SYMMETRY AND REDUCTIONS OF INTEGRABLE DYNAMICAL SYSTEMS: PEAKON AND THE TODA CHAIN SYSTEMS

Dublin Institute of Technology ARROW@DIT Articles School of Mathematics 014 SYMMETRY AND REDUCTIONS OF INTEGRABLE DYNAMICAL SYSTEMS: PEAKON AND THE TODA CHAIN SYSTEMS Vladimir Gerdjikov INRNE, Sofia, Bulgaria, gerjikov@inrne.bas.bg Rossen Ivanov Dublin Institute of Technology, rivanov@dit.ie Gaetano Vilasi Salerno University, Italy, vilasi@sa.infn.it Follow this and additional works at: http://arrow.dit.ie/scschmatart Part of the Mathematics Commons, Non-linear Dynamics Commons, Ordinary Differential Equations and Applied Dynamics Commons, and the Partial Differential Equations Commons Recommended Citation Gerdjikov, V. Ivanov,R. & Vilasi, G. (014) Symmetry and Reductions of Integrable Dynamical Systems: Peakon and the Toda Chain Systems, Romanian Astron. J., Vol. 4, No. 1, p. 37 48. This Article is brought to you for free and open access by the School of Mathematics at ARROW@DIT. It has been accepted for inclusion in Articles by an authorized administrator of ARROW@DIT. For more information, please contact yvonne.desmond@dit.ie, arrow.admin@dit.ie, brian.widdis@dit.ie. This work is licensed under a Creative Commons Attribution- Noncommercial-Share Alike 3.0 License

SYMMETRY AND REDUCTIONS OF INTEGRABLE DYNAMICAL SYSTEMS: PEAKON AND THE TODA CHAIN SYSTEMS VLADIMIR S. GERDJIKOV 1, ROSSEN I. IVANOV, GAETANO VILASI 3 1 Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria E-mail address: gerjikov@inrne.bas.bg School of Mathematical Sciences, Dublin Institute of Technology, Dublin 8, Ireland E-mail address: Rossen.Ivanov@dit.ie 3 Dipartimento di Scienze Fisiche E.R. Caianiello & INFN Sezione di Napoli, Salerno University, 84084 Fisciano, Salerno, Italy E-mail address: vilasi@sa.infn.it Abstract. We are analyzing several types of dynamical systems which are both integrable and important for physical applications. The first type are the so-called peak on systems that appear in the singular solutions of the Camassa-Holm equation describing special types of water waves. The second type are Toda chain systems, that describe molecule interactions. Their complexifications model soliton interactions in the adiabatic approximation. We analyze the algebraic aspects of the Toda chains and describe their real Hamiltonian forms. Key words: integrable dynamical systems, peakons, Toda lattices. 1. INTRODUCTION Many integrable dynamical systems possess various symmetries, expressed via invariant actions of certain finite groups transforming their variables. In the simplest case these are involutions, exchanging for example two sets of variables. The symmetry groups allow also two or more variables to be made equal (or dependent), thus reducing the overall number of the independent variables and simplifying the system and their solution. The reduced system of course inherits some of the properties of the original system. Most often these integrable systems possess Lax representation L t = [L,M]; the operators L and M usually take values in some underlying simple Lie algebra g. The Lax operators of the reduced systems may take values in an subalgebra of g which is invariant under a finite order automorphism of g (Mikhailov, 1981). An approach for obtaining new families of such systems is based on the construction of their real Hamiltonian forms (RHF) (Gerdjikov et al., 00, 004). This approach first complexifies the dynamical system and then picks up the corresponding RHF by applying a Cartan involution. This construction is specially transparent Romanian Astron. J., Vol. 4, No. 1, p. 37 48, Bucharest, 014

