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New interaction solutions of the Kadomtsev–Petviashvili equation

Liu Xi-Zhong, Yu Jun, Ren Bo, Yang Jian-Rong Citation:Chin. Phys. B, 2014, 23(10): 100201. doi: 10.1088/1674-1056/23/10/100201

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Prof. Xu Cen-Ke Department of Physics，University of California, Santa Barbara, CA 93106, USA 薛其坤 教授, 院士 Prof. Academician Xue Qi-Kun 清华大学物理系, 北京 100084 Department of Physics, Tsinghua University, Beijing 100084, China 叶 军 教授 Prof. Ye Jun Department of Physics, University of Colorado, Boulder, Colorado 80309- 0440，USA 张振宇 教授 Prof. Z. Y. Zhang Oak Ridge National Laboratory, Oak Ridge, TN 37831–6032, USA 编编编 辑辑辑 Editorial Staff 王久丽 Wang Jiu-Li 章志英 Zhang Zhi-Ying 蔡建伟 Cai Jian-Wei 翟 振 Zhai Zhen 郭红丽 Guo Hong-Li Chin. Phys. B Vol. 23, No. 10 (2014) 100201

New interaction solutions of the Kadomtsev–Petviashvili equation∗

Liu Xi-Zhong(刘希忠)a), Yu Jun(俞 军)a)†, Ren Bo(任 博)a), and Yang Jian-Rong(杨建荣)b)

a)Institute of Nonlinear Science, University, Shaoxing 312000, China b)Department of Physics and Electronics, Shangrao Normal University, Shangrao 334001, China

(Received 24 January 2014; revised manuscript received 8 April 2013; published online 10 August 2014)

The residual symmetry relating to the truncated Painleve´ expansion of the Kadomtsev–Petviashvili (KP) equation is nonlocal, which is localized in this paper by introducing multiple new dependent variables. By using the standard Lie group approach, new symmetry reduction solutions for the KP equation are obtained based on the general form of Lie point symmetry for the prolonged system. In this way, the interaction solutions between solitons and background waves are obtained, which are hard to find by other traditional methods.

Keywords: Kadomtsev–Petviashvili equation, localization procedure, residual symmetry, Backlund¨ transfor- mation, symmetry reduction solution PACS: 02.30.Jr, 02.30.Ik, 05.45.Yv, 47.35.Fg DOI: 10.1088/1674-1056/23/10/100201

1. Introduction terms of symmetry theory. However, the BT-related symme- [12–17] In nonlinear science, many effective ways exist to derive tries are nonlocal, and thus difficult to use directly in exact solutions, including soliton solutions and different wave constructing exact solutions of nonlinear systems. To over- solutions for integrable systems. Among which, the symmetry come that obstacle, one direct way is to localize it by intro- analysis plays an important role. The application of the sym- ducing new dependent variables so that the BT-related sym- [18–20] metry group in dealing with differential equations can be dated metry becomes localized in the new system. Recently, back to Sophus Lie,[1] and various symmetry methods have it was found that the symmetry related to the Painleve´ trun- been developed to find exact solutions for different partial dif- cated expansion is just the residue with respect to the singu- [21] ferential equations since then. As we know, once the Lie point lar manifold, which is called residual symmetry. In fact, symmetry group for a nonlinear system is obtained, one can hierarchies of symmetries, including nonlocal symmetries to reduce the system by using both the classical and nonclassical the Kadomtsev–Petviashvili (KP) equation, are pretty system- [22] Lie group methods[2–7] and the symmetry reduction solutions atically generated from the Lax formulation. Here, in this are consequently derived. The Darboux transformations (DT) paper, we just consider the residual symmetry of the KP equa- and the Backlund¨ transformations (BT) are two other effective tion and use it to construct interacting exact solutions between methods,[8–10] from which, in principle, one can obtain new solitons and different nonlinear waves, which are difficult to solutions from the known ones. However, one can only obtain obtain by other approaches. multiple soliton solutions by taking the seed solution as con- The KP equation stant. For the seed solutions taken as nonconstant nonlinear (ut − 6uux + uxxx)x + 3uyy = 0 (1) waves, it is difficult to find new solutions by using BT or DT. The integrability of a nonlinear system depends on dif- was first derived to study the evolution of long ion–acoustic ferent senses, such as possessing a Lax pair, having infinitely waves of small amplitude propagating in plasmas under the many symmetries, etc.[11] As for testing the Painleve´ integra- effect of long transverse perturbations.[23] The KP equation bility of a nonlinear system, the Painleve´ analysis plays an es- was widely considered as a natural extension of the classical sential role. For any Painleve´ integrable system, a truncated Korteweg–de Vries equation to two spatial dimensions, and expansion exists which relates one solution to another, so it is a has wide applications in almost all physical fields, such as non- particular form of BT. In many cases, the BT for an integrable linear optics,[24] ion–acoustic waves in plasmas, ferromagnet- system has been obtained by various ways, while the corre- ics, Bose–Einstein condensation, and string theory. sponding symmetry is rarely studied. To study the BT-related The paper is organized as follows. In Section 2, we first symmetry and use it to construct exact solutions of a nonlinear obtain the residual symmetry of the KP equation by using the system by a symmetry reduction method would open a new Painleve´ truncated expansion method, then localize it by in- window for studying the properties of integrable systems in troducing multiple new dependent variables. For the enlarged ∗Project supported by the National Natural Science Foundation of China (Grant Nos. 11347183, 11275129, 11305106, 11405110, and 11365017) and the Natural Science Foundation of Zhejiang Province of China (Grant Nos. Y7080455 and LQ13A050001). †Corresponding author. E-mail: [email protected] © 2014 Chinese Physical Society and IOP Publishing Ltd http://iopscience.iop.org/cpb http://cpb.iphy.ac.cn 100201-1 Chin. Phys. B Vol. 23, No. 10 (2014) 100201

