Multi-component bi-hamiltonian Dirac integrable equations

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1 Chaos, Solitons an Fractals 9 (009) Multi-component bi-hamiltonian Dirac integrable equations Wen-Xiu Ma * Department of Mathematics an Statistics, University of South Floria, Tampa, FL , USA Accepte January 00 Communicate by Prof. M. El Naschie Abstract A specific matrix iso-spectral problem of arbitrary orer is introuce an an associate hierarchy of multi-component Dirac integrable equations is constructe within the framework of zero curvature equations. The bi-hamiltonian structure of the obtaine Dirac hierarchy is presente be means of the variational trace ientity. Two examples in the cases of lower orer are compute. Ó 00 Elsevier Lt. All rights reserve. 1. Introuction The integrability of nonlinear ifferential equations [1] is an important problem in the fiel of ifferential equations. It is in the irection of solving the problem to look for integrable ifferential equations an to explore characteristics of integrability. Integrability of orinary ifferential equations an 1+1 imensional partial ifferential equations, particularly scalar equations, was extensively stuie, an various criteria such as the inverse scattering transform, the Hamiltonian formulation an the Painlevé series were propose for testing integrability [,]. However, we lack a thorough theory for etermining integrability of multi-component ifferential equations an higher-imensional ifferential equations. Both the multiplicity an the imension bring a iversity of mathematical structures of integrable equations. The integrable couplings are one of the examples which show rich mathematical properties that multi-component integrable equations possess [4 ]. The way of using semi-irect sums of Lie algebras [8,9] provies a thought essential to analyze an classify multi-component integrable equations. In this paper, we are going to construct a multi-component Dirac integrable hierarchy from an arbitrary orer matrix spectral problem, which possesses a bi-hamiltonian formulation. The paper is organize as follows. In Section, a new higher-orer matrix iso-spectral problem is introuce an the associate Lax integrable equations of multicomponents are compute. In Section, an integrable hierarchy among the resulting Lax integrable equations is worke out an prove to possess a bi-hamiltonian structure using the variational trace ientity. This implies that the the resulting Dirac hierarchy possesses infinitely many commuting symmetries an conserve ensities. A few concluing remarks are given in Section 4. * Corresponing author. Tel.: ; fax: aress: mawx@math.usf.eu /$ - see front matter Ó 00 Elsevier Lt. All rights reserve. oi: /j.chaos

2 . Lax pairs an associate integrable equations We consier a matrix iso-spectral problem: W.-X. Ma / Chaos, Solitons an Fractals 9 (009) / x ¼ U/ ¼ Uðu; kþ/; U ¼ kj q ; J q T ¼ 0 1 ; ð:1þ r 1 0 where k is a spectral parameter, r is a skew-symmetric (i.e., r T = r) matrix of arbitrary orer an u = p(q,r) is a vector potential. Note that p is a kin of arrangement of all entries in q an r into a vector. To erive associate integrable equations, we first solve the stationary zero curvature equation V x ¼½U; V Š ð:þ of the spectral problem (.1). We assume that a solution V can be given by V ¼ aj b ; b T c where a is scalar an c T = c. Then we