38 Vladimir S. GERDJIKOV, Rossen I. IVANOV, Gaetano VILASI for systems that possess Lax pairs. In Sections and 3 we study a special involution of the so-called peakon systems. Section 4 analyzes the construction of RHF of Toda chain models related to affine twisted Kac-Moody algebras (Helgasson, 001).. THE CAMASSA-HOLM EQUATION AND THE PEAKON SYSTEM The Camassa-Holm (CH) equation (Camassa et al., 1993) among his many and rich properties and structures is famous for possessing the so-called peakon solutions. These are solilary waves with a shape, given by the Green function G(x) of the Helmholtz operator 1 x, i.e. (1 x)g(x) = δ(x) (1) The solution with decaying properties is G(x) = 1 exp( x ). The CH equation has the form (all variables are real) m t + mu x + m x u = 0, m = u u xx, () and is integrable due to the Lax pair ( 1 ) Ψ xx = 4 + λm Ψ, ( 1 ) Ψ t = λ u Ψ x + u x Ψ, (3) The N-peakon solution is N u(x,t) = p k (t)g(x x k (t)), N m(x,t) = p k (t)δ(x q k (t)). (4) The peakon parameters p k (t),q k (t) satisfy an integrable Hamiltonian system q k = H N p k where the Hamiltonian is given by H N = 1 ṗ k = H N q k, (5) N p i p j G(q i q j ). (6) The Lax representation L = [L,M] of (5) is: L jk = p j p k α(q j q k ), M jk = κ p j p k α (q j q k ), (7) Here α (x) denotes derivative with respect to the argument of the function α(x) = G(x/). The peakon Hamiltonian H N may be expressed as a function of the invariants of the matrix L as H N = tr L + (tr L). (8)

3 Symmetry and Reductions of Integrable Dynamical Systems 39 The fact that G(x) is an even function leads to the following properties: (i) The N coordinates q j keep their initial ordering with time. Hence, we can assume natural ordering q 1 q... q N. Physically this means that the particles do not cross each other; (ii) The conjugated momenta p j keep their initial sign. Thus, there are no ambiguities in the expressions p i p j. The peakon-antipeakon case can be easily solved, and the solution is u(x,t) = p(g(x + q/) G(x q/)), where m(x,t) = p(δ(x + q/) δ(x q/)), (9) q = log sech (ct) p = ± c tanh(ct). (10) Note that this is an odd-function solution of the CH equation. More details can be found e.g. in (Holm et al., 009, 010; Ragnisco et al., 1996; Beals et al., 1999; Holm et al., 005). It is well known that geometrically the CH equation is a geodesic flow equation on the diffeomorphism group of the circle, (Holm et al., 1998, 009; Constantin et al., 003). It is known that there are more-general odd-function CH solutions (Constantin, 000), and thus there must exist a multi-peakon odd-function solution with p k = p k, q k = q k, (11) where k = N + 1 k, see the next section. This is related to the presence of a symmetry, more specifically a Z - automorphism. Let us assume that N = n and the variables are divided into two groups, positive and negative as follows q 1 < q <... < q n < 0 < q n+1 <... < q n ; p 1 < p <... < p n < 0 < p n+1 <... < p n. The action of the Z - automorphism is which exchanges the two groups of variables. p k p k, q k q k, (1) 3. PEAKON SYSTEM WITH Z SYMMETRY AND RELATION TO THE TODA CHAIN The Lax pair of the peakon system takes values in the algebra g sl(n). There is a Z - automorphism, given by (1). This allows the following reduction of the peakon system. The structure of L and M under the invariance with respect to the Z - involution p k = p k, q k = q k where k = N + 1 k, and thus N = n is: ( ) L1 ia L = ia T SL 1 S ( ) M1 ia M = ia T, S = SM 1 S n E k,n+1 k, (13)

