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A tri-Hamiltonian formulation of a new soliton hierarchy associated with so(3,R)
Applied Mathematics Letters 39 (2015) 28–30
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
A tri-Hamiltonian formulation of a new soliton hierarchy associated with so(3, R) Solomon Manukure ∗ , Wen-Xiu Ma, Emmanuel Appiah Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA
article
abstract
info
Article history: Received 31 July 2014 Accepted 8 August 2014 Available online 23 August 2014
A third Hamiltonian operator is presented for a new hierarchy of bi-Hamiltonian soliton equations, thereby showing that this hierarchy is tri-Hamiltonian. Additionally, an inverse hierarchy of common commuting symmetries is also presented. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Hereditary recursion operator Hamiltonian operators Tri-Hamiltonian structure
A new soliton hierarchy presented in [1] reads:
ut m
a = Km = m,x cm,x
=Φ
m
a0,x c0,x
=Φ
m
px , qx
m ≥ 0,
(1)
where the sequence {am , bm , cm |m ≥ 1} is determined by am =
1 2
cm−1,x + pbm ,
cm = qbm − am−1,x
and
q bm,x = − cm−1,x − pam−1,x , 2
with initial values a0 = p,
b0 = 1,
c0 = q.
The hereditary recursion operator Φ is given by
−∂ p∂ −1 p
Φ= −∂ − ∂ q∂ −1 p
1
1 ∂ − ∂ p ∂ −1 q 2 2 , 1 − ∂ q ∂ −1 q
∂=
∂ . ∂x
(2)
2
This hierarchy is derived from an isospectral problem associated with the real special orthogonal Lie algebra, so(3, R), through the zero curvature formulation (see [1] for details). The spectral matrix is a modification of the Dirac spectral matrix and hence similar in structure to the Kaup–Newell spectral matrix which is also a modification of the AKNS spectral matrix. However, this hierarchy (1) is not equivalent to the Kaup-Newell soliton hierarchy (see [1] for details). It is known that the Kaup-Newell soliton hierarchy possesses a tri-Hamiltonian structure [2]. In this letter, we similarly formulate a tri-Hamiltonian structure for this hierarchy (1) and also present an inverse hierarchy of commuting symmetries
∗
Corresponding author. Tel.: +1 8135624035. E-mail address: [email protected] (S. Manukure).
http://dx.doi.org/10.1016/j.aml.2014.08.004 0893-9659/© 2014 Elsevier Ltd. All rights reserved.
S. Manukure et al. / Applied Mathematics Letters 39 (2015) 28–30
29
based on the inverse of the recursion operator Φ defined by (2). It is important to remark that a bi-Hamiltonian structure for this hierarchy (1) is already presented in [1]. We begin by introducing the following family of Hamiltonian operators:
Iα,β =
1 −1 α∂ − q∂ q
−1 + q∂
2 1 + p ∂ −1 q
−1
β∂ − 2p∂
p
−1
,
(3)
p
where α and β are any constants. This family of operators is a simple generalization of the second Hamiltonian operator (M = Φ J) of the Dirac soliton hierarchy associated with so(3, R) (see [3] for details). The proof that Iα,β is Hamiltonian is rather complicated, but similar to that presented in [4]. A special choice leads to a Hamiltonian pair
I =
−1 + q∂ −1 p
1
−1 − q∂ q
2
1 + p∂
−1
−2p∂
q
−1
1
∂ J = 2 0
,
p
0
∂
.
(4)
Now, from the condition, II −1 = I −1 I = I2 , where I2 is the 2 × 2 identity matrix, we obtain, through a simple computation, the inverse operator of I as follows:
−2p∂ −1 p
1 + p ∂ −1 q
−1 − q∂ −1 p
− q∂ −1 q
I −1 =
1
.
(5)
2
As a result, the above Hamiltonian pair gives rise to a hereditary symmetry operator Φ = JI −1 , which is exactly the hereditary recursion operator defined by (2). Let us now assume that 1
1
− ∂ p∂ −1 p∂ 2
M = ΦJ =
1
1
− ∂ − ∂ q∂ 2
2
2
−1
2
p∂
1
2
,
∂ 2 − ∂ p∂ −1 q∂ 1
− ∂ q∂
−1
2
q∂
(6)
and the three operators I , J = Φ I and M = Φ 2 I constitute a Hamiltonian triple, i.e. any linear combination of I , J and M is also Hamiltonian. This is a consequence of a Hamiltonian pair with an invertible recursion operator [5]. It then follows from Magri’s scheme of bi-Hamiltonian formulation [6] and the theory of Hamiltonian methods for evolution equations [7] that the soliton hierarchy (1) has a tri-Hamiltonian formulation ut m = K m = I
δ Hm+1 δ Hm δ Hm−1 =J =M , δu δu δu
m ≥ 1,
(7)
where I and J are defined by (6), M is defined by (8) and the conserved functionals (see [1] for details) also given by
H0 =
p2 +
q2 2
dx,
Hm =
−
2pam + 4bm+1 + qcm 2m
dx,
m ≥ 1.
