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Dear authors, Thank you very much for your contribution to Chinese Physics B. Your paper has been published in Chinese Physics B, 2014, Vol.23, No.7. Attached is the PDF offprint of your published article, which will be convenient and helpful for your communication with peers and coworkers. Readers can download your published article through our website http://www.iop.org/cpb or http://cpb.iphy.ac.cn What follows is a list of related articles published recently in Chinese Physics B.

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Chin. Phys. B Vol. 23, No. 7 (2014) 070201

Symmetries and variational calculation of discrete Hamiltonian systems∗ Xia Li-Li(夏丽莉)a)b) , Chen Li-Qun(陈立群)b)c)d)† , Fu Jing-Li(傅景礼)e) , and Wu Jing-He(吴旌贺)a) a) Department of Physics, Henan Institute of Education, Zhengzhou 450046, China b) Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China c) Department of Mechanics, Shanghai University, Shanghai 200444, China d) Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai 200072, China e) Institute of Mathematical Physics, Zhejiang Sci-Tech University, Hangzhou 310018, China (Received 17 October 2013; revised manuscript received 26 January 2014; published online 16 May 2014)

We present a numerical simulation method of Noether and Lie symmetries for discrete Hamiltonian systems. The Noether and Lie symmetries for the systems are proposed by investigating the invariance properties of discrete Lagrangian in phase space. The numerical calculations of a two-degree-of-freedom nonlinear harmonic oscillator show that the difference discrete variational method preserves the exactness and the invariant quantity.

Keywords: discrete Hamiltonian systems, discrete variational integrators, symmetry, conserved quantity PACS: 02.20.Sv, 02.20.Qs, 11.30.–j, 45.20.Jj

DOI: 10.1088/1674-1056/23/7/070201

1. Introduction Noether theorem reveals that the conserved quantities can be obtained by the corresponding Noether symmetries. [1] The Noether symmetry has gained considerable attention from many researchers, such as Djukic and Vujanovic, [2] and Sarlet and Cantrijn. [3] Lutzky [4,5] proposed the definition of Lie symmetry on the basis of extended Lie groups which leave the differential equation invariant. Mei [6] derived a new type of symmetry (called the form invariance or Mei symmetry) which is different from the Noether symmetry and Lie symmetry. Symmetries are the intrinsic and fundamental features of the differential equations of mathematical physics. [7,8] Discrete systems defined by difference equations provide us with a potentially powerful way to deal with model dynamics in real-world problems. [9] Due to the theoretical significance of the symmetries and the first integrals, there have been many attempts to develop discrete symmetries and conserved quantities. Logan [10] and Dorodnitsyn [11] constructed the discrete Noether theorem by studying the difference Lagrangian systems. Dorodnitsyn and Kozlov [12] also gave the Noether theorem of difference Hamiltonian systems. Fu [13] devised a discrete variational principle and a method to build first integrals for finite-dimensional Lagrange–Maxwell mechanicoelectrical systems and extended discrete Noether symmetries to mechanico-electrical dynamical systems. Levi et al. [14] proposed a method to find the Lie point symmetry transformations, acting simultaneously on difference equations and lattices. Xia and Chen [15] studied the Mei symmetries and con-

structed the discrete versions of Noether conserved quantities for non-conservative Hamiltonian systems. Variational integrators often perform better than their non-variational counterparts. Simulations and animations using these integrators usually have great physical and visual fidelity with low computing cost. [16] Cadzow [17] presented a discrete variational principle and obtained the discrete Euler– Lagrange equation. Wendlandt and Marsden [18] developed the numerical implementation of discrete mechanics on the basis of discrete variational principle. Extensive research has been extended to study the discrete variational principle, which has been reviewed by Marsden et al., [19] Kane et al., [20] and Cort´es and Mart´ınez. [21] Guo and his cooperators [22] obtained the difference discrete variational principle, which both keeps the advantages of Lie’s and Veselov’s methods. This method can be applied to the Lagrangian mechanical systems, [23] the Hamiltonian mechanical systems, [24] the nonholonomic systems, [25] and the field theory. [26] However, there has been no sufficient investigation of the symmetries and discrete variational calculation based on the difference discrete variational principle. Liu [27] studied the discrete variational calculation of Birkhoffian systems and reached the conclusion that the discrete method is as effective as a symplectic scheme. Zhang et al. [28] investigated the invariance properties of the discrete Lagrangian in phase space by discretizing the phase (𝑝, 𝑞) and showed the discrete version of Noether’s theorem for Hamiltonian systems. However, the numerical simulation method of solving symmetric problems is lacking. In the present work,

∗ Project

supported by the Key Program of National Natural Science Foundation of China (Grant No. 11232009), the National Natural Science Foundation of China (Grant Nos. 11072218, 11272287, and 11102060), the Shanghai Leading Academic Discipline Project, China (Grant No. S30106), the Natural Science Foundation of Henan Province, China (Grant No. 132300410051), and the Educational Commission of Henan Province, China (Grant No. 13A140224). † Corresponding author. E-mail: [email protected] © 2014 Chinese Physical Society and IOP Publishing Ltd http://iopscience.iop.org/cpb   http://cpb.iphy.ac.cn

070201-1

Chin. Phys. B Vol. 23, No. 7 (2014) 070201 we study the difference discrete variational principle and the Lie point symmetry on the base of the Lie group transformation on the discretization of the phase space (t, 𝑝, 𝑞) and apply it to a harmonic oscillator system. This paper also demonstrates that the Lie symmetry of the Hamiltonian systems can lead to discrete Noether conserved quantities indirectly. Finally, the numerical curves of these symmetries of discrete Hamilton systems can be obtained. This paper is organized as follows In Section 2, the Hamilton’s equations and the variational integrators are recalled on the discretization of both time and space. In Section 3, the Lie group transformations acting both on the difference equations and the lattice are considered. The determining equations of Lie symmetry are derived for the systems by using the Lie point symmetries. In Section 4, the discrete version of gauge functions and the Noether conserved quantities are constructed via the Logan’s method. In Section 5, the numerical calculation of a two-degree-of-freedom nonlinear harmonic oscillator shows that the difference discrete variational method preserves a symplectic form and invariant quantity. Section 6 draws the conclusion of the present paper.