40 Vladimir S. GERDJIKOV, Rossen I. IVANOV, Gaetano VILASI 4 where E km is n n matrix (E km ) sp = δ ks δ mp ; L and M are n n matrices; L is symmetric and M is anti-symmetric. This means that their n n blocks satisfy: L 1 = L T 1, M 1 = M T 1, A T = SAS. (14) The particular entries of A, M 1 and L 1 can be recovered from (7). The equations that follow from L = [L,M] are L 1 =[L 1,M 1 ] + AA T, Finally, one can verify that L and M satisfy the involutions ΣLΣ 1 = L T, ΣMΣ 1 = M T, Σ = A = (L 1 M 1 )A + AS(M 1 + L 1 )S. (15) ( ) 0 S. (16) S 0 This is a reduction of sl(n) to the symplectic algebra sp(n) and provides oddfunction solutions of the CH equation. In (Ragnisco et al., 1996) it is proven the interesting fact that if p k,q k evolve according to the Hamiltonian H N, the Flashka variables A j = 1 p j p j+1 e j 1 e j, B j = 1 p j 1 e j 1 e j (1 e j 1 )(1 e j ), e j = e ±(q j+1 q j )/, evolve according to the Hamiltonian H = lndetλ, where Λ is the tri-diagonal matrix: Λ = n (17) n 1 B k E kk + A k (E k,k+1 + E k+1,k ). (18) Thus, the peakon lattice is one of the commuting flows of the TC hierarchy (Toda, 1967, 1989; Manakov, 1974; Flaschka, 1974). For N = n with eq. (11) we have A k = A N k, B k = B N k+1, (19) i.e. Λ reduces from sl(n) to its subalgebra sp(n), see e.g. (Gerdjikov et al., 1998). 4. THE TODA CHAINS AND THEIR REAL HAMILTONIAN FORMS It is well known that to each simple Lie algebra one can relate a Toda chain (TC) system. These systems are known to describe molecule interactions (Toda, 1967). There are two main classes of Toda chains: i) the conformal ones (Toda, 1967) and ii) the affine ones (Olshanetzky et al., 1983; Bogoyavlensky, 006). In the 1990-ies it was discovered that the complexifications of the conformal TC related to the algebras sl(n) model soliton interactions in the adiabatic approximation (Gerdjikov et al., 1996; Uzunov et al., 1996; Gerdjikov et al., 1998). Below

5 Symmetry and Reductions of Integrable Dynamical Systems 41 we analyze the algebraic aspects of the Toda chains and describe their families of real Hamiltonan forms. They can be viewed as special cases of the affine Toda field theories widely discussed in the literature (see e.g. (Mikhailov, 1981; Mikhailov et al., 1981) and the references therein). It will be clear below that they are related mostly to the normal real form of the Lie algebra g. Our aim here is to: 1. generalize the TC to complex-valued dynamical variables q C = q 0 + i q 1, and. outline the procedure for constructing the RHF of these TC models. The TC can be written down as finite-dimensional Hamiltonian system: dq k dt = {q k,h}, dp k dt = {p k,h}, H TC = 1 r ( p(t), p(t)) + n k e ( q(t),αk). Here α k, k = 1,...,r are the simple roots of the simple Lie algebra g, α 0 is the minimal root and n k are the positive integers for which α 0 = r n kα k, see (Helgasson, 001). The canonical momenta p and coordinates q satisfy canonical Poisson brackets: k=0 (0) {q k (t),p j (t)} = δ jk. (1) Remark 1 Changing the variables q k q k + ω k (α k,α k ) lnn k where ω k are the fundamental weights of g can make all n k in eq. (0) equal to 1. Next we impose the involution C on the phase space M {q k,p k } n satisfying: 1) C(F (p k,q k )) = F (C(p k ),C(q k )), ) C ({F (p k,q k ),G(p k,q k )}) = {C(F ),C(G)}, () 3) C(H(p k,q k )) = H(p k,q k ). It is important also that the Hamiltonian H(p k,q k ) is an analytic function of the dynamical variables q k (t) and p k (t). The complexification of the TC is rather straightforward. The resulting complex TC can be written down as standard Hamiltonian system with twice as many dynamical variables q a (t), p a (t), a = 0,1: p C (t) = p 0 (t) + i p 1 (t), q C (t) = q 0 (t) + i q 1 (t), (3) {q 0 k (t),p0 j(t)} = {q 1 k (t),p1 j(t)} = δ kj. (4)