(8)
A direct computation (e.g. using a computer algebra system) verifies this tri-Hamiltonian formulation (7). The second and last formulations form the bi-Hamiltonian structure presented in [1] while the first, as shown, is a consequence of the decomposition J = Φ I of the Hamiltonian operator J (see, e.g., [2]). Now the inverse of the recursion operator Φ , given by
Φ
−1
−q∂ −1 q∂ −1 = −1 2∂ + 2p∂ −1 q∂ −1
∂ −1 + q∂ −1 p∂ −1 , −2p∂ −1 p∂ −1
(9)
is also a hereditary recursion operator (see eg., [8]). As a result, an inverse hierarchy of soliton equations is presented as follows: ut m = K − m = Φ
−m
K0 = Φ
−m
px , qx
m ≥ 1,
(10)
with infinitely many commuting symmetries:
[K−m , K−n ] = K−′ m (u)[K−n ] − K−′ n (u)[K−m ] = 0,
m, n ≥ 1.
The first and second symmetry systems are
pt qt
q = 2p
and
pt qt
−q∂ −1 q∂ −1 q + 2∂ −1 p + 2q∂ −1 p∂ −1 p = . 2∂ −1 q + 2p∂ −1 q∂ −1 q − 4p∂ −1 p∂ −1 p
(11)
30
S. Manukure et al. / Applied Mathematics Letters 39 (2015) 28–30
The second system is nonlocal. Actually, except the first system all other systems in the inverse hierarchy are nonlocal. We remark that a similar construction is presented in [2] for the Kaup-Newell system. The success in obtaining the inverse hierarchy was due to the existence of the inverse of the recursion operator (9). Such an inverse has not been found for some well-known soliton hierarchies including the WKI hierarchy [9] and the Jaulent–Miodek hierarchy [10], and as a result these hierarchies are not known to have inverse hierarchies. The tri-Hamiltonian formulation of our hierarchy adds to the few famous examples of soliton hierarchies that possess tri-Hamiltonian structures, some of which include the coupled KdV systems [11], the Toda lattice [12], the Volterra lattice [13] and the Kaup-Newell system [14]. Acknowledgments The work was supported in part by NSF under the Grant DMS-1301675, NNSFC under the Grant 11371326, 11271008 and 61072147, Zhejiang Innovation Project of China (Grant No. T200905), and the First-class Discipline of Universities in Shanghai and the Shanghai Univ. Leading Academic Project (No. A.13-0101-12-004). The authors are also grateful to W.Y. Zhang, M. McAnally, C. X. Li, S. M. Yu, Y. Zhou, X. Gu and S. Shen for their valuable comments and discussions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
S. Manukure, W.X. Ma, (preprint, 2014). W.X. Ma, R.G. Zhou, J. Phys. A: Math. Gen. 32 (1999) L239–L242. W.Y. Zhang, W.X. Ma, Int. J. Theor. Phys. (2014) http://dx.doi.org/10.1007/s10773-014-2172-z. W.X. Ma, R.G. Zhou, J. Math. Phys. 40 (1999) 4419–4428. B. Fuchssteiner, A.S. Fokas, Physica D 4 (1981/82) 47–66. F. Magri, J. Math. Phys. 19 (1978) 1156–1162. P.J. Olver, Applications of Lie groups to differential equations, in: Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1986. B. Fuchssteiner, Prog. Theor. Phys. 68 (1982) 1082–1104. M. Wadati, K. Konno, Y.H. Ichikawa, J. Phys. Soc. Japan 47 (1979) 1698. H.Y. Ruan, S.Y. Lou, J. Phys. Soc. Japan 62 (1993) 1917–1921. M. Antonowicz, A.P. Fordy, Physica D 28 (1987) 345–357. B.A. Kupershmidt, Discrete Lax equations and differential-difference calculus, Astérisque 123 (1985) Soc. Math. France, Inst. Henri. Poincare, Paris. T. Tsuchida, Y. Kajinaga, M. Wadati, J. Phys. Soc. Japan 66 (1997) 2608–2617. D.J. Kaup, A.C. Newell, An exact solution for a derivative nonlinear Schrodinger equation, J. Math. Phys. 19 (1978) 798–801.