−1 KP system, the finite group transformation theorem is conse- and φ in Eq. (4), it is interesting to find that u1 satisfies the quently obtained by using Lie’s first principle. In Section 3, linearized KP equation with u2 as a solution. So, u1 is a sym- the general form of the Lie point symmetry group, which in- metry of the KP equation, which is called residual symmetry. cludes the localized residual symmetry as a special case for Vanishing the coefficients of φ −6, φ −5, and φ −4 in the enlarged KP system, is obtained. The similarity reduc- Eq. (4), we have tions for the system are given according to the standard Lie 2 point symmetry approach with interacting solutions between u0 = 2φx , u1 = −2φxx (5) solitons and nonlinear waves. The last section is devoted to a and short summary and discussion. 1 u = φ −2(−3φ 2 + 4φ φ + φ φ + 3φ 2). (6) 2 6 x xx x xxx x t y 2. Localization of residual symmetry and new Backlund¨ transformation Vanishing the coefficient of φ −3 or φ −2 in Eq. (4) and using Eqs. (5), (6), we get the Schwarz form of the KP equation It is well known that the KP equation is Painleve-´  2    integrable, which means a truncated expansion exists φt 3 φy φy + {φ;x} + 2 + 3 = 0, (7) α φx 2 φx x φx y i−α u = ∑ uiφ , (2) i=0 where φ 3 φ 2 x xxx xx where φ is the singular manifold and α is a positive integer. {φ; } = − 2 . φx 2 φx Substituting Eq. (2) into Eq. (1) and balancing the nonlinear The Schwarz form (7) is invariant under the Mobious¨ transfor- and dispersion terms, we have the truncated Painleve´ expan- mation sion in the form a1φ + b1 u0 u1 φ → , a1a2 6= b1b2, (8) u = + + u . (3) a2φ + b2 φ 2 φ 2 which means equation (7) possesses three symmetries σ = Now, substituting Eq. (3) into the KP Eq. (1), we have φ d1, σφ = d2φ, and 2 3u2yy − 6u2x − 6u2u2xx + u2xxxx + u2xt 2 σφ = d3φ (9) −1 + φ (3u1yy − 6u1u2xx + u1xt − 6u2u1xx −2 with arbitrary constants d , d , and d . It is obvious that the − 12u1xu2x + u1xxxx) + φ (−4u1xφxxx − u1φxt 1 2 3 u residual symmetry σ = u1 is related to the Mobious¨ transfor- − u1φxxxx − 6u1xxφxx − 6u1yφy + 6u2u1φxx mation symmetry (9) by the linearized equation (6). − u1xφt + 12u1φxu2x − 3u1φyy + 3u0yy + 12u2u1xφx Clearly, the residual symmetry − 6u0u2xx + u0xt − 6u2u0xx − 6u1u1xx − 4u1xxxφx 2 −3 σu = −2φxx (10) − 12u0xu2x + u0xxxx − u1t φx − 6u1x) + φ (−12u0xxφxx 2 2 2 − 12u2u1φx + 6u1φy + 8u1φxφxxx + 12u1xxφx with φ satisfying Eq. (7) is nonlocal, which is just the genera-