4 Vladimir S. GERDJIKOV, Rossen I. IVANOV, Gaetano VILASI 6 The corresponding Hamiltonian and symplectic form equal HCTC C Re H TC ( p 0 + i p 1, q 0 + i q 1 ) = 1 ( p 0, p 0 ) 1 r ( p 1, p 1 ) + n k e ( q 0,α k ) cos(( q 1,α k )), ω C = (d p 0 id q 0 d p 1 d q 1 ). The family of RHF then are obtained from the CTC by imposing an invariance condition with respect to the involution C C where by we denote the complex conjugation. The involution C splits the phase space M C into a direct sum M C M C + M C where k=0 (5) M C + = M 0 im 1, M C = im 0 M 1, (6) The phase space of the RHF is M R M C +. By M 0 and M 1 we denote the eigensubspaces of C, i.e. C(u a ) = ( 1) a u a for any u a M a. Each involution C satisfying 1) - 3) can be related to a RHF of the TC. Condition 3) means that C must preserve the system of admissible roots of g; so C must correspond to Z -symmetry of the extended Dynkin diagrams of g, see (Helgasson, 001; Olive et al., 1986). In (Gerdjikov et al., 00, 004) we gave examples of RHF for TC related to the simple Lie algebras. Here, beside the examples of the sl(5) and so(5)-tc, we analyze the TC related to the Kac-Moody algebra A () 4. 4.1. RHF OF TODA CHAINS RELATED TO A r sl(r + 1,C) LIE ALGEBRAS We start with some basic facts about the algebra g A r sl(r +1) (Helgasson, 001). The set of roots of the algebra is {α = e j e k }, 1 j k r +1. The set of simple roots is π Ar {α k = e k e k+1,k = 1,...,r}. The Cartan-Weyl basis in the typical representation of sl(r + 1) is given by r + 1 r + 1-matrices: H k = E k,k E k+1,k+1, E α = E j,k (7) where we used the matrices (E jk ) mn = δ jm δ kn. It is well known that each positive root e k e j, k < j is a linear combination of simple roots with non-negative coefficients: α e k e j = α k + α + + α j 1. The sum of the coefficients in this expansion is known as the height of the root: ht α = j k. Next we fix up the involution C by: C(q k ) = q r+ k, C(p k ) = p r+ k, k = 1,...,r, (8) The coordinates in M ± are given by p k = p + k + ip k, q k = q + k + iq k : q ± k = 1 (q k q r+ k ), p ± k = 1 (p k p r+ k ), (9)