Euler equation. Equation (3) is the discrete energy conservation law associated to Eqs. (1) and (2). It defines the lattice on which the canonical Hamiltonian equations are discretized. In the continuous limit, the lattice equation disappears. The discrete flow determined by Eqs. (1)–(3) preserves a discrete version of extended canonical two-form ω. Chen et al. [24] have proved that the integrator determined by Eqs. (1) and (2) is a symplectic energy integrator. One proof can be achieved by a straightforward calculation. We can also draw the conclusion via an example of numerical calculations in Section 5. There are different types of equations according to different Lagrangian schemes for Eqs. (1)–(3). Suppose that Ld is obtained by approximating the action function associated with a given Lagrangian [29] Ldα (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ) qk+1 − qk = ((1 − α)pk + α pk+1 ) tk+1 − tk − H ((1 − α)qk + αqk+1 , (1 − α)pk + α pk+1 ) ,

(4)

where h ∈ R+ is the time step and α ∈ [0, 1] is a real parameter. Equations (1) and (2) can be written as (1 − α)(pk−1 − pk ) + α(pk − pk+1 )

2. Discrete variational integrators in Hamiltonian formalism Let Q denote the configuration space with coordinates qi , and T ∗ Q the phase space with coordinate (pi , qi ), where i = 1, 2, . . . , n. Consider the extended phase space R × T ∗ Q with coordinates (t, qi , pi ). We use P × P for discrete version of R × T (T ∗ Q). Here P is the discrete version of R × T ∗ Q. A discrete L is defined as Ld : P × P → R. According to the discrete variational principle in total variation of the correspond action, the discrete Hamiltonian canonical equations and the discrete energy conservation law are [24]

− (tk+1 − tk )D2 H(tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ) − (tk − tk−1 )D5 H(tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) = 0, (5) (1 − α)(qk+1 − qk ) + α(qk − qk−1 ) − (tk+1 − tk )D3 H(tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ) − (tk − tk−1 )D6 H(tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) = 0. (6) Equation (3) can be expressed as qk+1 − qk tk+1 − tk − Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ) qk − qk−1 − ((1 − α)pk−1 + α pk ) tk − tk−1 + Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) = 0, ((1 − α)pk + α pk+1 )

(tk+1 − tk )D2 Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ) + (tk − tk−1 )D5 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) = 0, (1)

which is the energy-conserved quantity

(tk+1 − tk )D3 Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 )

H(tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 )

+ (tk − tk−1 )D6 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) = 0, (2)

− H(tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) = 0.

(tk+1 − tk )D1 Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 )

(8)

Different discrete schemes can be obtained by choosing α correctly. Dorodnitsyn and Kozlov [12] obtained the difference Hamiltonian equations in the case of α = 1/2 in Eqs. (5)–(7).

+ (tk − tk−1 )D4 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) − Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ) + Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) = 0,

(7)

(3)

where Di denotes the partial derivative of Ld with respect to the i-th argument. The 2n equations (Eqs. (1) and (2)) are first-order difference equations, which are the discrete Hamilton canonical equations, also called variational integrators. In the continuous limit, that is, the time t ∈ T ' R, the generalized coordinates qs (s = 1, 2, . . . , n) and ps vary continuously with time, both equations (1) and (2) become the differential

3. Lie symmetry of discrete Lagrangian in phase space The first steps in the construction of Lie symmetries for difference equations were taken by Maeda in 1980, [30] and later extended by Levi and Winternitz, [31] and Dorodnitsyn and Winternitz. [32] Based on these, the Lie point symmetry of the discrete Hamiltonian systems is proposed.

070201-2

Chin. Phys. B Vol. 23, No. 7 (2014) 070201 In what follows, the case of one independent variable tk and depended variable qi,k and pi,k will be considered tk∗ = t + ετk , q∗i,k = qi,k + εξi,k , p∗i,k = pi,k + εηi,k ,

(9)

where ε is a small parameter, the infinitesimal generators τk = τ(tk , qi,k , pi,k ), ξi,k = ξi (tk , qi,k , pi,k ), and ηi,k = ηi (tk , qi,k , pi,k ) are the sequences depending on tk , qi,k , and pi,k . The prolongation of the Lie group operator (5) is expressed as follows: [32] ∂ ∂ ∂ ∂ + ξi,k + ηi,k + τk−1 ∂tk ∂ qi,k ∂ pi,k ∂tk−1 ∂ ∂ ∂ + ξi,k−1 + ηi,k−1 + τk+1 ∂ qi,k−1 ∂ pi,k−1 ∂tk+1 ∂ ∂ + ξi,k+1 + ηi,k+1 , (10) ∂ qi,k+1 ∂ pi,k+1

Pr X = τk

a direct study of the invariance properties of the discrete Lagrangian Ld = L(tk ,tk+1 , qk , qk+1 , pk , pk+1 ) in phase space. Definition 2 If there exists a sequence g(tk ,tk+1 , qk , qk+1 , pk , pk+1 ) such that δ Ld (ετ, εξ , εη) = ε∆ g(tk ,tk+1 , qk , qk+1 , pk , pk+1 ), (15) where δ Ld = (hk+1 D2 Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ) + hk D5 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ))δ qi,k  + ∆ −δ qi,k−1 D2 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) + (hk+1 D3 Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ) + hk D6 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ))δ pi,k  + ∆ −δ pi,k−1 D3 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k )

where τk+1 = τ(tk+1 , qk+1 , pk+1 ), ξi,k+1 = ξi (tk+1 , qk+1 , pk+1 ), ηi,k+1 = ηi (tk+1 , qk+1 , pk+1 ), τk−1 = τ(tk−1 , qk−1 , pk−1 ), ξi,k−1 = ξi (tk−1 , qk−1 , pk−1 ), and ηi,k−1 = ηi (tk−1 , qk−1 , pk−1 ). For simplification, we obtain

+ (hk+1 D1 Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ) + hk D4 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) − Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 )

Π1 = (tk+1 − tk )D2 Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 )

+ Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ))δti,k

+ (tk − tk−1 )D5 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ), (11)

 + ∆ −δti,k−1 D1 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) ,

Π2 = (tk+1 − tk )D3 Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 )

(16)

+ (tk − tk−1 )D6 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ), (12) Π3 = (tk+1 − tk )D1 Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ) + (tk − tk−1 )D4 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) − Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ) + Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ).

(13)

The requirement is that equations (11)–(13), which are invariant under the considered group, are translated into Pr X ∏ |∏1 =0,∏2 =0,∏3 =0 = 0, ρ = 1, 2, 3.

(14)

ρ

Equation (14) is the discrete determining equations of Lie symmetry for Hamiltonian systems. Definition 1 If the infinitesimal generators τk , ξi,k , and ηi,k satisfy the determining equation (14), then the symmetry is the Lie symmetry of discrete Hamiltonian systems (1)–(3).

