 η 1b η 2 f η 2 c η 2 b Balancing coefficients of each ower of λ in the above equality leads to 1 2 η 1c m + η 2 g m 1 ( η 1 a m+1 η 2 e m+1 )dx =(γ m) 2 η 1 b m η 2 f m, m 0. ū η 2 c m η 2 b m The case of m = 1 tells that γ = 0, and thus we have η1 a m+2 + η 2 e m+2 u m + 1 dx = 1 2 η 1c m+1 + η 2 g m η 1 b m+1 η 2 f m+1 η 2 c m+1 η 2 b m+1., m 0. (83) 116

14 Consequently, we obtain the Hamiltonian structures for the soliton hierarchy of integrable coulings (77): 2b m+1 ū tm = q 2c m+1 = K m = r 2 f m+1 = J H m, m 0, (84) ū s 2g m+1 t m with the Hamiltonian oerator and the Hamiltonian functionals η η 2 J 2 0 = 2 0 η2 0 η, (85) 1 η η 2 0 η 1 η 2 2 H η1 a m+2 + η 2 e m+2 m = dx, m 0. (86) m Liouville integrability It is direct but lengthy to show by comte algebra systems that J defined by (85) and [ ] η 2 2 Φ 1 0 M = Φ J = [ ] [ ] [ η 2 Φ 2 η Φ 1 + η 1 Φ η ] (87) with Φ being defined as in (39) and Φ 1 and Φ 2 being defined by (79), constitute a Hamiltonian air. It means that any linear combination N of J and M satisfies K T N (ū)[ N S] Tdx+ cycle( K, S, T )=0 (88) for all vector fields K, S and T. This imlies that Φ is hereditary [18], i.e., it satisfies Φ (ū)[ Φ K] S Φ Φ (ū)[ K] S = Φ (ū)[ Φ S] K Φ Φ (ū)[ S] K (89) for all vector fields K and S. The hereditary roerty (89) is equivalent to for an arbitrary vector field K, where the Lie derivative L K Φ is defined by L Φ K Φ = ΦL K Φ (90) (L K Φ) S = Φ[ K, S] [ K, Φ S]. (91) It is known (see, e.g., [50]) that an autonomous oerator Φ = Φ(ū,ū x, ) is a recursion oerator of an evolution equation ū t = K iff the oerator Φ needs to satisfy Obviously, we have L K 0 Φ = 0, and thus L K Φ = 0. (92) L K m Φ = ΦL K m 1 Φ = = Φ m L K 0 Φ = 0, m 1. (93) 117

15 This shows that the oerator Φ is a common hereditary recursion oerator for the soliton hierarchy of integrable coulings (77). Now, it follows that the soliton hierarchy of integrable coulings (77) is bi-hamiltonian and Liouville integrable, and thus, it ossesses infinitely many commuting symmetries and conserved densities. In articular, we have the Abelian symmetry algebra: [ K k, K l ]= K k (ū)[ K l ] K l (ū)[ K k ]=0, k,l 0, (94) and the two Abelian algebras of conserved Hamiltonian functionals: { H k, H l } J = ( H ) k T J H l dx = 0, k,l 0, (95) ū ū and ( { H k, H H ) k T l } M = M H l dx = 0, k,l 0. (96) ū ū These conserved Hamiltonian functionals generate infinitely many conservation laws for each system of integrable coulings in the hierarchy (77). Such differential olynomial tye conservation laws can also be comuted systematically by comuter algebra systems (see, e.g., [51]) or generated from a Riccati equation from the associated satial sectral roblem (see, e.g., [52, 53]). 5. SUPER VARIATIONAL IDENTITIES If the underlying Lie algebra ḡ is sueralgebra, we have similar suer variational identities [26]. Let ḡ(λ) be a loo sueralgebra over a suercommutative ring. Then the continuous suer variational identity on ḡ(λ) holds: W, U γ dx = λ u λ λ λ γ W, U, (97) and the discrete suer variational identity on ḡ(λ) holds: u W, U n Z λ = λ γ λ λ γ W, U, (98) where γ is a fixed constant, U,W ḡ(λ) satisfy W x =[U,W] or (EW)U =UW, and, is a non-degenerate, symmetric and ad-invariant bilinear form on the loo sueralgebra ḡ(λ). When the suertrace str is non-degenerate, the above suer variational identities reduce to the suertrace identities [54]: str(ad W ad U )dx = λ γ u λ λ λ γ (ad W ad U ), (99) and u str(ad W ad U )=λ γ n Z