7 Symmetry and Reductions of Integrable Dynamical Systems 43 where k = 1,...,r, i.e., dimm + = dimm = r. Then the densities HTC R, ωr TC for the RHF of TC equal: H1,TC R = 1 r ( p + k p ) ( r 1 q k + e (q+ k q+ k+1 )/ k cos ) q k+1 + e q+ r / cos ( q r 1 q r ), (30) ω R 1,TC = r ( dp + k dq+ k dp k ) dq k. where p ± k = d q± k /dt. If we put q j = 0, p j = 0 we get the reduced TC related to the algebra B r (or to the height 1 Kac-Moody algebra B r (1) (Olshanetzky et al., 1983). In what follows, for simplicity we will assume r = ; generalization for r > is straightforward. Thus we start with sl(5) whose Coxeter number is 5. Let us introduce in it grading using its Coxeter automorphism: ( ) πi r C 1 = exp 5 H ρ, ρ = ω s = e 1 + e e 4 e 5. (31) s=1 Grading of sl(5) according to C 1 splits it into 5 linear subspaces sl(5) 4 s=0 g (s), (3) where g (s) is the eigensubspace of C 1 corresponding to the eigenvalue ω s 1, ω 1 = exp(πi/5). More specifically g (k) is the set of roots of height k: g (0) h, g (k) l.c.{e j e j+k }, k = 1,...,4 (33) and j + k is taken modulo 5. The Lax pair of the corresponding TC is given by (Manakov, 1974; Flaschka, 1974): L 1 = 1 5 4 p k H ek + a s (E αs + E αs ) + c 0 a 0 (E α0 + E α0 ), s=1 s=1 (34) 4 M 1 = a s (E αs E αs ) + c 0 a 0 (E α0 E α0 ), s=1 where a k = 1/exp((α k, q)/). The parameter c 0 above takes values 0 and 1 corresponding to the two types of Toda chains: conformal and affine respectively. The Lax equation dl 1 /dt = [L 1,M 1 ] is equivalent to the set of equations: da k dt = (p k+1 p k )a k, dp k dt = (a k a k 1 ), (35)

44 Vladimir S. GERDJIKOV, Rossen I. IVANOV, Gaetano VILASI 8 where k = 1,...,5 and k + 1 is taken modulo 5. These are the TC equations for the sl(5)-algebra with Hamiltonian: H 1 tr L := 5 p k 4 + e (αk, q) + c 0 e (α0, q). (36) 4.. REAL HAMILTONIAN FORMS OF B so(5) TODA CHAIN Our next example is related to the so(5) Lie algebra. It can be obtained from sl(5) by imposing invariance with respect to the external automorphism V corresponding to the symmetry of the Dynkin diagram of sl(5): V (X) = S 0 X T S 1 0, S 0 = 5 ( 1) s+1 E s,6 s, (37) The set of roots of so(5) is {±(e 1 ± e ),±e 1,±e }. The Cartan-Weyl basis which is compatible with the automorphism V, takes the form: s=1 H + 1 = E 11 E 55, H + = E E 44, E + kj = E kj S 0 E jk S 0 = E kj ( 1) k+j E j, k, (38) E 1 + and E+ 14 correspond to the roots e 1 e and e 1 + e respectively; E 13 + and E+ 3 correspond to e 1 and e. The Weyl generators for the negative roots are obtained by transposition. Again we can introduce grading using the Coxeter automorphism. However, now the Coxeter number h = 4 and the automorphism is provided by ( ) πi C = exp 4 H ρ, ρ = e 1 + e, (39) Now the grading is according to the height of the roots of so(5) modulo 4. The minimal (resp. maximal) roots is e 1 e (resp. e 1 + e ). If c 0 = 1 we have L = 1 p 1H 1 + + 1 p H + + a 1(E e1 e + E e1 +e ) + a (E e + E e ) + a 0 (E e1 e + E e1 +e ), M = a 1 (E e1 e E e1 +e ) + a (E e E e ) + c 0 a 0 (E e1 e E e1 +e ). The Hamiltonian of the so(5) TC is (40) H := tr L = 1 ( p 1 + p ) + 4(a 1 + a + a 0). (41) where a 1 = 1 exp((q 1 q )/), a = 1 exp(q /) and a 0 = 1 exp( (q 1 + q )/).