+ 24u2u0xφx + 24u1xφxφxx − 8u0xxxφx + 24u1u1xφx tor of the BT (3). 2 The residual symmetry of the KP equation is localized as − 2u0t φx − 2u0xφt + 6u1φxx − 6u0φyy − 6u0u1xx 2 a Lie point symmetry + 24u0φxu2x − 8u0xφxxx + 6u1φxx − 2u0φxxxx − 12u0xu1x

+ 12u2u0φxx + 2u1φt φx − 2u0φxt − 12u0yφy σu = h, (11a) −4 2 − 6u1u0xx) + φ (−6u0x + 72u0xφxφxx + 24u0φxφxxx σg = φg, (11b) 2 2 2 2 2 + 36u0xxφx − 36u1φx φxx − 18u1φx + 36u0u1xφx σh = g + φh, (11c) 2 2 φ 2 + 18u φ − 36u u φ + 6u φt φx + 36u u φx 0 xx 2 0 x 0 1 0x σφ = , (11d) 3 2 2 − 24u1xφx + 18u0u1φxx + 18u0φy − 6u0u0xx) −5 2 2 3 for the prolonged system + φ (12u0φxx − 72u0u1φx − 96u0xφx 2 4 − 144u0φx φxx + 48u0u0xφx + 24u1φx ) (ut − 6uux + uxxx)x + 3uyy = 0, (12a) −6 2 2  1 + φ 6u0φx (2φx − u0) = 0. (4) u = φ −2(−3φ 2 + 4φ φ + φ φ + 3φ 2), (12b) 6 x xx x xxx x t y For the arbitrariness of φ, the coefficients of different powers g = φx, (12c) 0 of φ should equal zero. By vanishing the coefficients of φ h = gx, (12d) 100201-2 Chin. Phys. B Vol. 23, No. 10 (2014) 100201 ∂ ∂ ∂ which is a special case of integrable couplings in Ref. [25]. +G + H + Φ , (20) These results can be easily verified by substituting Eq. (11) ∂g ∂h ∂φ into the linearized equations (12) as which means that equation (12) is invariant under the follow-