have " # qb T þ bq T kj b aj q þ qc br ½U; V Š¼ : kb T J aq T J rb T þ cq T q T b þ b T q þ½r; cš Therefore, the stationary zero curvature Eq. (.) becomes a x J ¼ bq T qb T ; b x ¼ kj b aj q þ qc br; c x ¼ b T q q T b þ½r; cš: Let us seek a formal solution of the type V ¼ aj b ¼ X1 b T c k¼0 V k k k ; V k ¼ a kj b k ðb k Þ T c k ð:aþ ð:bþ ð:cþ ; ð:4þ where (c k ) T = c k, k P 0. Then Eq. (.) recursively efine all a k, b k an c k, k P 0, with the initial values satisfying b 0 ¼ 0; a 0;x ¼ 0; c 0;x ¼½r; c 0 Š: Now for any integer n P 1, we introuce V ðnþ ¼ðk n V Þ þ þ D n ¼ Xn V j k n j þ D n ; D n ¼ 0 0 ; ð:5þ 0 n j¼0 n being skew-symmetric an of the same size as c, an efine the time evolution law for the eigenfunction /: / tn ¼ V ðnþ / ¼ V ðnþ ðu; kþ/: ð:6þ Setting U 1 = Uj k=0, we can compute that ððk n V Þ þ Þ x ½U; ðk n V Þ þ Š¼V n;x ½U 1 ; V n Š¼ 0 b n;x þ a n J q qc n þ b n r ; 0 0 q n D n;x ½U; D n Š¼D n;x ½U 1 ; D n Š¼ ; n q T n;x ½r; n Š where n P 1. It follows then that the compatibility conitions of (.1) an (.6) for all n P 1, i.e., the zero curvature equations U tn V ðnþ x þ½u; V ðnþ Š¼0; n P 1; ð:þ associate with the Lax pairs of U an V (n), n P 1, give rise to a hierarchy of Lax integrable evolution equations ¼ b n;x þ a n J q qc n þ b n r q n ; r tn ¼ n;x ½r; n Š; n P 1: ð:8þ

3 84 W.-X. Ma / Chaos, Solitons an Fractals 9 (009) 8 8. Multi-component Dirac integrable hierarchy.1. Integrable hierarchy When r = 0, we have to take n,x =0,nP1. Then, the Lax integrable hierarchy (.8) becomes ¼ b n;x þ a n J q qc n q n ; n P 1: ð:1þ To obtain Hamiltonian integrable equations, we further take n ¼ 0; n P 1: ð:þ The corresponing integrable hierarchy (.1) now reas ¼ b n;x þ a n J q qc n ; n P 1: ð:þ Owing to r = 0,(.) implies a n;x J ¼ b n q T qb T n ; J b nþ1 ¼ b n;x þ a n J q qc n ; c n;x ¼ b T n q qt b n ; n P 1: ð:4þ This tells that the resulting hierarchy (.) can be rewritten as ¼ X n ðqþ ¼J c b nþ1 ¼ J b nþ1 ¼ M c b n ¼ b n;x þ½o 1 ðb n q T qb T n ÞŠq qo 1 ðb T n q qt b n Þ; n P 1; ð:5þ where J c an M c enote the compact form of two matrix operators J an M: ðpðqþþ tn ¼ Jpðb nþ1 Þ¼Mpðb n Þ; n P 1: ð:6þ On one han, note that the inner prouct between two vectors p(a) an p(b) is ðpðaþ; pðbþþ ¼ trða T BÞx; where A an B enote matrices of the same size as q. Obviously, J an M are skew-symmetric. That is, for any two matrices A an B of the same size as q, we have ðpðaþ; pðo c BÞÞ¼ ðpðo c AÞ; pðbþþ; when O c = J c or O c = M c. Since the operator J is inepenent of the epenent variables in q, J automatically satisfies the Jacobi ientity ðpðaþ; pððj c Þ 0 ½J c BŠCÞÞ þ cycleða; B; CÞ ¼0; where (J c ) 0 enotes the Gateaux erivative, an A,B an C are arbitrary matrices of the same size as q. Therefore, J is a Hamiltonian operator. Moreover, it can be irectly verifie that J an M form a Hamiltonian pair, i.e., any linear combination of J an M is still Hamiltonian. On the other han, let us recall the variational trace ientity u V ; ou ok x ¼ k c o ok kc V ; ou ou ; ð:þ where the Killing form hp,qi = tr(pq) an the constant c ¼ k ln jhv ; V ij (see [10 1]). In our case, we have k V ; ou ¼ b; V ; ou ¼ a: ð:8þ oq ok Accoringly, using the variational trace ientity, we obtain H q e n ¼ b n ; eh n ¼ a nþ1 x; n P 1: n It naturally follows that the integrable hierarchy (.5) has a bi-hamiltonian structure: ¼ J c q e H nþ1 ¼ M c q e H n ; n P 1: ð:10þ The bi-hamiltonian structure guarantees [14] that ð:9þ