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
A tri-Hamiltonian formulation of a new soliton hierarchy associated with so(3, R) Solomon Manukure ∗ , Wen-Xiu Ma, Emmanuel Appiah Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA
article
abstract
info
Article history: Received 31 July 2014 Accepted 8 August 2014 Available online 23 August 2014
A third Hamiltonian operator is presented for a new hierarchy of bi-Hamiltonian soliton equations, thereby showing that this hierarchy is tri-Hamiltonian. Additionally, an inverse hierarchy of common commuting symmetries is also presented. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Hereditary recursion operator Hamiltonian operators Tri-Hamiltonian structure
A new soliton hierarchy presented in [1] reads:
ut m
a = Km = m,x cm,x
=Φ
m
a0,x c0,x
=Φ
m
px , qx
m ≥ 0,
(1)
where the sequence {am , bm , cm |m ≥ 1} is determined by am =
1 2
cm−1,x + pbm ,
cm = qbm − am−1,x
and
q bm,x = − cm−1,x − pam−1,x , 2
with initial values a0 = p,
b0 = 1,
c0 = q.
The hereditary recursion operator Φ is given by
−∂ p∂ −1 p
Φ= −∂ − ∂ q∂ −1 p
1
1 ∂ − ∂ p ∂ −1 q 2 2 , 1 − ∂ q ∂ −1 q
∂=
∂ . ∂x
(2)
2
This hierarchy is derived from an isospectral problem associated with the real special orthogonal Lie algebra, so(3, R), through the zero curvature formulation (see [1] for details). The spectral matrix is a modification of the Dirac spectral matrix and hence similar in structure to the Kaup–Newell spectral matrix which is also a modification of the AKNS spectral matrix. However, this hierarchy (1) is not equivalent to the Kaup-Newell soliton hierarchy (see [1] for details). It is known that the Kaup-Newell soliton hierarchy possesses a tri-Hamiltonian structure [2]. In this letter, we similarly formulate a tri-Hamiltonian structure for this hierarchy (1) and also present an inverse hierarchy of commuting symmetries
∗
Corresponding author. Tel.: +1 8135624035. E-mail address: [email protected] (S. Manukure).
http://dx.doi.org/10.1016/j.aml.2014.08.004 0893-9659/© 2014 Elsevier Ltd. All rights reserved.
S. Manukure et al. / Applied Mathematics Letters 39 (2015) 28–30
29
based on the inverse of the recursion operator Φ defined by (2). It is important to remark that a bi-Hamiltonian structure for this hierarchy (1) is already presented in [1]. We begin by introducing the following family of Hamiltonian operators:
Iα,β =
1 −1 α∂ − q∂ q
−1 + q∂
2 1 + p ∂ −1 q
−1
β∂ − 2p∂
p
−1
,
(3)
p
where α and β are any constants. This family of operators is a simple generalization of the second Hamiltonian operator (M = Φ J) of the Dirac soliton hierarchy associated with so(3, R) (see [3] for details). The proof that Iα,β is Hamiltonian is rather complicated, but similar to that presented in [4]. A special choice leads to a Hamiltonian pair
I =
−1 + q∂ −1 p
1
−1 − q∂ q
2
1 + p∂
−1
−2p∂
q
−1
1
∂ J = 2 0
,
p
0
∂
.
(4)
Now, from the condition, II −1 = I −1 I = I2 , where I2 is the 2 × 2 identity matrix, we obtain, through a simple computation, the inverse operator of I as follows:
−2p∂ −1 p
1 + p ∂ −1 q
−1 − q∂ −1 p
− q∂ −1 q
I −1 =
1
.
(5)
2
As a result, the above Hamiltonian pair gives rise to a hereditary symmetry operator Φ = JI −1 , which is exactly the hereditary recursion operator defined by (2). Let us now assume that 1
1
− ∂ p∂ −1 p∂ 2
M = ΦJ =
1
1
− ∂ − ∂ q∂ 2
2
2
−1
2
p∂
1
2
,
∂ 2 − ∂ p∂ −1 q∂ 1
− ∂ q∂
−1
2
q∂
(6)
and the three operators I , J = Φ I and M = Φ 2 I constitute a Hamiltonian triple, i.e. any linear combination of I , J and M is also Hamiltonian. This is a consequence of a Hamiltonian pair with an invertible recursion operator [5]. It then follows from Magri’s scheme of bi-Hamiltonian formulation [6] and the theory of Hamiltonian methods for evolution equations [7] that the soliton hierarchy (1) has a tri-Hamiltonian formulation ut m = K m = I
δ Hm+1 δ Hm δ Hm−1 =J =M , δu δu δu
m ≥ 1,
(7)
where I and J are defined by (6), M is defined by (8) and the conserved functionals (see [1] for details) also given by
H0 =
p2 +
q2 2
dx,
Hm =
−
2pam + 4bm+1 + qcm 2m
dx,
m ≥ 1.