σφ,x − σg = 0, (13a) ing transformation

σg,x − σh = 0, (13b) {x,y,t,u,g,h,φ} → {x + εX,y + εY,t + εT,u 3 2 φxσφ,xφt + 6σuφx − (σφ,t + 4σφ,xxx)φx +εU,g + εG,h + εH,φ + εΦ} (21) + 2(3φxxσφ,xx + 2σφ,xφxxx − 3φyσφ,y)φx 2 2 − 6σφ,xφxx + 6σφ,xφy = 0, (13c) with an infinitesimal parameter ε. Equivalently, the symmetry 2 in the form (21) can be written as a function form (σφ,xφxx + φxσφ,xx)φt − (σφ,xxxx + σφ,xt + 3σφ,yy)φx  + − 2φxt σφ,x + (σφ,t + 4σφ,xxx)φxx + 4σφ,xxφxxx σu = Xux +Yuy + Tut −U, (22a)  2 − 2φxxxxσφ,x − 6φyyσφ,x φx − 9φxxσφ,xx σg = Xgx +Ygy + Tgt − G, (22b) 2 + 4σφ,xφxxφxxx + 3σφ,xxφy + 6φxxφyσφ,y = 0. (13d) σh = Xhx +Yhy + Tht − H, (22c) In other words, the prolonged system (12) has the Lie point σφ = Xφx +Yφy + Tφt − Φ. (22d) symmetry vector 2 Substituting Eq. (22) into Eq. (13) and eliminating u , g , g , 2 φ x x t V = h∂u + gφ∂g + (g + hφ)∂h + ∂φ , (14) 2 ht , φx, φt , and uyy in terms of the prolonged system (12), we which localizes the residual symmetry. get more than 900 determining equations for the functions X, Now, let us proceed to study the finite transformation Y, T, U, G, H, and Φ. Calculated by computer algebra, the form of Eq. (14), which can be stated in the following theo- solutions in general forms are rem. 1 1 1 Theorem 1 If {u,g,h,φ} is a solution of the prolonged X = f x − f y2 − f y + f , 2 1t 12 1tt 6 2t 3 system (12), then so is {uˆ,gˆ,hˆ,φˆ} with 3 Y = f y + f ,T = f + c , (2g2 − 2hφ + uφ 2)ε2 + (4h − 4uφ)ε + 4u 1t 2 2 1 1 uˆ = 2 , (15a) (εφ − 2) 1 1 2 U = − f1tt x + f1ttt y 4g 12 72 gˆ = , (15b) 1 1 (εφ − 2)2 + f y − f − u f − hc , 36 2tt 6 3t 1t 2 4h 8εg2 hˆ = − , (15c) 1 (εφ − 2)2 (εφ − 2)3 G = − g f1t − c2gφ + c3g, 2 2φ 2 φˆ = − , (15d) H = −c2g − h f1t − c2hφ + c3h, − εφ 2 1 Φ = − c φ 2 + c φ + c , (23) with arbitrary group parameter ε. 2 2 3 4 Proof Using Lie’s first theorem on vector (14) with the where f , f , f are arbitrary functions of t and c , c , c , c corresponding initial condition, i.e., 1 2 3 1 2 3 4 are arbitrary constants. du ˆ(ε) = hˆ(ε), uˆ(0) = u, (16) It is noted that, if we take f1 = f2 = f3 = c1 = c3 = c4 = 0 dε and c2 = 1 in Eq. (23), the obtained general form of the sym- dg ˆ(ε) = φˆ(ε)gˆ(ε), gˆ(0) = g, (17) metry is degenerated into the special form in Eq. (14). The dε time-related x translation invariance and t translation invari- ˆ dh(ε) 2 = gˆ(ε) + φˆ(ε)hˆ(ε), hˆ(0) = h, (18) ance symmetries can also be obtained by setting f1 = f2 = dε c1 = c2 = c3 = c4 = 0, f3 6= 0, and f1 = f2 = f3 = c2 = c3 = dφˆ(ε) φˆ(ε)2 = , φˆ(0) = φ, (19) c4 = 0, c1 = 1, respectively. dε 2 Consequently, the symmetries in Eqs. (22) can be written we find the solutions (15). Thus the theorem is proved. as 1 1 1  3. New symmetry reductions of the KP equation σ = f x − f y2 − f y + f u + ( f y + f )u u 2 1t 12 1tt 6 2t 3 x 1t 2 y The Lie point symmetry of the prolonged KP system (12) 3  1 + f + c u + f x has the general form 2 1 1 t 12 1tt ∂ ∂ ∂ ∂ 1 1 1 V = X +Y + T +U − f y2 − f y + f + u f + hc , ∂x ∂y ∂t ∂u 72 1ttt 36 2tt 6 3t 1t 2 100201-3 Chin. Phys. B Vol. 23, No. 10 (2014) 100201 1 1 1  x 2 2 + − m , (32) σg = f1t x − f1tt y − f2t y + f3 gx + ( f1t y + f2)gy 1/3 2 2 12 6 (3 f1 + 2c1) 3   y 3 1 η = − 2m . (33) + f + c g + g f + c gφ − c g, 2/3 1 2 1 1 t 2 1t 2 3 (3 f1 + 2c1) 1 1 1  σ = f x − f y2 − f y + f h + ( f y + f )h In Eq. (32), we have made h 2 1t 12 1tt 6 2t 3 x 1t 2 y Z 1 3  m = dt (34) + f + c h + c g2 + h f + c hφ − c h, 3 3 f + 2c 2 1 1 t 2 1t 2 3 1 1   1 1 2 1 and σφ = f1t x − f1tt y − f2t y + f3 φx + ( f1t y + f2)φy 2 12 6 6 f = − 1t x   Ω 1/3 3 1 2 (3 f1 + 2c1) + f1 + c1 φt + c2φ − c3φ − c4. (24) 2 2  f 2 f 2  + 1tt − 1t y2 1/3 4/3 Similarity reduction solutions of the enlarged system can (3 f1 + 2c1) (3 f1 + 2c1) 1/3 4/3 be found by setting σu = σg = σh = σφ = 0 in Eq. (24), which +2[3(3 f1 + 2c1) m1t f1t + (3 f1 + 2c1) m1tt ]y is equivalent to solving the corresponding characteristic equa- 2 2 +2(3 f1 + 2c1) m1t − 4(3 f1 + 2c1)m2t (35) tions dx dy dt du dg dh dφ for simplicity. ======. (25) X Y T U G H Φ Substituting Eqs. (28), (29), (30), and (31) into the pro- In the following part of the paper, three nontrivial cases longed KP system, we get the corresponding symmetry reduc- are considered regarding the symmetry reduction. tion equations