λ λ λ γ (ad W ad U ), (100) where ad a b =[a,b]. A second class of the owerful suer variational identities are bi-suertrace identities for constructing suer Hamiltonian dark equations: u [str ( W0 U 1 λ ) ( U 0 )] + str W1 dx = λ γ λ λ λ γ[ str ( U 1 W 0 ) ( U 0 )] + str W1 in the continuous case, and u [ ( U 1 ) ( U 0 )] str W0 + str W1 = λ γ n Z λ λ λ λ γ[ str ( U 1 ) ( U 0 )] W 0 + str W1 (102) in the discrete case. More generally, the suer variational identities can reduce to the last-comonent-trace identities: u N i=0 str ( U N j ) W i dx = λ γ λ λ λ γ N i=0 (101) str ( U N i ) W i, (103) 118

16 and u n Z N i=0 str ( U N i ) W i = λ γ λ λ λ γ N i=0 str ( U N i ) W 0, (104) which hel generate Hamiltonian structures of suer multi-integrable coulings. If a sectral matrix U = U(u,λ) is of order 2, we can make a suer generalization to construct a suer soliton hierarchy: Ū = U(u,λ)+αE 3 + βe 4 = U(u,λ) α β β α 0, (105) where E 3 and E 4 are odd generators of the suer sl(2), u is a vector of commuting variables, and α and β are anticommuting variables. Alications of the suer-trace identities lead to suer integrable systems and suer-symmetric integrable systems (see, e.g., [54]-[57]). We can also form semi-direct sums of Lie sueralgebras and take new enlarged sectral matrices from the resulting semi-direct sums of Lie sueralgebras, to construct suer integrable coulings. More secifically, we can make with the Lie roduct: ḡ(λ)= g g c (106) W =[Ū, V ]=[U +U c,v +V c ]=W +W c, Ū, V ḡ(λ), (107) where W =[U,V ] g, W c =[U,V c ]+[U c,v ]+[U c,v c ] g c. (108) Alications of the suer variational identities such as the bi-suertrace identities will yield suer Hamiltonian structures for suer integrable coulings. The rocedure for constructing suer integrable coulings is similar to the one in the classical case, and one only needs to ay secial attention to the anticommuting fermionic variables in deriving integrable coulings. 6. CONCLUDING REMARKS We roosed a few classes of matrix Lie algebras consisting of block matrices to generate integrable coulings, and successfully constructed two new tye hierarchies of integrable coulings for the AKNS equations. The resented enlarged matrix loo algebras rovide key elements to construct integrable coulings, and we oint out that the construction idea by using irreducible reresentations can also be alied to the other existing soliton hierarchies like the Dirac hierarchy (see, e.g., [58]) and the Kau-Newell hierarchy (see [59]). Integrable coulings bring insiration and insights into the general structure of integrable systems with multicomonents. It will be very helful in building an exhaustive list of integrable systems to collect articular examles of integrable coulings, both linear and nonlinear. The theory of integrable coulings generates various and comlicated recursion oerators in block matrix form, and the mathematics behind integrable coulings is dee and interesting. It is extremely imortant to exlore more secific examles of integrable coulings in establishing mathematical structures that integrable systems, articularly multi-comonent ones, ossess. There are many further questions on integrable coulings and their solution theories. We list some of them for interested readers as follows. Para-Grassmann zero curvature equations: It is not clear how to generalize a rocedure for constructing integrable systems by zero curvature equations to the ara-grassmann case. Even in the secial suer-symmetric case of D = 1 and N = 1, it is not known how to solve D x W =[U,W], D x = θ + θ x - suer derivative, for W such that the suer-symmetric zero curvature equation U t D x V +[U,V ]=0 generates suer-symmetric integrable systems. The general ara-grassmann case definitely brings more difficulties. 119

