The generalized Kupershmidt deformation for integrable bi-hamiltonian systems Yuqin Yao Tsinghua University, Beijing, China (with Yunbo Zeng July, 2009
catalogue 1 Introduction 2 KdV6 3 The generalized Kupershmidt deformed KdV hierarchy 4 The generalized Kupershmidt deformed Camassa-Holm 5 The generalized Kupershmidt deformed Boussinesq
Introduction It is an effective approach to construct a new integrable system starting from a bi-hamiltonian system. Fuchssteiner and Fokas (1981 showed that compatible symplectic structures lead to hereditary symmetries, which provides a method to construct a hierarchy of exactly solvable evolution s. Olver and Rosenau(1996 demonstrated that most integrable bi-hamiltonian systems are governed by a compatible trio of Hamiltonian structures, and their recombination leads to integrable hierarchies of nonlinear s. Kupershmidt (2008 proposed the Kupershmidt deformation of the bi-hamiltonian systems.
KdV6 Recently, KdV6 attract more attentions. Karasu-Kalkani et al applied the Painleve analysis to the class of 6th-order nonlinear wave and they have found 4 cases that pass the Painleve test. Three of these were previously known, but the 4th one turned out to be new ( 3 x + 8u x x + 4u xx (u t + u xxx + 6u 2 x = 0. (1 This, as it stands, does not belong to any recognizable theory. In the variables v = u x, w = u t + u xxx + 6u 2 x, (1 is converted to v t + v xxx + 12vv x w x = 0, w xxx + 8vw x + 4wv x = 0, (2a (2b which is referred as KdV6.
KdV6 The authors found Lax pair and an auto-bäcklund transformation for KdV6, but they were unable to find higher symmetries and asked if higher conserved densities and a Hamiltonian formalism exist for KdV6. Kundu A, Sahadevan R et al show that KdV6 possess infinitely many generalized symmetries, conserved quantities and a recursion operator. Kupershmidt described KdV6 as a nonholonomic perturbations of bi-hamiltonian systems. By rescaling v and t in (2, one gets u t = 6uu x + u xxx w x, w xxx + 4uw x + 2wu x = 0, (3a (3b
KdV6 which can be converted into u t = B 1 ( δh 3 δu B 1(ω, B 2 (ω = 0, (4a (4b where B 1 = = x, B 2 = 3 + 2(u + u (5 are the two standard Hamiltonian operators of the KdV hierarchy and H 3 = u 3 u2 x 2. (4 is called the Kupershmidt deformed system. In general, for a bi-hamiltonian system u tn = B 1 ( δh n+1 δu = B 2( δh n δu (6 where B 1 and B 2 are the standard Hamiltonian operators.
KdV6 The Kupershmidt deformation of the bi-hamiltonian system (6 is constructed as follows u tn = B 1 ( δh n+1 δu B 1(ω, B 2 (ω = 0. (7 This deformation is conjectured to preserve integrability and the conjecture is verified in a few representative cases(kupershmidt, 2008 We show that the KdV6 is equivalent to the Rosochatius deformation of KdV with self-consistent sources. We also give the t-type bi-hamiltonian formalism of KdV6 and some new solutions.
The generalized Kupershmidt deformed KdV hierarchy The KdV hierarchy read where u tn = B 1 ( δh n+1 δu = B 2( δh n, n = 1, 2, (8 δu B 1 = = x, B 2 = 3 + 2(u + u H n+1 = 2 2n + 1 Ln u, L = 1 4 2 u + 1 2 1 u x. For N distinct real λ j, consider the spectral problem ϕ jxx + (u λ j ϕ j = 0, j = 1, 2,, N. It is easy to find that δλ j δu = ϕ2 j.
The generalized Kupershmidt deformed KdV hierarchy We generalize Kupershmidt deformation of KdV hierarchy u tn = B 1 ( δh n+1 δu B 1( N ω j, (9a (B 2 λ j B 1 (ω j = 0, j = 1, 2,, N. (9b Since ω j is at the same position as δh n+1 δu, it is reasonable to take ω j = δλ j δu.
The generalized Kupershmidt deformed KdV hierarchy So the generalized Kupershmidt deformation for a bi-hamiltonian systems is proposed as follows u tn = B 1 ( δh n+1 δu N δλ j δu, (B 2 λ j B 1 ( δλ j = 0, j = 1, 2,, N. δu (10b (10a
The generalized Kupershmidt deformed KdV hierarchy From (10b, we can obtain ϕ jxx + (u λ j ϕ j = µ j ϕ 3, j where µ j, j = 1, 2,, N are integrable constants. When n = 2, (10 gives rise to the generalized Kupershmidt deformed KdV u t = 1 4 (u xxx + 6uu x N (ϕ 2 j x, (11a ϕ jxx + (u λ j ϕ j = µ j ϕ 3, j = 1, 2,, N (11b j which is just the Rosochatius deformation of KdV with self-consistent sources.