Case 1 f1 6= 0 and c1 6= 0. 2 ∆ Φ − c2G = 0, (36) In this case, without loss of generality, we rewrite the ar- ξ 2 ∆ Φξξ − c2H = 0, (37) bitrary functions f2 and f3 as −3Φ2 + 8∆ 2Φ4 + 6Φ2U − (1 + 4Φ )Φ + 3Φ2 = 0, (38) 5/3 η ξ ξ ξξξ ξ ξξ f2 = (3 f1 + 2c1) m1t , (26) 1 and f = − (3 f + 2c )4/3(3 f + 2c )m2 − m  (27) 3 3 1 1 1 1 1t 2t 2 2 4 2 −3Φξξ Φη − 4∆ Φξξ Φ + (3Φηη + Φξξξξ )Φ with m1 and m2 being arbitrary functions of t. ξ ξ 3 Now, by solving Eq. (25), we get − (1 + 4Φξξξ )Φξξ Φξ + 3Φξξ = 0. (39)   q ∆ tanh ∆(m3 + Φ) + c3 2 By solving out G, H, and U from Eqs. (36), (37), (38) φ = , ∆ = 2c2c4 + c3, (28) c2 and then substituting them into Eq. (31), we have the follow- G g = , (29) ing theorem. 1/3 2   (3 f1 + 2c1) cosh ∆(m3 + Φ) Theorem 2 If Φ is a solution of the reduction equa- 2     2c2G sinh ∆(m3 + Φ) − H∆ cosh ∆(m3 + Φ) tion (39), then the u given by h = − , (30) (3 f + 2c )2/3 cosh3  (m + ) 1 1 ∆ ∆ 3 Φ 1 u = 1   2 u = 2 2/3 2 1 2    36(3 f1 + 2c1) 1 + tanh ∆(m3 + Φ) Φ 2/3 2 2 1 ξ 36(3 f1 + 2c1) ∆ 1 + tanh ∆(m3 + Φ) 2 2     2 2 4 2 1 × (18Φη − 48∆ Φξ + ΩΦξ × ∆ 2(Ω + 36U)tanh4 ∆(m + Φ) 2 3 1    2 4 1 +6(4Φξξξ + 1)Φξ − 18Φξξ )tanh ∆(m3 + Φ) −144c H∆ tanh3 ∆(m + Φ) 2 2 2 3 1      −144Φ2Φ ∆ tanh3 ∆(m + Φ) 2 2 2 2 1 ξ ξξ 2 3 + 2∆ (Ω + 36U) + 288c2G tanh ∆(m3 + Φ) 2 2 2 4 2    +2(18Φη + 96∆ Φξ + ΩΦξ + 6(4Φξξξ + 1)Φξ 1 2 −144c2H∆ tanh ∆(m3 + Φ) + ∆ (Ω + 36U) , (31) 1  2 −18Φ2 )tanh2 ∆(m + Φ) ξξ 2 3 where Φ, U, G, and H are group invariants functions of simi- 1  −144Φ2Φ ∆ tanh ∆(m + Φ) + 18Φ2 larity variables ξ and η, which can be expressed as ξ ξξ 2 3 η f 1  = 1t y2 + (3 f + 2c )1/3m y − 2 4 + 2 + ( + ) − 2 ξ 4/3 1 1 1t 48∆ Φξ ΩΦξ 24Φξξξ 6 Φξ 18Φξξ (40) 6(3 f1 + 2c1) 3 100201-4 Chin. Phys. B Vol. 23, No. 10 (2014) 100201 is a solution of the KP equation. The corresponding symmetry reduction equations can be From Eq. (40), we can obtain some interesting solutions. obtained by substituting Eqs. (42), (43), (44), and (45), into a Even Φ is a trivial solution of Eq. (39). As an illustration, we prolonged KP system (12). The results are take the solution of Eq. (39) as Φ = ξ + η and then we find that 2 0 0 3∆ φx0 + 2c2g = 0, (47) 1 u = 2 0 0   2 3∆ φx0x0 − 2c2h = 0, (48) 2/3 2 1 36(3 f1 + 2c1) 1 + tanh ∆(m3 + ξ + η) 0 2 0 2 2 2 2 04 2 12φx0 f2t φt f2 + 216 f2 ∆ f2t φx0     2 0 3 02 2 0 2 2 2 4 1 −9 (4 f2 