On complete integrability of the Mikhailov-Novikov-Wang system

ON COMPLETE INTEGRABILITY OF THE MIKHAILOV–NO VIK OV–W ANG SYSTEM PETR VOJ ˇ C ´ AK Abstra ct. W e obtain compatible Hamiltonian and symp lectic structure for a new tw o- compon ent fifth-order integrable system recently found b y Mikh ailo v, Novik ov and W ang, and show that this system p ossesses a hereditary recursion op erator and infi nitely many co m- muting symmetries and conserv ation laws, as well as infi nitely many compatible Hamiltonian and symplectic structures, and is therefore completely integra b le. The system in question admits a red uction to the K aup–Kup ershmidt equation. P ACS 02.30 .Ik In the course of an ongoing classification of integ rab le p olynomial evolutio n systems in t wo indep end en t and t w o dep end en t v ariables Mikhailo v, Novik ov and W a n g [8] (see also [9]) hav e found a system u t = − 5 3 u 5 − 10 v v 3 − 15 v 1 v 2 + 10 uu 3 + 25 u 1 u 2 − 6 v 2 v 1 + 6 v 2 u 1 +12 uv v 1 − 12 u 2 u 1 , v t = 15 v 5 + 30 v 1 v 2 − 30 v 3 u − 45 v 2 u 1 − 35 v 1 u 2 − 10 v u 3 − 6 v 2 v 1 +6 v 2 u 1 + 12 u 2 v 1 + 12 v uu 1 . (1) Here and b elo w u i = ∂ i u ∂ x i , v j = ∂ j v ∂ x j . Note that (1) is one of just tw o new int egrable systems found in [8], and therefore it is n atural to explore its prop erties in order to find out whether it enjo ys any features not present in th e other h igher-ord er integ rab le systems. Up on setting v ≡ 0 the s y s tem (1 ) r educes [8] to the well- kn o wn K aup–Kup ershmidt equa- tion, see e.g. [4, 15] and references therein for more details on the latter. Using the so-called symb olic method Mikhailo v et al. [8] p ro ve d that the system (1) p os- sesses infi nitely m an y generalized symm etries (in the sense of [10]) of orders m ≡ 1 , 5 mo d 6. Ho w ever, this r esult alone neither pro vides an explicit constru ction for the symmetries in ques- tion nor do es it necessarily ent ail the existence of a recursion op erator, see e.g. [1, 14] and references therein. On the other h and, n o r ecursion op erator, symp lectic or (bi-)Hamiltonian structure for (1) was found so far. In view of the imp ortan t role play ed b y these quantitie s in establishing in tegrabilit y , see e.g. [2, 10] and references therein, it is natural to ask whether (1) admits an y suc h quantit ies at all, and it is the goal of the pr esen t pap er to sh o w that this is ind eed the ca se and th us (1) indeed is a complete ly integ r ab le system. T o th is end we ha ve first foun d a few lo w-order s y m metries and co sym m etries of (1 ) and subsequently constructed nonlocal parts of the op erators in question (and then the operators p er se ) using these q u an tities, cf. e.g. [5, 7, 11]. As a result, w e arrive at the follo wing assertion. Key wor ds and phr ases. R ecursion operators, Hamiltonian structure, symp lectic structure, Mikhailov– Novik ov–W ang system, integrable systems, symmetries, bi-Hamiltonian sy stems. 