9 Symmetry and Reductions of Integrable Dynamical Systems 45 The set of equations: da 1 dt = (p da 1 p )a 1, dt = p da 0 a, dt = (p 1 + p )a 0, (4) dp 1 dt = 4(a 3 a dp 1), dt = 4(a 1 a + a 0), If we put p k = dq k /dt then we get the affine Toda chain related to so(5): d q ( ) 1 dt = e q 1 q e (q 1+q ), d (43) q dt = e(q 1 q ) e q + e (q 1+q ). The RHF of this Toda chain can be generated using the Z -automorphism V of the extended Dynkin diagram which acts as follows: V e 1 = e 1, V e = e. The Hamiltonian for the RHF of this TC is given by (compare with (41)): H RHF = 1 ( p 1 + p ) + e q + e q cos(q 1 ). (44) 4.3. REAL HAMILTONIAN FORMS OF A () 4 TODA CHAIN The last example corresponds to the grading of the height (twisted) Kac- Moody algebra A () 4. The grading in g sl(5) is introduced by C 3 C 1 V, where C 1 is the Coxeter automorphism of sl(5) (31) and V is induced by the symmetry of the Dynkin diagram. This means that V e k = e 6 k. Note that V preserves the subspaces g (s) in (3). As a result each of these subspaces is split into g (s) = g (s) g (s) + so that V X(s) ± = ±X(s) ± for all X(s) ± g(s) ±. Obviously sl(5) is split now into 10 linear subspaces of C 3 and C3 10 = 11. So the eigenvalues of C 3 are ω p 0 where ω = exp(πi/10). In what follows we will need the basis in g (0) +, g(1) and g( 1). g (0) + l. c.{h+ 1,H+ } (45) Let us now introduce basis which is compatible with the automorphism V, namely: E + kj = E kj S 0 E jk S 0 = E kj ( 1) k+j E j, k, V (E+ kj ) = E+ kj, E kj = E kj + S 0 E jk S 0 = E kj + ( 1) k+j E j, k, V (E kj ) = E kj. (46) Obviously, using V each of the subspaces splits into g (k) = g (k) + g(k), g(3) = l. c.{e 14,E 31,E 4 }, g() = l. c.{e 13,E 41,E 4 }, (47) Since C 3 (g (0) + ) g(0) +, C 3(g (3) ) ω 0g (3) and C 3(g () ) ω 1 0 g(), the Lax pair is: L = b 1 H 1 b H + a 1 (E 14 + E 41 ) + a (E 13 + E 31 ) + a 0(E 4 + E 4 ), M = a 1 (E 14 E 41 ) + a (E 13 E 31 ) + a 0( E 4 + E 4 ). (48)

46 Vladimir S. GERDJIKOV, Rossen I. IVANOV, Gaetano VILASI 10 The set of equations: If we put da 1 dt = 1 (p 1 p )a 1, dp 1 dt = 4(a 0 a 1), da dt = p a, dp dt = 4(a 1 a ), da 0 dt = 1 p a, (49) p k = dq k dt, a 1 = 1 e(q 1 q )/, a = 1 e q 1/, a 0 = 1 eq 1, (50) then we get the affine Toda chain related to the twisted Kac-Moody algebra A () 4 : d q 1 dt = eq 1 q + e q 1, The Hamiltonian is equal to tr L and is given by: d q = e q 1 q 4e q. (51) dt H = 1 (p 1 + p ) + e q 1 q + e q 1 + e q. (5) 5. CONCLUSIONS The examples of Toda chains can be generalized to include also the other twisted Kac-Moody algebras. The RHF of these TC are generated by automorphisms, which are symmetries of the extended Dynkin diagrams. For the algebras B r such symmetry exists only for r =. Another problem is to study the RHF of other classes of dynamical systems, such as Hénon-Heiles-type systems (Kostov et al., 010), the complex symmetric Hamiltonian systems proposed in (Ivanov, 006) etc. ACKNOWLEDGEMENTS VG and RI acknowledge financial support from the Irish Research Council (Ireland). VG thanks A. Stefanov and S. Varbev for useful discussions. GV acknowledges financial support from the Agenzia Spaziale Italiana (ASI). REFERENCES R. Beals, D. Sattinger and J. Szmigielski, Multi-peakons and a theorem of Stieltjes, Inv. Problems 15 (1999), L1 L4; arxiv: solv-int/9903011v1. O. I. Bogoyavlensky. On Perturbations of the Periodic Toda Lattice. Commun. math. Phys. 51, 0109 (1976)

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