4. Noether symmetry and conserved quantities Conservation laws of the dynamical mechanical systems can always be obtained by certain symmetries. Lie symmetries can have the Noether conserved quantities under certain conditions. [7] Using the analogy with the continuous case, many results [12,28,33] based on symmetries of discrete functions provide a simple and clear way to construct first integrals of difference Hamiltonian equations. In this section, we propose the discrete Noether theorem in Hamiltonian formalism Ld = L(tk ,tk+1 , qk , qk+1 , pk , pk+1 ). A systematic procedure for the establishment of the first integrals of the discrete Hamiltonian canonical equation can be developed from

where hk+1 = tk+1 − tk and hk = tk − tk−1 for each k, then the discrete Lagrangian in phase space is difference invariance with respect to the infinitesimal transformation (9). Conservation laws have a deep relevance because they express the conservation of physical quantities, such as mass, momentum, angular momentum, energy, and electrical charge. They are also important due to their use in investigating integrability, stability of solutions, or in implementing efficient numerical methods of integration. The point of the Noether theorem is that every symmetry (transformation) corresponds to a conserved quantity. The discrete analogues of Noether theorem for difference equations were reported in Refs. [9]– [12], [28], and [33]. Theorem 1 If the discrete Lagrangian Ld in phase space is difference invariance with respect to the infinitesimal transformations and equations (1)–(3) hold, then one has IN,k = ξi,k−1 D2 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) + ηi,k−1 D3 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) + τD1 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) + g. (17) The difference equations (17) are called the discrete counterparts of Noether conservation laws for the discrete Hamiltonian systems. If a discrete regular function gi,k = g(tk ,tk+1 , qi,k , qi,k+1 , pi,k , pi,k+1 ) exists such that the infinitesimal transformation generators τk , ξi,k , and ηi,k satisfy the discrete Noether identity (15), then the systems possess the discrete conserved quantities (17).

070201-3

Chin. Phys. B Vol. 23, No. 7 (2014) 070201 − Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 )

Proof The variation of Ld is the linear part of the increments of ε

+ Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ))δti,k  + ∆ −δti,k−1 D1 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) = ε∆ g.

∗ Ld (tk∗ ,tk+1 , q∗k , q∗k+1 , p∗k , p∗k+1 ) − Ld (tk ,tk+1 , qk , qk+1 , pk , pk+1 ).

(21)

(18) Utilizing Eqs. (1)–(3) yields

Denote this variation by δ Ld (δtk , δ qi,k , δ pi,k ), then

 ∆ δ qi,k−1 D2 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k )

δ Ld (δtk , δ qi,k , δ pi,k ) ∂ Ld (tk ,tk+1 , qi,k , qi,k+1 , pi,k , pi,k+1 ) = ∂ε ε=0 + D3 Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 )δ qi,k

 + ∆ δ pi,k−1 D3 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k )  + ∆ δti,k−1 D1 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) + ∆ g = 0. (22)

+ D4 Ld (tk+1 , qi,k+1 , pi,k+1 ,tk+2 , qi,k+2 , pi,k+2 )δ qi,k+1 That is,

+ D5 Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 )δ pi,k + D6 Ld (tk+1 , qi,k+1 , pi,k+1 ,tk+2 , qi,k+2 , pi,k+2 )δ pi,k+1

∆ (δ qi,k−1 D2 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k )

+ D1 Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 )δtk

+ δ pi,k−1 D3 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k )

+ D2 Ld (tk+1 , qi,k+1 , pi,k+1 ,tk+2 , qi,k+2 , pi,k+2 )δtk+1 .

+ δti,k−1 D1 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) + g) = 0.

(19)

(23)

A procedure analogous to the integration by parts in classical calculus of variations can be performed on Eq. (19) using the method of the monograph. [34] For Ld = L(tk ,tk+1 , qk , qk+1 , pk , pk+1 ), δ Ld (δtk , δ qi,k , δ pi,k ) can be expressed as

Equation (17) can be obtained. In the Hamiltonian formulation, the Lie point symmetry can have Noether conserved quantity under certain conditions. The discrete Noether theorem of the Hamiltonian systems are proposed. Theorem 2 If the infinitesimal transformation generators τk , ξi,k , and ηi,k satisfy the determining equation (14), the discrete regular function gi,k = gi (tk ,tk+1 , qi,k , qi,k+1 , pi,k , pi,k+1 ) exists and the discrete Noether identity (15) is satisfied, then the Lie point symmetry of the Hamiltonian systems possess the discrete conserved quantities (17).

δ Ld = (hk+1 D2 Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ) + hk D5 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ))δ qi,k  + ∆ −δ qi,k−1 D2 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) + (hk+1 D3 Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ) + hk D6 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ))δ pi,k  + ∆ −δ pi,k−1 D3 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) + (h+ D1 Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ) + h− D4 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) − Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ) + Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ))δti,k  + ∆ −δti,k−1 D1 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) . (20) Considering Eqs. (15) and (20), we have

5. Example The harmonic oscillator is an extremely important physical problem. Many potentials look like a harmonic oscillator near their minimum. Many more physical systems can, at least approximately, be described in terms of harmonic oscillator models. In this paper, a coupled nonlinear two-degreeof-freedom harmonic oscillator will be studied to illustrate the theory. The Lagrangian function of the coupled nonlinear harmonic oscillator is 1 1 L = mq˙2i + γ cos qi + K(q1 − q2 )2 , i = 1, 2, 2 2

(hk+1 D2 Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ) + hk D5 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ))δ qi,k  + ∆ −δ qi,k−1 D2 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) + (hk+1 D3 Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ) + hk D6 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ))δ pi,k  + ∆ −δ pi,k−1 D3 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) + (hk+1 D1 Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ) + hk D4 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k )

(24)

where m is a particle of mass, and γ and K are constants. This system’s configuration is determined by generalized coordinates qi . Consider the discretization of the uniform mesh. According to the Verlet scheme, [35,36] the difference Lagrangian function of a two-degree-of-freedom nonlinear harmonic oscillator can be constructed   qi,k+1 − qi,k h Ld = (pi,k+1 + pi,k ) − Hd , i = 1, 2, (25) 2 hk+1 070201-4

Chin. Phys. B Vol. 23, No. 7 (2014) 070201  1 1 2 2 + (q1,k − q2,k ) + (q1,k+1 − q2,k+1 ) . 2 2

where  1 Hd = (pi,k+1 )2 + (pi,k )2 − cos qi,k − cos qi,k+1 2 1 1 (26) + (q1,k − q2,k )2 + (q1,k+1 − q2,k+1 )2 . 2 2 The time step hk+1 ∈ R+ , m = 1, K = −1, and γ = 1. The difference equations (1) and (2) for the discrete nonlinear harmonic oscillator have the difference discrete variational algorithm. On the discretization of the uniform mesh, they can be rewritten as p1,k+1 + p1,k − 2 p1,k + p1,k−1 h − D2 Hd (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ) + 2 2 h + D5 Hd (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) = 0, (27) 2 q1,k+1 − q1,k 2 q1,k+1 − q1,k h − D3 Hd (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ) + 2 2 h − D6 Hd (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) = 0, (28) 2 p2,k+1 + p2,k − 2 p2,k + p2,k−1 h − D2 Hd (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ) + 2 2 h − D5 Hd (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) = 0, (29) 2 q2,k+1 − q2,k 2 q2,k+1 − q2,k h − D3 Hd (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ) + 2 2 h − D6 Hd (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) = 0. (30) 2 Under the infinitesimal transformation, tk∗ = t + ε, q∗i,k = qi,k , p∗i,k = pi,k .