f3t φ + f2 f2tt φ − 8 f2 u + 4 f3 ) f2t × (24 − 48∆ + Ω)tanh ∆(m3 + ξ + η) 2 2 0 2 02 −4(2 f2 f3t f3 + f2 f2tt φ f3) f2t + 4 f2 f2tt f3 φx0 2 2 +2(24 + 96∆ + Ω)tanh 2 2 0 0 −24( f f f − f f f + 2 f f φ 0 0 0 )φ 0    2t 2 3t 2tt 2 3 2t 2 x x x x 1 2 2 02 2 × ∆(m3 + ξ + η) + 24 − 48∆ + Ω (41) −4(1 − 9 f φ 0 0 ) f = 0, (49) 2 2 x x 2t with ξ,η expressed by Eqs. (32) and (33), is a nontrivial solu- and tion of the KP equation. Case 2 f = c = 0 and f 6= 0. 1 1 2 0 0 2 0 2 2 2 0 2 04 0 0 0 f f − f 0 0 f 0 Similar to Case 1, we get the second-type symmetry re- 6φx φx x 2t φt 2 54 2 ∆ φx x 2t φx 03 3 0 2 2 0 2 02 duction solutions for the enlarged KP system −3φx0 f2t f2 − 6(φx0t f2 − f2 φx0x0x0x0 ) f2t φx0   0  0 0 2 2 0 0 0 1 ∆(6 f3 − f2t y − 3 f2t f2φ ) −12(φx0x0 f2t f2 f3t − φx0x0 f2tt f2 f3 + 2 f2t f2 φx0x0 φx0x0x0 )φx0 φ = c3 − ∆ tanh , (42) c2 2 f2t f2 2 03 0 2 +2(9 f2 φx0x0 − φx0x0 ) f2t = 0. (50) g0 g = , (43) ∆(6 f − f y − 3 f f φ 0) cosh2 3 2t 2t 2 Similar to theorem 2, solving out g0, h0, and φ 0 from 2 f f 2t 2 Eqs. (47), (48), (49) and substituting them into Eq. (45), we 1 h =  0  have the following theorem. 3 ∆(6 f3 − f2t y − 3 f2t f2φ ) ∆ cosh Theorem 3 If φ 0 is a solution of the symmetry reduction 2 f2t f2   0  equation (50), then the u given by 02 ∆(6 f3 − f2t y − 3 f2t f2φ ) × 2c2g sinh 2 f2t f2  0   0  9 2 02 2 ∆(6 f3 − f2t y − 3 f2t f2φ ) 0 ∆(6 f3 − f2t y − 3 f2t f2φ ) u = − ∆ φx0 tanh +∆h cosh , (44) 2 2 f2t f2 2 f2t f2  0  0 ∆(6 f3 − f2t y − 3 f2t f2φ ) 2  0  +3 0 0 tanh 2c2 02 2 ∆(6 f3 − f2t y − 3 f2t f2φ ) ∆φx x 2 f2t f2 u = 2 g tanh ∆ 2 f2t f2 1  0 2 0 2 2 2 2 04  0  − 12φ 0 f φ f + 216 f ∆ f φ 0 2c2 0 ∆(6 f3 − f2t y − 3 f2t f2φ ) 2 2 02 x 2t t 2 2 2t x + h tanh 72 f2 f2t φx0 ∆ 2 f2t f2 2 2 2 02 +(12 f2 f3t y − 36 f3 − f2 f2tt y ) f2t φx0 1  0 − (12y + 36φ f2) f3t 2 2 0 0 2 −24( f t f f t − f tt f f + 2 f f φ 0 0 0 )φ 0 72 f2 f2t 2 2 3 2 2 3 2t 2 x x x x 0 0 0  2 2 02 2  +(3φ f2 − y)(y + 3φ f2) f2tt − 72u f2 f2t −4(1 − 9 f2 φx0x0 ) f2t (51) 0 2 −36(2 f3t + f2tt f2φ ) f2t f3 + 36 f2tt f3 , (45) is a solution of the KP equation. where u0 ≡ u0(x0,t),g0 ≡ g0(x0,t),h0 ≡ h0(x0,t),φ 0 ≡ φ 0(x0,t) Case 3 f = f = c = 0. with 1 2 1 In this case, executing a similar procedure as in Cases 1 0 f2t 2 f3 x = −x − y + y. (46) and 2, we can find the third type of group-invariant solutions 12 f2 f2