17 Oen question on linear ordinary differential equations: It was shown recently that for a general linear ordinary differential equation (ODE) with continuous coefficients y (n) + a 1 (x)y (n 1) + a 2 (x)y (n 2) + + a n 1 (x)y + a n (x)y = 0, where y (i) denotes the i-th order derivative of y = y(x) with resect to x, there is a solution y(x) such that y(x), y (x),,y (n 1) (x) are linearly indeendent over the interval where solutions exist, and this hels recognize solution structures of linear ODEs and generate subsaces of solutions invariant under the flows of given systems of evolution equations [60]. There is, however, an unsolved roblem about the reresentation of the general solutions of linear systems of ODEs with continuous coefficients. Let n 2, I =(a,b) R and t 0 I. Consider a Cauchy roblem { x (t)=a(t)x(t), x(t 0 )=x 0 R n, where A(t) is an n n matrix whose entries are continuous on I. It is known that the general solution is given by t x(t)=e t0 A(s)ds x0, if the coefficient matrix commutes with its integral, i.e., we have t [A(t), A(s)ds]=0, t I. t 0 An oen question is whether this commutativity condition is necessary to guarantee that the vector function x(t) defined above solves the given Cauchy roblem [61]. Criterion for multivariate olynomials with one zero: Hirota bilinear equations have recently been generalized by a technique of assigning signs to derivatives and the linear suerosition rincile was used to classify bilinear equations and determine their linear subsaces of solutions [62]. The related analysis resents an oen question on multivariate olynomials [63]: how to determine if a real multivariate olynomial has one and only one zero? There are many such olynomials, for examle, x 2 + y 2 2x + 2y + 2, zero (x,y)=(1, 1); 5x 2 + 2xy + 2y 2 12x 6y + 9, zero (x,y)=(1,1). Note that all such multivariate olynomials satisfy the requirement in Hilbert s 17th roblem [64]. So, the above roblem is more general than Hilbert s 17th roblem. Conjecture on integrability of commuting soliton equations: There are infinitely many functionally indeendent symmetries generated from a recursion oerator in a hierarchy of commuting soliton equations. Those symmetries generate an infinite number of one-arameter Lie grous of solutions to each member in the given soliton hierarchy. We conjecture that those infinitely many one-arameter Lie grous of solutions form a dense subset of solutions in the whole solution set to each member in the soliton hierarchy. More secifically, let us denote a hierarchy of commuting soliton equations by u tm = K m (u), m 0. For a given member u tn = K n (u), we assume that symmetry K m Lie grou of solutions S m (ε m ), ε m I m =(a m,b m ) R. Let T n be the set of solutions to the n-th member u tn = K n, and make a metric sace (T n (D),d) with a bounded domain D: T n (D)={ f D f T n }, d( f,g)=su (t,x) D f (t,x) g(t,x). Is the union m=0 S m(ε m ) dense in the metric sace (T n (D),d) with any bounded domain D for each member u tn = K n in the soliton hierarchy? 120