1 2 Theorem 1. Th e system (1) p ossesses a Hamilton ian op er ator P = D 3 x − 6 5 uD x − 3 5 u 1 − 6 5 v D x − 3 5 v 1 − 6 5 v D x − 3 5 v 1 3 D 3 x − ( 18 5 u + 12 5 v ) D x − 9 5 u 1 − 6 5 v 1 ! , (2) a symple ctic op e r ator S = S 11 + 6 5 γ 21 D − 1 x ◦ γ 11 + 6 5 D − 1 x ◦ γ 21 S 12 + 6 5 γ 21 D − 1 x ◦ γ 12 + 6 5 D − 1 x ◦ γ 22 S 21 + 6 5 γ 22 D − 1 x ◦ γ 11 S 22 + 6 5 γ 22 D − 1 x ◦ γ 12 ! , (3) and a her e ditary r e cursion op er ator R = P ◦ S that c an b e written as R = R 11 + G 11 D − 1 x ◦ γ 11 + G 12 D − 1 x ◦ γ 21 R 12 + G 11 D − 1 x ◦ γ 12 + G 12 D − 1 x ◦ γ 22 R 21 + G 21 D − 1 x ◦ γ 11 + G 22 D − 1 x ◦ γ 21 R 22 + G 21 D − 1 x ◦ γ 12 + G 22 D − 1 x ◦ γ 22 ! , (4) wher e S 11 = − D 3 x + 6 uD x + 3 u 1 , S 12 = − 6 v D x + 3 v 1 , S 21 = − 6 v D x − 9 v 1 , S 22 = 9 D 3 x − ( 54 5 u − 36 5 v ) D x − 27 5 u 1 + 18 5 v 1 , γ 11 = 1 , γ 12 = 0 , γ 21 = u 2 − 12 5 u 2 + 6 5 v 2 , γ 22 = − 6 5 v 2 + 12 5 uv − 3 v 2 , R 11 = D 6 x − 36 5 uD 4 x − 108 5 u 1 D 3 x − ( 147 5 u 2 − 324 25 u 2 + 252 25 v 2 ) D 2 x − (21 u 3 − 216 5 uu 1 + 36 v v 1 ) D x − 39 5 u 4 + 738 25 uu 2 − 666 25 v v 2 + 621 25 u 2 1 − 423 25 v 2 1 − 864 125 u 3 + 864 125 uv 2 − 216 125 v 3 , R 12 = 84 5 v D 4 x + 102 5 v 1 D 3 x + ( 63 5 v 2 − 576 25 uv + 252 25 v 2 ) D 2 x +( 21 5 v 3 + 576 25 v v 1 − 144 5 v u 1 − 396 25 uv 1 ) D x − 216 125 uv 2 + 432 125 u 2 v + 36 5 v v 2 − 234 25 u 2 v − 18 5 uv 2 − 36 5 u 1 v 1 + 126 25 v 2 1 + 3 5 v 4 , R 21 = 84 5 v D 4 x + 402 5 v 1 D 3 x − ( 576 25 uv − 729 5 v 2 + 252 25 v 2 ) D 2 x − ( 648 25 v u 1 + 1908 25 uv 1 − 657 5 v 3 + 432 25 v v 1 ) D x + 216 125 v 2 u − 1782 25 uv 2 − 108 25 v v 2 − 486 25 v u 2 + 297 5 v 4 + 378 25 v 2 1 + 432 125 u 2 v − 1656 25 u 1 v 1 , R 22 = − 27 D 6 x + 324 5 uD 4 x + ( 648 5 u 1 − 324 5 v 1 ) D 3 x + ( 252 25 v 2 − 972 25 u 2 − 486 5 v 2 + 729 5 u 2 ) D 2 x +(81 u 3 − 54 v 3 − 1944 25 uu 1 + 684 25 v v 1 − 648 25 v u 1 + 648 25 uv 1 ) D x − 486 25 uu 2 + 432 125 uv 2 − 324 25 v u 2 + 324 25 uv 2 + 198 25 v v 2 − 54 5 v 4 + 153 25 v 2 1 − 243 25 u 2 1 + 81 5 u 4 − 216 125 v 3 , G 11 = − 6 5 u 5 − 36 5 v v 3 − 54 5 v 1 v 2 + 36 5 uu 3 + 18 u 1 u 2 − 108 25 v 2 v 1 + 108 25 v 2 u 1 + 216 25 uv v 1 − 216 25 u 2 u 1 , G 21 = 54 5 v 5 + 108 5 v 1 v 2 − 108 5 v 3 u − 162 5 v 2 u 1 − 126 5 v 1 u 2 − 36 5 v u 3 − 108 25 v 2 v 1 + 108 25 v 2 u 1 + 216 25 u 2 v 1 + 216 25 v uu 1 , G 12 = 18 25 u 1 , G 22 = 18 25 v 1 . This result can b e r eadily verified by straigh tforward b ut tedious computation. 