(8)
A direct computation (e.g. using a computer algebra system) verifies this tri-Hamiltonian formulation (7). The second and last formulations form the bi-Hamiltonian structure presented in [1] while the first, as shown, is a consequence of the decomposition J = Φ I of the Hamiltonian operator J (see, e.g., [2]). Now the inverse of the recursion operator Φ , given by
Φ
−1
−q∂ −1 q∂ −1 = −1 2∂ + 2p∂ −1 q∂ −1
∂ −1 + q∂ −1 p∂ −1 , −2p∂ −1 p∂ −1
(9)
is also a hereditary recursion operator (see eg., [8]). As a result, an inverse hierarchy of soliton equations is presented as follows: ut m = K − m = Φ
−m
K0 = Φ
−m
px , qx
m ≥ 1,
(10)
with infinitely many commuting symmetries:
[K−m , K−n ] = K−′ m (u)[K−n ] − K−′ n (u)[K−m ] = 0,
m, n ≥ 1.
The first and second symmetry systems are
pt qt
q = 2p
and
pt qt
−q∂ −1 q∂ −1 q + 2∂ −1 p + 2q∂ −1 p∂ −1 p = . 2∂ −1 q + 2p∂ −1 q∂ −1 q − 4p∂ −1 p∂ −1 p
(11)
30
S. Manukure et al. / Applied Mathematics Letters 39 (2015) 28–30
The second system is nonlocal. Actually, except the first system all other systems in the inverse hierarchy are nonlocal. We remark that a similar construction is presented in [2] for the Kaup-Newell system. The success in obtaining the inverse hierarchy was due to the existence of the inverse of the recursion operator (9). Such an inverse has not been found for some well-known soliton hierarchies including the WKI hierarchy [9] and the Jaulent–Miodek hierarchy [10], and as a result these hierarchies are not known to have inverse hierarchies. The tri-Hamiltonian formulation of our hierarchy adds to the few famous examples of soliton hierarchies that possess tri-Hamiltonian structures, some of which include the coupled KdV systems [11], the Toda lattice [12], the Volterra lattice [13] and the Kaup-Newell system [14]. Acknowledgments The work was supported in part by NSF under the Grant DMS-1301675, NNSFC under the Grant 11371326, 11271008 and 61072147, Zhejiang Innovation Project of China (Grant No. T200905), and the First-class Discipline of Universities in Shanghai and the Shanghai Univ. Leading Academic Project (No. A.13-0101-12-004). The authors are also grateful to W.Y. Zhang, M. McAnally, C. X. Li, S. M. Yu, Y. Zhou, X. Gu and S. Shen for their valuable comments and discussions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
S. Manukure, W.X. Ma, (preprint, 2014). W.X. Ma, R.G. Zhou, J. Phys. A: Math. Gen. 32 (1999) L239–L242. W.Y. Zhang, W.X. Ma, Int. J. Theor. Phys. (2014) http://dx.doi.org/10.1007/s10773-014-2172-z. W.X. Ma, R.G. Zhou, J. Math. Phys. 40 (1999) 4419–4428. B. Fuchssteiner, A.S. Fokas, Physica D 4 (1981/82) 47–66. F. Magri, J. Math. Phys. 19 (1978) 1156–1162. P.J. Olver, Applications of Lie groups to differential equations, in: Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1986. B. Fuchssteiner, Prog. Theor. Phys. 68 (1982) 1082–1104. M. Wadati, K. Konno, Y.H. Ichikawa, J. Phys. Soc. Japan 47 (1979) 1698. H.Y. Ruan, S.Y. Lou, J. Phys. Soc. Japan 62 (1993) 1917–1921. M. Antonowicz, A.P. Fordy, Physica D 28 (1987) 345–357. B.A. Kupershmidt, Discrete Lax equations and differential-difference calculus, Astérisque 123 (1985) Soc. Math. France, Inst. Henri. Poincare, Paris. T. Tsuchida, Y. Kajinaga, M. Wadati, J. Phys. Soc. Japan 66 (1997) 2608–2617. D.J. Kaup, A.C. Newell, An exact solution for a derivative nonlinear Schrodinger equation, J. Math. Phys. 19 (1978) 798–801.