The generalized Kupershmidt deformed KdV hierarchy The Lax pair is ( ψ1 ψ 2 ( ψ1 ψ 2 V = 1 2 x t = U = V ( ψ1 ψ 2 ( ψ1 ψ 2 ( 0 1, U = λ u 0, ux λ 2 u 2 λ uxx 4 u2 2 + 1 2 ϕ 2 j, (12a 4 λ + u 2 N ( N 1 ϕj ϕ jx ϕ 2 j λ λ j ϕ 2 jx + µ j ϕ ϕ 2 j ϕ. (12b jx j u x 4
The generalized Kupershmidt deformed Camassa-Holm The Camassa-Holm (CH read δh 1 m t = B 1 δu = B δh 0 2 δu = 2u xm um x, m = u u xx + ω (13 where B 1 = + 3, B 2 = m + m H 0 = 1 (u 2 + u 2 2 xdx, H 1 = 1 (u 3 + uu 2 2 xdx.
The generalized Kupershmidt deformed Camassa-Holm We have δλ j δm = λ jϕ 2 j. The generalized Kupershmidt deformed CH is constructed as follows m t = B 1 ( δh 1 N δm 1 δλ j λ j δm = 2u xm um x + (B 2 1 B 1 ( 1 δλ j = 0, j = 1, 2,, N. λ j λ j δm (14b N [(ϕ 2 j x (ϕ 2 j xxx ], (14a
The generalized Kupershmidt deformed Camassa-Holm (14b gives ϕ jxx = 1 ϕ 4 j 1 mλ 2 jϕ j + µ j. ϕ 3 j So Eq.(14 gives the Kupershmidt deformed Camassa-Holm N m t = 2u xm um x + [(ϕ 2 j x (ϕ 2 j xxx], (15a ϕ jxx = 1 4 ϕ j 1 2 mλ jϕ j + µ j, j = 1, 2,, N (15b ϕ 3 j which is called as the RD-CHESCS. Eq.(15 has the lax pair ( ( ( ψ1 ψ1 0 1 = U, U = 1 ψ 2 ψ x 2 1 λm 0 4 2 ( ( ψ1 ψ1 = V, ψ 2 t ψ 2 ( u x2 1 V = u λ u 1 + muλ ux 4 4λ 2 2 (16a ( N λλ j ϕj ϕ jx ϕ 2 j λ λ j ϕ 2 jx + µ j ϕ ϕ 2 j ϕ jx j
The generalized Kupershmidt deformed Boussinesq The Boussinesq is ( ( v δh2 ( δh1 = B δv w 1 δh 2 = B δv 2 δh 1 δw δw where t B 1 = ( 0 0 ( =, 2w x 2 3 vv x 1 6 w xxx, (17 B 2 = 1 ( 2 3 + 2v + v x 3w + 2w x 3 3w + w x 1 6 ( 5 + 5v 3 + 15 2 vx 2 + 9 vxx + 4v 2 + v 2 xxx + 4vv x are the two standard Hamiltonian operators of the Boussinesq and H 1 = wdx, H 2 = ( 1 12 v x 2 1 9 v 3 + w 2 dx.
The generalized Kupershmidt deformed Boussinesq From the following spectral problem and its adjoint spectral problem ϕ jxxx + vϕ jx + ( 1 2 v x + wϕ j = λϕ j, (18a ϕ jxxx + vϕ jx + ( 1 2 v x wϕ j = λϕ j, j = 1, 2,, N. (18b we have δλ j δv = 3 2 (ϕ jxϕ j ϕ j ϕ jx, δλ j δw = 3ϕ jϕ j. The generalized Kupershmidt deformed Boussinesq ( v w t (B 2 λ j B 1 ( δh2 = B 1 ( δv δh 2 ( δλj δv δλ j δw δw N ( δλj δv δλ j δw, (19a = 0, j = 1, 2,, N. (19b
The generalized Kupershmidt deformed Boussinesq By the complicated computation, from (19 we obtain the generalized Kupershmidt deformed Boussinesq N v t = 2w x 3 (ϕ j ϕ j x, (20a w t = 1 6 (4vv x + v xxx 3 2 (ϕ jxxϕ j ϕ j ϕ jxx, ϕ jxxx + vϕ jx + ( 1 2 v x + wϕ j = λ j ϕ j, ϕ jxxx + vϕ jx + ( 1 2 v x wϕ j = λ j ϕ j, j = 1, 2,, N (20b (20c (20d which just is the Boussinesq with self-consistent sources.