3 Note that w e can write the nonlo cal terms of recursion op erator (4) and symp lectic op erator (3) in the form (cf. [6, 11, 13]) R − = 2 X α =1 G α ⊗ D − 1 x ◦ γ α , (5) S − = 6 5 ( γ T 1 ⊗ D − 1 x ◦ γ 2 + γ T 2 ⊗ D − 1 x ◦ γ 1 ) , (6) where G α = G 1 α G 2 α ! , γ α = ( γ α 1 , γ α 2 ) . This enables one to rewrite the quan tities P , S and R in the so-called Guthrie form [3, 12]. Let us also men tion that th e recursion op erator R ca n b e emplo ye d for the construction of a zero-curv ature representat ion for (1), and we inte n d to add ress this issue in our futur e work. Setting v ≡ 0 in P , S , and R and extracting the app ropriate en tries th er eof we read- ily reco v er the Hamiltonia n , symplectic and recurs ion op erator for the Kaup–Ku p ershm idt equation, cf. e.g. [15] and referen ces therein. Moreo v er, it readily follo ws from Th eorem 1 that the system (1) has, as usu ally is the case for in tegrable s ystems (see e.g. [2]), infinite hierarc h ies of compatible Hamiltonian operators R k ◦ P an d symplectic op erators S ◦ R k , k = 0 , 1 , 2 . . . In particular, this m eans th at (1) is a m ulti-Hamiltonian system. While the Hamiltonian op erator P is lo cal, it is straigh tforwa r d to v erify that all Hamilton- ian op er ators of th e f orm R k ◦ P , k = 1 , 2 , . . . , are n onlo cal. W e conjecture that P is the only lo cal Hamiltonian stru cture for the Mikhailo v–No viko v–W ang system (1). Note also that all symplectic stru ctures S ◦ R k , k = 0 , 1 , 2 , . . . , in cluding S itself, are nonlo cal. F urthermore, it is p ossible to construct t wo infinite sequences of conserved functionals H 1 ,k and H 2 ,k giv en b y the formula δ H i,k = ( R ∗ ) k ( δ H i ) , (7) where H 1 = − 5 3 Z udx, H 2 = Z  5 6 u 2 1 − 5 2 v 2 1 + 4 3 u 3 + 2 3 v 3 − 2 uv 2  dx, R ∗ = S ◦ P is th e f ormal adjoint of R and δ sta n ds for the v ariational deriv ativ e. It is readily seen that all fu nctionals H i,k are in inv olution w ith resp ect to Poisson brac k ets asso ciated with the Ha m iltonian stru ctures R s ◦ P for all s = 0 , 1 , 2 , . . . . By Pr op osition 2 of [13], for all functionals H i,k ≡ R ρ i,k dx , i = 0 , 1 , 2 , . . . , k = 1 , 2, their densities ρ i,k defined recursiv ely through (7) are lo cal, i.e. th ey dep end (at most) on x, t, u, v and a finite num b er of u j and v j . System (1) can b e written in the Hamilto n ian form u t v t ! = P δ H 2 . (8) 4 Let Q 1 = u 1 v 1 ! = P δ H 1 , Q 2 = P δ H 2 , (9) i.e. Q 2 is a column conta in ing the righ t-hand sides of (1 ). Th e recursion op erator (4) and the symmetries with the c h aracteristics Q 1 and Q 2 are readily ve r ified to meet the r equiremen ts of Theorem 1 fr om [13 ], and therefore the symm etries with the charact eristics Q i,j = R j ( Q i ), i = 1 , 2, j = 0 , 1 , 2 , . . . , are lo cal. In fact, it can b e sho wn that for an y giv en i and j the c haracteristic Q i,j dep end s only on u, v , u 1 , v 1 , . . . , u 1+4( i − 1)+6 j , v 