It is the Noether symmetry. The operator (31) is of both the Lie and Noether symmetries. From Theorem 1, the Noether symmetry of the discrete Hamiltonian systems can possess the discrete conserved quantities directly. From Theorem 2, the Lie symmetry can possess the discrete Noether conserved quantities indirectly. The discrete nonlinear harmonic oscillator possess the discrete conserved quantities IN,k = ξi,k−1 D2 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) + ηi,k−1 D3 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) + τD1 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) + g, (38) or 

1 2 (p + p2i,k ) − cos qi,k − cos qi,k−1 2 i,k−1  1 1 + (q1,k − q2,k )2 + (q1,k+1 − q2,k+1 )2 . 2 2

h I = 2

Hd (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ) = Hd (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) under the transformation operators. The difference conserved quantity (39) is consistent with the energy equation (3) of the system. Equations (1) and (2) are the result of the difference discrete variational principle for difference equations. It is the key point for the following numerical algorithm. In order to implement the variational integrators of the harmonic oscillator, the difference equations can be written as D5 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k )

The corresponding operator is

= −D2 Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ),

∂ . ∂t The determining equations of Lie symmetry are Pr X p1,k−1 − p1,k+1 − 2h sin q1,k − 2 q1,k − q2,k

= −D3 Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ). = 0, (33)

(40)

D6 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k )

(32)



(39)

Equation (3) has

(31)

X=

(37)

(41)

Taking bi,k = D5 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) and ci,k = D6 Ld (tk−1 , qi,k−1 , pi,k−1 ,tk , qi,k , pi,k ) for each k, this equation is simply written as



Pr X q1,k+1 − q1,k − 2hp1,k = 0,

(34)  Pr X p2,k−1 − p2,k+1 − 2h sin q2,k + 2(q1,k − q2,k ) = 0,

bi,k+1 = −D2 Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ), ci,k+1 = −D3 Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ),

(35) Pr X(q2,k+1 − q2,k − 2hp2,k ) = 0.

(36)

The operator (31) or (32) satisfies the discrete determining equations (33)–(36). It is the Lie symmetry. Direct verification shows that equation (25) is difference invariant with  h 1 2 g = (p + p2i,k ) − cos qi,k − cos qi,k−1 2 2 i,k−1

(42)

together with the next update bi,k+1 = D5 Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ), ci,k+1 = D6 Ld (tk , qi,k , pi,k ,tk+1 , qi,k+1 , pi,k+1 ).

(43)

By solving the implicit equation (42) for qi,k+1 and pi,k+1 and then evaluating the explicit equation (43) to

070201-5

Chin. Phys. B Vol. 23, No. 7 (2014) 070201 give bi,k+1 and ci,k+1 , so we update (qi,k , pi,k , bi,k , ci,k ) to (qi,k+1 , pi,k+1 , bi,k+1 , ci,k+1 ). According to the difference discrete variational integrators (27)–(30), the discrete results of the nonlinear harmonic oscillator are derived when we give the same initial conditions p1 = 0.01, q1 = 0.01, p2 = 0.011, q2 = 0.01, and h = 0.01. A numerical result of the preservation of energy is shown for a short time in Fig. 1 for the constant time step case t + − t = t − t − = const between the calculation implementations of variational Verlet and the four-step Runge–Kutta methods. The same pattern is observed if the simulation is carried out for essentially arbitrarily long times. The energy of the nonlinear harmonic oscillator is constant in the variational Verlet scheme. It is consistent with the conclusion that the energy of nonlinear harmonic oscillator system is conservative. It is immediately apparent from this figure that the variational Verlet is typical of symplectic methods compared with the Runge– Kutta scheme. Thus, the difference discrete variational algorithm is more accurate than the four-step Runge–Kutta method in Fig. 2 which shows that the computational result of the trajectories of phase space. The variational Verlet scheme is still periodically orbital, but is unstable from Runge–Kutta. Figure 3 shows the discrete Noether conserved quantity (39) of the nonlinear harmonic oscillator. From the computational result shown in Fig. 3, equation (39) is stable and conservative. The discrete Noether theorem proposes the conserved quantity (39) from the Noether symmetry. The numerical integration result is consistent with that obtained by the Noether theorem. It can follow that the discrete variational method is effective and reasonable for the system. The discrete variational method has theoretical significance for the numerical simulations.

Energy

-. -. . .

 .  p -. -. q -.-.

-.

.

(b)

Energy

-. -. -. -. -. . .

 .  p -. -. q -.-.

.

Fig. 2. Trajectories of the phase space computed with variational Verlet (a) and the Runge–Kutta methods (b).

(In-I)/10

-7

8 (a) 4 0 -4 -8 0

100

200 300 Time

400

500

100

200 300 Time

400

500

3 -9 (In-I)/10

-5 (Ek-E)/10

-. -.

variational verlet 4-RK

2 1 0

0

-3

-1 -2 (a) -3 0 2

-7

(a)

-.

(b)

3

(Ek-E)/10

-.

-6 0 100

200 300 Time

400

500

Fig. 3. (a) Error of the Noether conserved quantity (38). (b) The partial enlarged view.

variational verlet 4-RK

-2

6. Conclusion

-6

We investigate the discretization of Lagrangian in phase space via the difference discrete variational principle. The Lie and Noether symmetries are constructed for discrete Hamiltonian systems based on the discrete variational integrators. Both the Lie and Noether symmetries can lead to discrete Noether conserved quantity. The numerical calculations of the two-degree-of-freedom nonlinear harmonic oscillator demon-

-10 (b) -14 40

60

80

100 Time

120

140

Fig. 1. (a) Comparison of error of energies of coupled nonlinear harmonic oscillators. (b) The partial enlarged view.

070201-6

Chin. Phys. B Vol. 23, No. 7 (2014) 070201 strate that the difference discrete variational method preserves the invariant quantity.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

Noether A E 1918 Math. Phys. KI. 2 235 Djukic D D S and Vujanovic B D 1975 Acta Mech. 23 17 Sarlet W and Cantrijn F 1981 SIAM Rev. 23 467 Lutzky M 1979 J. Phys. A: Math. Gen. 12 973 Lutzky M 1979 Phys. Lett. A 72 86 Mei F X 2000 J. Beijing Inst. Technol. 9 120 Mei F X 1999 Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems (Beijing: Science Press) (in Chinese) Mei F X, Xu X J and Zhang Y F 2004 Acta Mech. Sin. 20 668 Dorodnitsyn V 2011 Applications of Lie Groups to Difference Equations (Boca Raton, FL: Chapman & Hall/CRC) Logan J D 1973 Aequat. Math. 9 210 Dorodnitsyn V 2001 Appl. Numer. Math. 39 307 Dorodnitsyn V and Kozlov R 2009 J. Phys. A: Math. Theor. 42 454007 Fu J L, Dai G D, Salvador J and Tang Y F 2007 Chin. Phys. 16 570 Levi D, Tremblay S and Winternitz P 2000 J. Phys. A: Math. Gen. 33 8507 Xia L L and Chen L Q 2012 Nonlinear Dyn. 70 1223 Grinspun E, Desbrun M, Polthier K, Schr¨oder P and Stern A 2006 Discrete Differential Geometry: An Applied Introduction — The 33rd International Conference and Exhibition on Computer Graphics and Interactive Techniques, July 30–August 3, 2006 Boston, USA, (ACM SIGGRAPH 2006 Course 1)