4 f eh k ; eh l g J ¼feH k ; eh l g M ¼ 0; k; l P 1; ð:11þ ½pðX k Þ; pðx l ÞŠ ¼ J u f eh kþ1 ; eh lþ1 g J ¼ 0; k; l P 1; ð:1þ where the Poisson bracket associate with a Hamiltonian operator O is given by! f eh ; ekg O ¼ eh u ; O! T ~K eh ¼ O ek u u u x; an the commutator of vector functions is efine by ½X ; Y Š¼X 0 ðuþ½y Š Y 0 ðuþ½x Š¼ o oe ðx ðu þ ey Þ Y ðu þ ex ÞÞj e¼0 : The equalities in (.11) an (.1) imply that each system in the hierarchy (.10) has infinitely many commuting symmetries an conserve ensities (see [14 16] for a general theory on the bi-hamiltonian formulation)... The case of scalar r When r is a 1 1 matrix, let us choose c = 0, an set q ¼ u 1 ; b n ¼ n ; n P 0; ð:1þ u e n where u 1 an u are scalar variables, an n an e n are scalar functions. Then, the recursion relation (.4) becomes 8 >< nþ1 ¼ e n;x þ u 1 a n ; e nþ1 ¼ n;x þ u a n ; >: a nþ1;x ¼ u nþ1 u 1 e nþ1 ; where n P 0, an thus, the Dirac integrable hierarchy (.5) (i.e., (.10)) reuces to the two-component Dirac hierarchy [1] associate with a Dirac spectral problem [18]: u tn ¼ u 1 ¼ e nþ1 ¼ J nþ1 ¼ J H u t n nþ1 e nþ1 u e nþ1 ¼ n;x þ u a n ¼ M n ¼ M H e n;x u 1 a n e n u e n ; n P 1; ð:14þ where the Hamiltonian functionals eh n ; n P 1, are efine by (.9) an the Hamiltonian pair reas J ¼ 0 1 " # ; M ¼ o þ u o 1 u u o 1 u 1 : 1 0 u 1 o 1 u o þ u 1 o 1 u 1 The corresponing hereitary recursion operator [19] of the hierarchy (.14) is " # U ¼ MJ 1 ¼ u o 1 u 1 o u o 1 u o þ u 1 o 1 u 1 u 1 o 1 u : We call the propose hierarchy (.5) the multi-component Dirac hierarchy, because the reuction of the matrix q to a vector yiels this two-component Dirac hierarchy. Upon choosing a 0 = 1 an setting every integration constant to zero, the first nonlinear system of integrable equations in the hierarchy (.14) is the Dirac nonlinear Schröinger equations: ( u 1;t ¼ u ;xx 1 u ðu 1 þ u Þ; u ;t ¼ u 1;xx þ 1 u 1ðu 1 þ u Þ: ð:15þ Its bi-hamiltonian structure is the following u t ¼ J u e H ¼ M u e H ; where two Hamiltonian functionals are 1 eh ¼ ðu 1;xu u 1 u ;x Þx; eh ¼ W.-X. Ma / Chaos, Solitons an Fractals 9 (009) u 1u 1;xx 1 u u ;xx 1 8 ðu 1 þ u Þ x:

5 86 W.-X. Ma / Chaos, Solitons an Fractals 9 (009) The case of matrix r When r is a matrix, let us set q ¼ u 1 u u u 4 ; b n ¼ n f n e n g n ; c n ¼ 0 h n h n 0 ; n P 0; ð:16þ where u i,16i65, are scalar variables, an n, e n, f n, g n an h n are scalar functions. Then, the recursion relation (.4) becomes 8 nþ1 ¼ e n;x þ u 1 a n u 4 h n ; e nþ1 ¼ n;x þ u a n þ u h n ; >< f nþ1 ¼ g n;x þ u a n þ u h n ; g nþ1 ¼ f n;x þ u 4 a n u 1 h n ; a nþ1;x ¼ u nþ1 þ u 4 f nþ1 u 1 e nþ1 u g nþ1 ; >: h nþ1;x ¼ u nþ1 þ u 4 e nþ1 u 1 f nþ1 u g nþ1 ; where n P 0, an so, the Dirac integrable hierarchy (.5) (i.e., (.10)) gives rise to u 1 e nþ1 nþ1 n;x þ u a n þ u h n u nþ1 u tn ¼ 6 ¼ ¼ J e nþ ¼ J H u e e n;x u 1 a n þ u 4 h n nþ1 ¼ 6 4 f n;x þ u 4 a n u 1 h n 5 ¼ M 6 4 u g nþ1 f nþ1 n e n f n 5 ¼ M H u e n ; n P 1; u 4 t n f nþ1 g nþ1 g n;x u a n u h n g n ð:1þ where the Hamiltonian pair is given by J ¼ an o þ p þ p p 1 þ p 4 p 4 p 1 p p p 1 þ p 4 o þ p 11 þ p 44 p 14 p 41 p 1 p 4 M ¼ 6 4 p 4 p 1 p 41 p 14 o þ p 44 þ p 11 p 4 þ p 1 p p p 1 p 4 p 4 þ p 1 o þ p þ p 5 ; with p ij = u i o 1 u j,16i,j64. The corresponing hereitary recursion operator [19] for the hierarchy (.1) is p 1 þ p 4 