1+4( i − 1)+6 j . Moreo v er, as the recursion operator (4) is hereditary and the symmetries with the c haracteristics Q 1 and Q 2 comm ute, so do the symmetries with c h aracteristics Q i,j for all i = 1 , 2 and all j = 0 , 1 , 2 , . . . F o r example, seve nth-order s y m metry of (1) has the c haracteristic Q 1 , 1 = ( Q 1 1 , 1 , Q 2 1 , 1 ) T , where Q 1 1 , 1 = u 7 − 42 5 uu 5 + 126 5 u 3 1 − 126 5 u 1 v 2 1 − 147 5 u 1 u 4 − 1386 25 v 2 u 1 v + 2268 25 uu 1 u 2 + 1512 125 uv 2 u 1 − 2016 125 u 3 u 1 − 252 125 v 3 u 1 + 1512 125 v 1 v u 2 + 504 25 u 2 u 3 − 756 25 uv v 3 − 756 125 v 2 uv 1 − 756 25 uv 1 v 2 + 252 25 v 2 v 3 + 84 5 v 2 v 3 + 21 v 1 v 4 + 84 5 v v 5 − 252 5 u 2 u 3 − 252 25 v 2 u 3 + 126 25 v 3 1 − 1134 25 v 1 u 2 v + 756 25 v v 1 v 2 , Q 2 1 , 1 = − 27 v 7 − 756 25 v 2 u 1 v + 252 25 v 2 v 3 + 126 25 v 3 1 + 189 v 4 u 1 + 483 5 u 4 v 1 − 378 5 u 2 1 v 1 + 1386 5 u 2 v 3 − 756 5 v 2 v 3 − 252 125 v 3 v 1 − 756 25 uv u 3 − 756 25 v 1 u 2 v + 1512 25 uv 1 v 2 + 882 25 v v 1 v 2 − 1134 25 u 1 v u 2 + 378 25 u 1 v 2 1 + 378 5 v 5 u − 252 25 v 2 u 3 − 3024 25 u 2 uv 1 − 378 5 v 1 v 4 − 4536 25 v 2 u 1 u + 1008 125 u 3 v 1 + 756 125 uv 2 u 1 − 1512 25 u 2 v 3 + 1134 5 u 3 v 2 + 1512 125 u 2 u 1 v + 84 5 u 5 v . (10) Let us also ment ion that the formulas (8 ) and (9) are sp ecial cases of a more general relation Q i,j = P δ H i,j , i = 1 , 2 , j = 0 , 1 , 2 , . . . , (11) and the flo ws associated with Q i,j with j > 1 are b i-Hamiltonian, as w e ha v e Q i,j = P δ H i,j = ˜ P δH i,j − 1 , i = 1 , 2 , j = 1 , 2 , . . . . (12) b y virtue of (7). Here ˜ P = R ◦ P . A cknowledgment s The author th anks Dr. A. S ergy ey ev for useful d iscussions. This researc h w as supp orted b y Silesian Univ ersity in Opa v a under the studen t grant pro ject S GS/18/20 10 and b y th e Ministry of Education, Y outh and Sp orts of Czec h Repu blic under the grant MSM47 81305904. Referen ces [1] F. Beukers, J.A. S anders, J.P . W ang, O n Integrabilit y of Systems of Evolution Equations, J. Diff. Eq. 172 (2001), 396–408 . [2] M. B laszak, M ulti-Hamiltonian The ory of Dynamic al Systems ( Heidelb erg: S pringer, 1998). 5 [3] G. A. Guthrie, Recursion operators and non-local symmetries, Pr o c. R oy. So c. 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Sergy eyev, Infinitely many lo cal higher symmetries without recursion operator or master symmetry: integ rability of the F oursov–Burgers system revisited, A cta Applic andae Mathemat i c ae 109 (2010) 273–2 81 (arXiv:0804.20 20 v3). [15] J. P . W ang, A list of 1 + 1 dimensional integrable equations and their prop erties, J. Nonline ar Math. Phys. 9 ( 2002) 213–233. Ma them a tical Institute, Silesian University in Op a v a, Na R ybn ´ ı ˇ cku 1, 746 01 Op a v a, Cz e ch Republic E-mail addr ess : Petr.Vojcak@ma th.slu.cz