[17] Cadzow J A 1970 Int. J. Control 11 393 [18] Wendlandt J M and Marsden J E 1997 Physica D: Nonlinear Phenomena 106 223 [19] Marsden J E, Patrick G W and Shkoller S 1998 Commun. Math. Phys. 199 351 [20] Kane C, Marsden J E and Ortiz M 1999 J. Math. Phys. 40 3353 [21] Cort´es J and Mart´ınez S 2001 Nonlinearity 14 1365 [22] Guo H Y, Wu K, Wang S H, Wang S K and Wei J M 2000 Commun. Theor. Phys. 34 307 [23] Guo H Y, Li Y Q and Wu K 2001 Commun. Theor. Phys. 35 703 [24] Chen J B, Guo H Y and Wu K 2003 J. Math. Phys. 44 1688 [25] McLachlan R and Perlmutter M 2006 J. Nonlinear Sci. 16 283 [26] Guo H Y and Wu K 2003 J. Math. Phys. 44 5978 [27] Liu S X, Liu C and Guo Y X 2011 Chin. Phys. B 20 034501 [28] Zhang H B, Chen L Q and Liu R W 2005 Chin. Phys. 14 1063 [29] Kane C, Marsden J E, Ortiz M and West M 1999 Variational Integrators and the Newmark Algorithm for Conservative and Dissipative Mechanical Systems (Ph.D. dissertation) (California: Caltech) [30] Maeda S 1980 Math. Japonica 25 405 [31] Levi D and Winternitz P 1991 Phys. Lett. A 152 335 [32] Dorodnitsyn V and Winternitz P 2000 Nonlinear Dyn. 22 49 [33] Bahar L Y and Kwatny H G 1987 Int. J. Nonlinear Mech. 22 125 [34] Miller K S 1968 The American Mathematical Monthly 75 630 [35] Marsden J E and West M 2001 Acta Numerica 10 357 [36] Hairer E, Lubich C and Wanner G 2003 Acta Numerica 12 399

070201-7

Chinese Physics B Volume 23

Number 7

July 2014

TOPICAL REVIEW — Magnetism, magnetic materials, and interdisciplinary research 077308

Exotic electronic states in the world of flat bands: From theory to material Liu Zheng, Liu Feng and Wu Yong-Shi

077501

Perpendicular magnetic tunnel junction and its application in magnetic random access memory Liu Hou-Fang, Syed Shahbaz Ali and Han Xiu-Feng

078704

Formation of multifunctional Fe3 O4 /Au composite nanoparticles for dual-mode MR/CT imaging applications Hu Yong, Li Jing-Chao, Shen Ming-Wu and Shi Xiang-Yang TOPICAL REVIEW — Statistical physics and complex systems

070501

Nonequilibrium thermodynamics and fluctuation relations for small systems Cao Liang, Ke Pu, Qiao Li-Yan and Zheng Zhi-Gang

070507

Level spacing statistics for two-dimensional massless Dirac billiards Huang Liang, Xu Hong-Ya, Lai Ying-Cheng and Celso Grebogi

070512

Nonequilibrium work equalities in isolated quantum systems Liu Fei and Ouyang Zhong-Can

070513

Equivalent formulations of “the equation of life” Ao Ping

070514

Sub-diffusive scaling with power-law trapping times Luo Liang and Tang Lei-Han

074501

Effective temperature and fluctuation-dissipation relation in athermal granular systems: A review Chen Qiong and Hou Mei-Ying

076402

Percolation on networks with dependence links Li Ming and Wang Bing-Hong

078701

RNA structure prediction: Progress and perspective Shi Ya-Zhou, Wu Yuan-Yan, Wang Feng-Hua and Tan Zhi-Jie

078702

Collective behaviors of suprachiasm nucleus neurons under different light–dark cycles Gu Chang-Gui, Zhang Xin-Hua and Liu Zong-Hua

078705

Proteins: From sequence to structure Zheng Wei-Mou

078901

Statistical physics of hard combinatorial optimization: Vertex cover problem

078902

Zhao Jin-Hua and Zhou Hai-Jun Statistical physics of human beings in games: Controlled experiments Liang Yuan and Huang Ji-Ping (Continued on the Bookbinding Inside Back Cover)

078903

A mini-review on econophysics: Comparative study of Chinese and western financial markets Zheng Bo, Jiang Xiong-Fei and Ni Peng-Yun

078905

Zero-determinant strategy: An underway revolution in game theory Hao Dong, Rong Zhi-Hai and Zhou Tao

078906

Attractive target wave patterns in complex networks consisting of excitable nodes Zhang Li-Sheng, Liao Xu-Hong, Mi Yuan-Yuan, Qian Yu and Hu Gang RAPID COMMUNICATION

073402

A double toroidal analyzer for scanning probe electron energy spectrometer Xu Chun-Kai, Zhang Pan-Ke, Li Meng and Chen Xiang-Jun

077505

Multiferroic properties in terbium orthoferrite Song Yu-Quan, Zhou Wei-Ping, Fang Yong, Yang Yan-Ting, Wang Liao-Yu, Wang Dun-Hui and Du You-Wei GENERAL

070201

Symmetries and variational calculation of discrete Hamiltonian systems Xia Li-Li, Chen Li-Qun, Fu Jing-Li and Wu Jing-He

070202

Non-autonomous discrete Boussinesq equation: Solutions and consistency Nong Li-Juan and Zhang Da-Juan

070203

Rogue-wave pair and dark-bright-rogue wave solutions of the coupled Hirota equations Wang Xin and Chen Yong

070204

Optimal switching policy for performance enhancement of distributed parameter systems based on event-driven control Mu Wen-Ying, Cui Bao-Tong, Lou Xu-Yang and Li Wen

070205

Impulsive effect on exponential synchronization of neural networks with leakage delay under sampleddata feedback control S. Lakshmanan, Ju H. Park, Fathalla A. Rihan and R. Rakkiyappan

070206

Co-evolution of the brand effect and competitiveness in evolving networks

070207

Guo Jin-Li An interpolating reproducing kernel particle method for two-dimensional scatter points Qin Yi-Xiao, Liu Ying-Ying, Li Zhong-Hua and Yang Ming