The generalized Kupershmidt deformed Boussinesq Lax representation L t = [ 2 + 2 N 3 v + ϕ j 1 ϕ j, L] Lψ = ( 3 + v + 1 2 v x + wψ = λψ, ψ t = ( 2 + 2 N 3 v + ϕ j 1 ϕ j ψ. (21a (21b (21c
The generalized Kupershmidt deformed Jaulent-Miodek The JM hierarchy is ( q r t n = B 1 ( bn+2 b n+1 where ( 0 2 B 1 = 2 q x 2q ( bn+2 b n+1 = B 1 ( δhn+1 δq δh n+1 δr ( δhn = B δq 2 δh n δr ( 2 0, B 2 = 0 r x + 2r 1 2 3 ( bn+1 = L b n, n = 1, 2, b 0 = b 1 = 0, b 2 = 1, H n = 1 n 1 (2b n+2 qb n+1.,
The generalized Kupershmidt deformed Jaulent-Miodek Similarly, the generalized Kupershmidt deformed JM hierarchy is constructed as follows ( q r t n = B 1 ( (B 2 λ j B 1 ( δhn+1 δq δh n+1 ( δλj δq δλ j δr δr + N ( δλj δq δλ j δr, (22a = 0, j = 1, 2,, N. (22b (22b leads to ϕ 2jx = ( λ 2 j + λ j q + rϕ 1j + µ j ϕ 3, j = 1, 2,, N. 1j
The generalized Kupershmidt deformed Jaulent-Miodek Then Eqs.(22 with n = 3 gives rise to the generalized Kupershmidt deformed JM q t = r x 3 2 qq x + 2 N ϕ 1j ϕ 2j, r t = 1 4 q xxx q x r 1 2 qr x + N [2(λ j qϕ 1j ϕ 2j 1 2 q xϕ 2 1j], ϕ 1jx = ϕ 2j, ϕ 2jx = ( λ 2 j + λ j q + rϕ 1j + µ j ϕ 3 1j which just is the RD-JMESCS (23a (23b j = 1, 2,, N (23c
The generalized Kupershmidt deformed Jaulent-Miodek Eq.(23 has the Lax representation (12a with ( 0 1 U = λ 2, + λq + r 0 V = 4 q x λ 1 2 q λ 3 1 2 qλ2 ( 1 2 q2 + rλ + 1 4 q xx 1 2 qr 1 4 q x + 1 ( 0 0 2 λ Φ 1, Φ 1 ΛΦ 1, Φ 1 q Φ 1, Φ 1 0 ( + 1 N 1 φ1j φ 2j φ 2 1j 2 λ λ j φ 2 2j + µ j φ φ 2 1j φ 2j 1j ( 1
The generalized Kupershmidt deformed Jaulent-Miodek Denote the inner product in R N by.,. and Φ i = (ϕ i1, ϕ i2,, ϕ in T, i = 1, 2, µ = (µ 1,, µ N T, Λ = diag(λ 1,, λ N Eq.(23 can be written as ( ( q 1 = B 8 q xx 3 4 qr 5 16 q3 + 1 2 ΛΦ 1, Φ 1 r 1 1 2 r 3 8 q2 + 1 2 Φ 1, Φ 1 t (24a ϕ 1jx = ϕ 2j, ϕ 2jx = λ 2 j ϕ 1j + qλ j ϕ 1j + rϕ 1j + µ j ϕ 3. (24b 1j Notices that Kernel of B 1 is (c 1 + 1 2 qc 2, c 2 T, we may rewrite (24a as 1 8 q xx 3 4 qr 5 16 q3 + 1 2 ΛΦ 1, Φ 1 = c 1 + 1 2 qc 2, 1 2 r 3 8 q2 + 1 2 Φ 1, Φ 1 = c 2 (25a c 1x = 1 2 t(r + 1 4 q2, c 2x = 1 2 tq. (25b