o p p p p p 4 þ p 1 o þ p U ¼ MJ 1 11 þ p 44 p 1 p 4 p 1 p 4 p 14 þ p 41 ¼ 6 4 p 41 p 14 p 4 þ p 1 p 4 þ p 1 o p 44 p 11 5 : p 1 p 4 p þ p o þ p þ p p 4 p 1 Upon choosing a 0 = 1 an h 0 = 0, an setting every integration constant to zero, the first nonlinear system of integrable equations in the hierarchy (.1) reas 8 u 1;t ¼ u ;xx 1 u ðu 1 þ u þ u þ u 4 Þþu ðu 1 u 4 u u Þ; >< u ;t ¼ u 1;xx þ 1 u 1ðu 1 þ u þ u þ u 4 Þþu 4ðu 1 u 4 u u Þ; u ;t ¼ u 4;xx 1 u 4ðu 1 þ u þ u þ u 4 Þ u ð:18þ 1ðu 1 u 4 u u Þ; >: u 4;t ¼ u ;xx þ 1 u ðu 1 þ u þ u þ u 4 Þ u ðu 1 u 4 u u Þ: It has a bi-hamiltonian structure: u t ¼ J H u e ¼ M H u e ;

6 W.-X. Ma / Chaos, Solitons an Fractals 9 (009) with two Hamiltonian functionals being given by 1 eh ¼ ðu 1;xu u 1 u ;x þ u ;x u 4 u u 4;x Þx; eh ¼ 1 u 1u 1;xx 1 u u ;xx 1 u u ;xx 1 u 4u 4;xx 1 8 ðu 1 þ u þ u þ u 4 Þ 1 ðu 1u 4 u u Þ x: If taking u = u 4 = 0, this system reuces to the Dirac nonlinear Schröinger Eq. (.15). 4. Concluing remarks A higher-orer matrix iso-spectral problem was introuce an an associate multi-component integrable hierarchy was presente. The obtaine Dirac integrable hierarchy was prove to possess a bi-hamiltonian structure by using the variational trace ientity. The recursion structure guarantees infinitely many commuting symmetries an conserve ensities. There are other ways to construct multi-component integrable equations such as perturbations [0,5,6], loop algebra methos [1,] an enlarging spectral problems base on semi-irect sums of loop algebras [,8,9]. The relate results an other existing theories [ 5] provie hints for classifying multi-component integrable equations an setting criteria for integrability. It is also interesting to fin conitions on matrix forms, even sufficient conitions, uner which matrix spectral problems can engener multi-component integrable equations with Hamiltonian structures. Acknowlegement Financial support from the National Natural Science Founation of China an the University of South Floria is acknowlege. References [1] Ma WX. In: The encyclopeia of nonlinear science. New York: Fitzroy Dearborn Publishers; 005. p [] Ablowitz MJ, Clarkson PA. Solitons, nonlinear evolution equations an inverse scattering. Lonon mathematical society lecture note series, vol Cambrige: Cambrige University Press; [] Degasperis A. Am J Phys 1988;66:486. [4] Ma WX, Fuchssteiner B. Phys Lett A 1996;1:49. [5] Ma WX. Meth Appl Anal 000;:1. [6] Ma WX. J Math Phys 00;4:1408. [] Ma WX. J Math Phys 005;46:050. [8] Ma WX, Xu XX, hang Y. Phys Lett A 006;51:15. [9] Ma WX, Xu XX, hang Y. J Math Phys 006;4: [10] Tu G. Sci Sinica Ser A 1986;9:18. [11] Guo FK, hang Y. J Phys A: Math Gen 005;8:85. [1] Ma WX, Chen M. J Phys A: Math Gen 006;9:108. [1] Guo FK, hang Y. Preprint [14] Magri F. J Math Phys 198;19:1156. [15] Olver PJ. Applications of Lie groups to ifferential equations. Grauate texts in mathematics, vol. 10. New York: Springer- Verlag; [16] Fokas AS. Stu Appl Math 198;:5. [1] Ma WX, Li KS. Inverse Probl 1996;1:L5. [18] Ma WX. Chin Ann Math B 199;18:9. [19] Fuchssteiner B. Nonlinear Anal 199;:849. [0] Ma WX, Fuchssteiner B. Chaos, Solitons & Fractals 1996;:1. [1] Guo FK, hang Y. J Math Phys 00;44:59. [] Ma WX. Phys Lett A 00;16:. [] van er Linen J, Capel HW, Nijhoff FW. Physica A 1989;160:5. [4] Siorenko Yu, Strampp W. J Math Phys 199;4:149. [5] Xia TC, Fan EG. J Math Phys 005;46:04510.