070208

Average vector field methods for the coupled Schr¨odinger KdV equations Zhang Hong, Song Song-He, Chen Xu-Dong and Zhou Wei-En

070301

Comparison between photon annihilation-then-creation and photon creation-then-annihilation thermal states: Non-classical and non-Gaussian properties Xu Xue-Xiang, Yuan Hong-Chun and Wang Yan

070302

Global entanglement in ground state of {Cu3 } single-molecular magnet with magnetic field Li Ji-Qiang and Zhou Bin

070303

Rise of quantum correlations in non-Markovian environments in continuous-variable systems

070304

Liu Xin and Wu Wei Optimal 1 → 𝑀 phase-covariant cloning in three dimensions Zhang Wen-Hai, Yu Long-Bao, Cao Zhuo-Liang and Ye Liu

(Continued on the Bookbinding Inside Back Cover) 070305

Symmetric quantum discord for a two-qubit state Wang Zhong-Xiao and Wang Bo-Bo

070306

Quantum correlations in a two-qubit anisotropic Heisenberg XY Z chain with uniform magnetic field Li Lei and Yang Guo-Hui

070307

Adiabatic tunneling of Bose–Einstein condensates with modulated atom interaction in a double-well potential Xin Xiao-Tian, Huang Fang, Xu Zhi-Jun and Li Hai-Bin

070308

Ground state of rotating ultracold quantum gases with anisotropic spin orbit coupling and concentrically coupled annular potential Wang Xin, Tan Ren-Bing, Du Zhi-Jing, Zhao Wen-Yu, Zhang Xiao-Fei and Zhang Shou-Gang

070502

Delay-dependent asymptotic stability of mobile ad-hoc networks: A descriptor system approach Yang Juan, Yang Dan, Huang Bin, Zhang Xiao-Hong and Luo Jian-Lu

070503

Mapping equivalent approach to analysis and realization of memristor-based dynamical circuit Bao Bo-Cheng, Hu Feng-Wei, Liu Zhong and Xu Jian-Ping

070504

Signal reconstruction in wireless sensor networks based on a cubature Kalman particle filter Huang Jin-Wang and Feng Jiu-Chao

070505

Space time fractional KdV Burgers equation for dust acoustic shock waves in dusty plasma with nonthermal ions Emad K. El-Shewy, Abeer A. Mahmoud, Ashraf M. Tawfik, Essam M. Abulwafa and Ahmed Elgarayhi

070506

PC synchronization of a class of chaotic systems via event-triggered control Luo Run-Zi and He Long-Min

070508

Partial and complete periodic synchronization in coupled discontinuous map lattices Yang Ke-Li, Chen Hui-Yun, Du Wei-Wei, Jin Tao and Qu Shi-Xian

070509

Distributed formation control for a multi-agent system with dynamic and static obstacle avoidances Cao Jian-Fu, Ling Zhi-Hao, Yuan Yi-Feng and Gao Chong

070510

Fault-tolerant topology in the wireless sensor networks for energy depletion and random failure Liu Bin, Dong Ming-Ru, Yin Rong-Rong and Yin Wen-Xiao

070511

Nonequilibrium behavior of the kinetic metamagnetic spin-5/2 Blume–Capel model

070701

¨ ut Temizer Um¨ Ferromagnetic materials under high pressure in a diamond-anvil cell: A magnetic study Wang Xin, Hu Tian-Li, Han Bing, Jin Hui-Chao, Li Yan, Zhou Qiang and Zhang Tao

070702

Mutator for transferring a memristor emulator into meminductive and memcapacitive circuits Yu Dong-Sheng, Liang Yan, Herbert H. C. Iu and Hu Yi-Hua ATOMIC AND MOLECULAR PHYSICS

073101

2 3 A typical slow reaction H(2 S) + S2 (𝑋 3 Σ− g ) → SH(𝑋 Π) + S( P) on a new surface: Quantum dynamics

calculations Wei Wei, Gao Shou-Bao, Sun Zhao-Peng, Song Yu-Zhi and Meng Qing-Tian (Continued on the Bookbinding Inside Back Cover)

073201

On-chip optical pulse shaper for arbitrary waveform generation Liao Sha-Sha, Yang Ting and Dong Jian-Ji

073301

Dynamical correlation between quantum entanglement and intramolecular energy in molecular vibrations: An algebraic approach Feng Hai-Ran, Meng Xiang-Jia, Li Peng and Zheng Yu-Jun

073401

Potential energy curves and spectroscopic properties of X2 Σ+ and A2 Π states of 13 C14 N Liao Jian-Wen and Yang Chuan-Lu ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID DYNAMICS

074101

A progressive processing method for breast cancer detection via UWB based on an MRI-derived model Xiao Xia, Song Hang, Wang Zong-Jie and Wang Liang

074201

Solar-blind ultraviolet band-pass filter based on metal–dielectric multilayer structures Wang Tian-Jiao, Xu Wei-Zong, Lu Hai, Ren Fang-Fang, Chen Dun-Jun, Zhang Rong and Zheng You-Dou

074202

Scintillation of partially coherent Gaussian–Schell model beam propagation in slant atmospheric turbulence considering inner- and outer-scale effects Li Ya-Qing, Wu Zhen-Sen, Zhang Yuan-Yuan and Wang Ming-Jun

074203

Entropy squeezing and atomic inversion in the 𝑘-photon Jaynes–Cummings model in the presence of the Stark shift and a Kerr medium: A full nonlinear approach H R Baghshahi, M K Tavassoly and A Behjat

074204

Electromagnetically induced grating in a four-level tripod-type atomic system Dong Ya-Bin and Guo Yao-Hua

074205

Application of thermal stress model to paint removal by Q-switched Nd:YAG laser Zou Wan-Fang, Xie Ying-Mao, Xiao Xing, Zeng Xiang-Zhi and Luo Ying

074206

All optical method for measuring the carrier envelope phase from half-cycle cutoffs Li Qian-Guang, Chen Huan, Zhang Xiu and Yi Xu-Nong

074207

Spectral energetic properties of the X-ray-boosted photoionization by an intense few-cycle laser Ge Yu-Cheng and He Hai-Ping

074208

Transversal reverse transformation of anomalous hollow beams in strongly isotropic nonlocal media Dai Zhi-Ping, Yang Zhen-Jun, Zhang Shu-Min, Pang Zhao-Guang and You Kai-Min

074209

Phase transition model of water flow irradiated by high-energy laser in a chamber Wei Ji-Feng, Sun Li-Qun, Zhang Kai and Hu Xiao-Yang

074301

Nonlinear impedances of thermoacoustic stacks with ordered and disordered structures Ge Huan, Fan Li, Xia Jie, Zhang Shu-Yi, Tao Sha, Yang Yue-Tao and Zhang Hui