18 If the answer is yes, the solution to a Cauchy roblem can be aroximated by the solutions generated from those Lie symmetries. Thus, soliton hierarchies resent good models of integrable nonlinear artial differential equations from a comutational oint of view, indeed. Criterion for existence of Hamiltonian structures: We know that Hamiltonian structures exist for the erturbation systems (see, e.g., [23], [65]-[68]). But some enlarged matrix loo algebras do not ossess any non-degenerate, symmetric and ad-invariant bilinear forms required in the variational identities and a secific such examle was given in subsection 4.1 (see also [69, 70] for other examles). Is there any concrete criterion which tells when there exist Hamiltonian structures for integrable coulings, even bi- and tri-integrable coulings? A concrete examle is the following bi-integrable couling u t = K(u), v t = K (u)[v], w t = K (u)[w], where K denotes the Gateaux derivative as before. Is there any Hamiltonian structure for this secific bi-integrable couling? ACKNOWLEDGMENTS The work was suorted in art by the State Administration of Foreign Exerts Affairs of China, the National Natural Science Foundation of China (Nos , , , , and ), Chunhui Plan of the Ministry of Education of China, Zhejiang Innovation Project of China (Grant No. T200905), the First-class Disciline of Universities in Shanghai and the Shanghai Univ. Leading Academic Disciline Project (Grant No. A ), the Natural Science Foundation of Shanghai (No. 09ZR ) and the Shanghai Leading Academic Disciline Project (No. J50101). The author is also grateful to E. A. Aiah, X. Gu, X. Y. Li, S. Manukure, H. Q. Zhang, and Q. L. Zhao for their stimulating discussions. REFERENCES 1. V. I. Arnold, Mathematical Methods of Classical Mechanics (2nd edition, Sringer-Verlag, New York, 1989). 2. J. Bour, J. Math. Pures Al. 20, (1855). 3. J. Liouville, J. Math. Pures Al. 20, (1855). 4. W. X. Ma, in: Encycloedia of Nonlinear Science, ed. A. Scott (Taylor & Francis, New York, 2005), M. A. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge University Press, Cambridge, 1991). 6. R. Hirota, The Direct Method in Soliton Theory (Cambridge University Press, Cambridge, 2004). 7. D. Y. Chen, Introduction to Solitons (Science Press, Beijing, 2006). 8. L. A. Dickey, Soliton Equations and Hamiltonian Systems (2nd edition, World Scientific, Singoore, 2003). 9. P. D. Lax, Comm. Pure Al. Math. 21, (1968). 10. W. X. Ma, Acta Math. Sci. 14, (1994). 11. A. S. Fokas, Inverse Problems 25, , 44. (2009). 12. V. G. Drinfel d and V. V. Sokolov, Soviet Math. Dokl. 23, (1981). 13. P. J. Olver, J. Math. Phys. 18, (1977). 14. F. Magri, J. Math. Phys. 19, (1978). 15. I. M. Gelfand and I. Ja Dorfman, Funktsional. Anal. i Prilozhen. 13, 13 30, 96 (1979). 16. B. Fuchssteiner and A. S. Fokas, Physica D 4, (1981/82). 17. P. J. Olver, Alications of Lie grous to differential equations (Sringer-Verlag, New York, 1986). 18. B. Fuchssteiner, Nonlinear Anal. 3, (1979). 19. M. J. Ablowitz, D. J. Kau, A. C. Newell, and H. Segur, Stud. Al. Math. 53, (1974). 20. W. X. Ma, Al. Math. Comut. 220, (2013). 21. W. X. Ma, X. X. Xu, and Y. F. Zhang, Phys. Lett. A 351, (2006). 22. W. X. Ma, X. X. Xu, and Y. F. Zhang, J. Math. Phys. 47, , 16. (2006). 23. W. X. Ma and B. Fuchssteiner, Chaos, Solitons & Fractals 7, (1996). 24. W. X. Ma, Methods Al. Anal. 7, (2000). 25. W. X. Ma, in: Secial Issue on Solitons, ed. W. X. Ma, Chin. Ann. Math. B 33, (2012). 26. W. X. Ma, J. H. Meng, and H. Q. Zhang, Global J. Math. Sci. 1, 1 17 (2012). 121

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