The generalized Kupershmidt deformed Jaulent-Miodek By introducing q 1 = q, p 1 = 1 8 qx, Eqs. (24b and (25b give rise to the t-type Hamiltonian form where R x = G 1 δf 1 δr, R = (Φ T 1, q 1, Φ T 2, p 1, c 1, c 2 T, (26a F 1 = 4p1 2 1 16 q4 1 1 2 q2 1c 2 + q 1c 1 c2 2 + 3 8 q2 1 Φ 1, Φ 1 1 q1 ΛΦ1, Φ1 2 and the + 1 2 Φ2, Φ2 + 1 2 Λ2 Φ 1, Φ 1 + c 2 Φ 1, Φ 1 1 4 t type Hamiltonian operator G 1 is given by 0 I (N+1 (N+1 0 0 G 1 = I (N+1 (N+1 0 0 0 0 0 0 1 0 0 N ϕ 4 1j + 1 2 1 2 t 2 t 0. N µ j ϕ 2 1j, (26b (26c
The generalized Kupershmidt deformed Jaulent-Miodek The Rosochatius deformation of MJM with self-consistent sources (RD-MJMSCS is defined as ( r q t = B 1 ( δ H 2 δũ + δλ ( δũ = B 1 1 2 q x q r + Φ 1, Φ 2 1 2 r 2 3 8 q2 + 1 2 r x + 1 2 Φ 1, Φ 1 (27a ϕ 1jx = r ϕ 1j + λ j ϕ 2j, ϕ 2jx = λ j ϕ 1j + q ϕ 1j + r ϕ 2j + λ j ϕ 3. 1j (27b where B 1 = ( 1 2 0 0 2, H 2 = 1 2 q x r 1 2 q r 2 1 8 q3. µ j
The generalized Kupershmidt deformed Jaulent-Miodek Since the Kernel of B 1 is ( c 1, c 2 T, let 1 2 qx q r + Φ 1, Φ 2 = c 1, 1 2 r 2 3 8 q2 + 1 2 rx + 1 2 Φ 1, Φ 1 = c 2, q 1 = q, p 1 = 1 2 r, R = ( ΦT 1, q 1, Φ T 2, p 1, c 1, c 2 T, then RD-MJMSCS (27 can be written as a t-type Hamiltonian system R x = G 1 δ F 1 δ R F 1 = 2 p 1 c 1 + q 1 c 2 + 2 p 2 1 q 1 + 1 8 q3 1 + 2 p 1 Φ 1, Φ 2 1 2 q1 Φ 1, Φ 1 (28a G 1 = + 1 2 Λ Φ 2, Φ 2 + 1 2 Λ Φ 1, Φ 1 + N µ j, (28b 2λ j ϕ 2 1j 0 I (N+1 (N+1 0 0 I (N+1 (N+1 0 0 0 0 0 2 t 0 0 0 0 1 2 t. (28c
The generalized Kupershmidt deformed Jaulent-Miodek The Miura map relating systems (26 and (28, i.e. R = M( R, is given by Φ 1 = Φ 1, Φ 2 = Λ Φ 2 + 2 p 1 Φ 1, q 1 = q 1, p 1 = 1 2 q 1 p 1 1 4 Φ 1, Φ 2 + 1 4 c 1, (29a (29b Denote c 1 = 1 2 F 1 + t p 1, c 2 = c 2. (29c M DR D R T where DR D R is the Jacobi matrix consisting of Frechet derivative of T M, M denotes adjoint of M.
The generalized Kupershmidt deformed Jaulent-Miodek The second Hamiltonian operator of Eq.(26 0 0 Λ 1 4 Φ1 1 2 Φ2 0 0 0 2Φ T 1 1 q1 4p1 t 0 2 G 2 = M G 1M 1 = Λ 2Φ 1 0 Φ2 g35 0 4 1 4 ΦT 1 1 2 q1 1 4 ΦT 1 2 t g45 0 8 1 2 ΦT 2 4p 1 t g 35 g 45 g 55 tq 1 0 0 0 0 q 1 t 2 t (30 where g 35 = 1 2 q1λφ1 1 2 Λ2 Φ 1 3 8 q2 1Φ 1 c 2Φ 1 + 1 µ1 Φ1 Φ1, Φ1 + (,, µ N T 4 ϕ 3 1j ϕ 3 1N g 45 = 1 2 c1 + 1 4 ΛΦ1, Φ1 3 8 q1 Φ1, Φ1 + 1 2 q1c2 + 1 8 q3 1 g 55 = t( 1 4 Φ1, Φ1 1 2 c2 + ( 1 4 Φ1, Φ1 1 2 c2 t 1 4 q1 tq1. Thus we get the bi-hamiltonian structure for Eq.(26a-(26c δf 1 R x = G 1 δr = δf 0 G2, F0 = 2c1. (31 δr
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