074302

Integrated physics package of a chip-scale atomic clock Li Shao-Liang, Xu Jing, Zhang Zhi-Qiang, Zhao Lu-Bing, Long Liang and Wu Ya-Ming

074401

Flow and heat transfer of a nanofluid over a hyperbolically stretching sheet A. Ahmad, S. Asghar and A. Alsaedi (Continued on the Bookbinding Inside Back Cover)

074701

Three-dimensional magnetohydrodynamics axisymmetric stagnation flow and heat transfer due to an axisymmetric shrinking/stretching sheet with viscous dissipation and heat source/sink Dinesh Rajotia and R. N. Jat

074702

Molecular dynamics simulations of the nano-droplet impact process on hydrophobic surfaces Hu Hai-Bao, Chen Li-Bin, Bao Lu-Yao and Huang Su-He

074703

Influence of limestone fillers on combustion characteristics of asphalt mortar for pavements Wu Ke, Zhu Kai, Wu Hao, Han Jun, Wang Jin-Chang, Huang Zhi-Yi and Liang Pei PHYSICS OF GASES, PLASMAS, AND ELECTRIC DISCHARGES

075201

Balmer-alpha and Balmer-beta Stark line intensity profiles for high-power hydrogen inductively coupled plasmas Wang Song-Bai, Lei Guang-Jiu, Liu Dong-Ping and Yang Si-Ze

075202

Mitigation of energetic ion debris from Gd plasma using dual laser pulses and the combined effect with ambient gas Dou Yin-Ping, Sun Chang-Kai, Liu Chao-Zhi, Gao Jian, Hao Zuo-Qiang and Lin Jing-Quan

075203

Characteristics of wall sheath and secondary electron emission under different electron temperatures in a Hall thruster Duan Ping, Qin Hai-Juan, Zhou Xin-Wei, Cao An-Ning, Chen Long and Gao Hong

075204

Atmospheric pressure plasma jet utilizing Ar and Ar/H2 O mixtures and its applications to bacteria inactivation Cheng Cheng, Shen Jie, Xiao De-Zhi, Xie Hong-Bing, Lan Yan, Fang Shi-Dong, Meng Yue-Dong and Chu Paul K

075205

Effect of passive structure and toroidal rotation on resistive wall mode stability in the EAST tokamak Liu Guang-Jun, Wan Bao-Nian, Sun You-Wen, Liu Yue-Qiang, Guo Wen-Feng, Hao Guang-Zhou, Ding Si-Ye, Shen Biao, Xiao Bing-Jia and Qian Jin-Ping

075206

Toroidicity and shape dependence of peeling mode growth rates in axisymmetric toroidal plasmas Shi Bing-Ren

075207

DD proton spectrum for diagnosing the areal density of imploded capsules on Shenguang III prototype laser facility Teng Jian, Zhang Tian-Kui, Wu Bo, Pu Yu-Dong, Hong Wei, Shan Lian-Qiang, Zhu Bin, He Wei-Hua, Lu Feng, Wen Xian-Lun, Zhou Wei-Min, Cao Lei-Feng, Jiang Shao-En and Gu Yu-Qiu

075208

Efficiency and stability enhancement of a virtual cathode oscillator Fan Yu-Wei, Li Zhi-Qiang, Shu Ting and Liu Jing

075209

Mode transition in homogenous dielectric barrier discharge in argon at atmospheric pressure Liu Fu-Cheng, He Ya-Feng and Wang Xiao-Fei

075210

Shockwave–boundary layer interaction control by plasma aerodynamic actuation: An experimental investigation Sun Quan, Cui Wei, Li Ying-Hong, Cheng Bang-Qin, Jin Di and Li Jun (Continued on the Bookbinding Inside Back Cover)

CONDENSED MATTER: STRUCTURAL, MECHANICAL, AND THERMAL PROPERTIES 076101

Small-angle X-ray analysis of the effect of grain size on the thermal damage of octahydro-1, 3, 5, 7tetranitro-1, 3, 5, 7 tetrazocine-based plastic-bounded expolsives Yan Guan-Yun, Tian Qiang, Liu Jia-Hui, Chen Bo, Sun Guang-Ai, Huang Ming and Li Xiu-Hong

076102

Quantum confinement and surface chemistry of 0.8–1.6 nm hydrosilylated silicon nanocrystals Pi Xiao-Dong, Wang Rong and Yang De-Ren

076103

Spectroscopic and scanning probe analysis on large-area epitaxial graphene grown under pressure of 4 mbar on 4H-SiC (0001) substrates Wang Dang-Chao and Zhang Yu-Ming

076104

Ferromagnetism on a paramagnetic host background in cobalt-doped Bi2 Se3 topological insulator Zhang Min, L¨u Li, Wei Zhan-Tao, Yang Xin-Sheng and Zhao Yong

076105

Physical properties of FePt nanocomposite doped with Ag atoms: First-principles study Jia Yong-Fei, Shu Xiao-Lin, Xie Yong and Chen Zi-Yu

076301

Effect of size polydispersity on the structural and vibrational characteristics of two-dimensional granular assemblies Zhang Guo-Hua, Sun Qi-Cheng, Shi Zhi-Ping, Feng Xu, Gu Qiang and Jin Feng

076401

Characteristics of phase transitions via intervention in random networks Jia Xiao, Hong Jin-Song, Yang Hong-Chun, Yang Chun, Shi Xiao-Hong and Hu Jian-Quan

076403

Electrical and optical properties of indium tin oxide/epoxy composite film Guo Xia, Guo Chun-Wei, Chen Yu and Su Zhi-Ping

076501

Dynamic thermo-mechanical coupled response of random particulate composites: A statistical two-scale method Yang Zi-Hao, Chen Yun, Yang Zhi-Qiang and Ma Qiang

076801

Fabrication of VO2 thin film by rapid thermal annealing in oxygen atmosphere and its metal–insulator phase transition properties Liang Ji-Ran, Wu Mai-Jun, Hu Ming, Liu Jian, Zhu Nai-Wei, Xia Xiao-Xu and Chen Hong-Da CONDENSED MATTER: ELECTRONIC STRUCTURE, ELECTRICAL, MAGNETIC, AND OPTICAL PROPERTIES

077101

Interaction and spin–orbit effects on a kagome lattice at 1/3 filling Liu Hai-Di, Chen Yao-Hua, Lin Heng-Fu, Tao Hong-Shuai and Wu Jian-Hua

077102

First-principles study of structural, electronic and optical properties of ZnF2 Wu Jian-Bang, Cheng Xin-Lu, Zhang Hong and Xiong Zheng-Wei

077103

Hybrid density functional studies of cadmium vacancy in CdTe Xu Run), Xu Hai-Tao, Tang Min-Yan and Wang Lin-Jun

077104

A theoretical investigation of the band alignment of type-I direct band gap dilute nitride phosphide alloy of GaNx Asy P1−x−y /GaP quantum wells on GaP substrates ¨ L Unsal, ¨ O B G¨on¨ul and M Temiz (Continued on the Bookbinding Inside Back Cover)

077105

Influence of temperature on strain-induced polarization Coulomb field scattering in AlN/GaN heterostructure field-effect transistors L¨u Yuan-Jie, Feng Zhi-Hong, Lin Zhao-Jun, Guo Hong-Yu, Gu Guo-Dong, Yin Jia-Yun, Wang Yuan-Gang, Xu Peng, Song Xu-Bo and Cai Shu-Jun

077201

Design consideration and fabrication of 1.2-kV 4H-SiC trenched-and-implanted vertical junction fieldeffect transistors Chen Si-Zhe and Sheng Kuang

077202

A novel solution-based self-assembly approach to preparing ultralong titanyl phthalocyanine sub-micron wires Zhu Zong-Peng, Wei Bin, Zhang Jian-Hua and Wang Jun

077301

Lattice structures and electronic properties of CIGS/CdS interface: First-principles calculations Tang Fu-Ling, Liu Ran, Xue Hong-Tao, Lu Wen-Jiang, Feng Yu-Dong, Rui Zhi-Yuan, and Huang Min

077302

Efficiency of electrical manipulation in two-dimensional topological insulators Pang Mi and Wu Xiao-Guang

077303

Effect of annealing on performance of PEDOT:PSS/n-GaN Schottky solar cells Feng Qian, Du Kai, Li Yu-Kun, Shi Peng and Feng Qing

077304

Non-recessed-gate quasi-E-mode double heterojunction AlGaN/GaN high electron mobility transistor with high breakdown voltage Mi Min-Han, Zhang Kai, Chen Xing, Zhao Sheng-Lei, Wang Chong, Zhang Jin-Cheng, Ma Xiao-Hua and Hao Yue

077305

Effect of alumina thickness on Al2 O3 /InP interface with post deposition annealing in oxygen ambient Yang Zhuo, Yang Jing-Zhi, Huang Yong, Zhang Kai and Hao Yue

077306

A low specific on-resistance SOI LDMOS with a novel junction field plate Luo Yin-Chun, Luo Xiao-Rong, Hu Gang-Yi, Fan Yuan-Hang, Li Peng-Cheng, Wei Jie, Tan Qiao and Zhang Bo

077307

High dV /dt immunity MOS controlled thyristor using a double variable lateral doping technique for capacitor discharge applications Chen Wan-Jun, Sun Rui-Ze, Peng Chao-Fei and Zhang Bo

077401

Formation of epitaxial Tl2 Ba2 Ca2 Cu3 O10 superconducting films by dc-magnetron sputtering and triple post-annealing method Xie Wei, Wang Pei, Ji Lu, Ge De-Yong, Du Jia-Nan, Gao Xiao-Xin, Liu Xin, Song Feng-Bin, Hu Lei, Zhang Xu, He Ming and Zhao Xin-Jie

077502

Modulation of magnetic properties and enhanced magnetoelectric effects in MnW1−𝑥 Mo𝑥 O4 compounds Fang Yong, Zhou Wei-Ping, Song Yu-Quan, L¨u Li-Ya, Wang Dun-Hui and Du Yu Wei

077503

Substituting Al for Fe in Pr(Al𝑥 Fe1−𝑥 )1.9 alloys: Effects on magnetic and magnetostrictive properties Tang Yan-Mei, Chen Le-Yi, Wei Jun, Tang Shao-Long and Du You-Wei

(Continued on the Bookbinding Inside Back Cover)

077504

Degradation of ferroelectric and weak ferromagnetic properties of BiFeO3 films due to the diffusion of silicon atoms Xiao Ren-Zheng, Zhang Zao-Di, Vasiliy O. Pelenovich, Wang Ze-Song, Zhang Rui, Li Hui, Liu Yong, Huang Zhi-Hong and Fu De-Jun

077601

An electron spin resonance study of Eu doping effect in La4/3 Sr5/3 Mn2 O7 single crystal He Li-Min, Ji Yu, Wu Hong-Ye, Xu Bao, Sun Yun-Bin, Zhang Xue-Feng, Lu Yi and Zhao Jian-Jun

077801

What has been measured by reflection magnetic circular dichroism in Ga1−𝑥 Mn𝑥 As/GaAs structures? He Zhen-Xin, Zheng Hou-Zhi, Huang Xue-Jiao, Wang Hai-Long and Zhao Jian-Hua

077802

Pure blue and white light electroluminescence in a multilayer organic light-emitting diode using a new blue emitter Wei Na, Guo Kun-Ping, Zhou Peng-Chao, Yu Jian-Ning, Wei Bin and Zhang Jian-Hua

077901

Self-organized voids revisited: Experimental verification of the formation mechanism Song Juan, Ye Jun-Yi, Qian Meng-Di, Luo Fang-Fang, Lin Xian, Bian Hua-Dong, Dai Ye, Ma Guo-Hong, Chen Qing-Xi, Jiang Yan, Zhao Quan-Zhong and Qiu Jian-Rong INTERDISCIPLINARY PHYSICS AND RELATED AREAS OF SCIENCE AND TECHNOLOGY

078101

Microwave absorption properties of a double-layer absorber based on nanocomposite BaFe12 O19 /α-Fe and nanocrystalline α-Fe microfibers Shen Xiang-Qian, Liu Hong-Bo, Wang Zhou, Qian Xin-Ye, Jing Mao-Xiang and Yang Xin-Chun

078102

Improved interfacial and electrical properties of GaSb metal oxide semiconductor devices passivated with acidic (NH4 )2 S solution Zhao Lian-Feng, Tan Zhen, Wang Jing and Xu Jun

078401

Hybrid phase-locked loop with fast locking time and low spur in a 0.18-µm CMOS process Zhu Si-Heng, Si Li-Ming, Guo Chao, Shi Jun-Yu and Zhu Wei-Ren

078402

Four-dimensional parameter estimation of plane waves using swarming intelligence Fawad Zaman, Ijaz Mansoor Qureshi, Fahad Munir and Zafar Ullah Khan

078703

Image reconstruction from few views by ℓ0 -norm optimization

078904

Sun Yu-Li and Tao Jin-Xu Row–column visibility graph approach to two-dimensional landscapes Xiao Qin, Pan Xue, Li Xin-Li, Mutua Stephen, Yang Hui-Jie, Jiang Yan, Wang Jian-Yong and Zhang Qing-Jun GEOPHYSICS, ASTRONOMY, AND ASTROPHYSICS

079401

Experimental verification of the parasitic bipolar amplification effect in PMOS single event transients He Yi-Bai and Chen Shu-Ming