g00 g = − , (52) ∆(x + φ 00) cosh2 2 f3 1 h = ∆(x + φ 00) ∆(x + φ 00) ∆(x + φ 00) ∆ cosh3 sinh + cosh 2 f3 2 f3 2 f3   00   00   00  002 00 2 ∆(x + φ ) 00 ∆(x + φ ) ∆(x + φ ) × 2c2g − h ∆ cosh − h ∆ sinh cosh , (53) 2 f3 2 f3 2 f3 100201-5 Chin. Phys. B Vol. 23, No. 10 (2014) 100201   00   1 ∆(x + φ ) In this paper, we have shown that the residual symme- φ = c3 + tanh ∆ , (54) c2 2 f3 try can be used to construct interaction solutions for nonlinear 2  00  2c2 002 2 ∆(x + φ ) equations. It is meaningful to try to use other forms of non- u = 2 g tanh ∆ 2 f3 local symmetries, which can be found from Backlund¨ trans-  00  2c2 002 00 ∆(x + φ ) formation, the bilinear forms, and negative hierarchies, the − 2 (2c2g − ∆h )tanh ∆ 2 f3 nonlinearizations,[26,27] and self-consistent source,[28] etc., to f − 3t x + u00 (55) construct new exact solutions. Despite the bright prospect, 6 f3 some open questions still exist which deserve to be probed. with group-invariant functions u00 ≡ u00(y,t), g00 ≡ g00(y,t), First, it still remains unclear what kind of nonlocal symmetries h00 ≡ h00(y,t), φ 00 ≡ φ 00(y,t). must have close prolongations and can be applied to construct The corresponding symmetry reduction equations are exact solutions. Secondly, different nonlocal symmetries, for 2 00 example the residual symmetry and the DT-related symmetry, ∆ + 2c2 f3g = 0, (56) lead to different symmetry reduction solutions, so what is the −∆ 3 + 2c f 2h00 = 0, (57) 2 3 link between them? Can the nonlocal symmetries be classified 2 2 002 00 2 00 2 −2∆ + 3 f3 φy − 6u f3 − f3 f3t φ + f3 φt = 0, (58) so that symmetries in one kind leads to essentially the same 00 −3 f3φyy + f3t = 0. (59) kind of interaction solutions? All in all, applying nonlocal symmetries to construct exact solutions for nonlinear systems Again, after solving out g00, h00, and u00 by Eqs. (56), (57), is adventurous but meaningful. (58) and substituting them into Eq. (55), then we have the fol- lowing theorem Theroem 4 If φ 00 is a solution of the symmetry reduction References equation (59), then the u given by [1] Lie S 1891 Vorlesungen ber Differentialgleichungen mit Bekannten Infinitesimalen Transformationen (Leipzig: Teuber) (reprinted by ∆ 2 ∆(x + φ 00) 1 Chelsea: New York; 1967) u = tanh2 + φ 002 2 f 2 2 f 2 y [2] Olver P J 1993 Application of Lie Groups to Differential Equation, 3 3 Graduate Texts in Mathematics, 2nd edn. (NewYork: Springer-Verlag) 2 1 00 f3t 00 ∆ [3] Bluman G W and Kumei S 1989 Symmetries and Differential Equation + φt − (x + φ ) − 2 (60) (Berlin: Springer-Verlag) 6 6 f3 3 f 3 [4] Wu J L and Lou S Y 2012 Chin. Phys. B 21 120204 is a solution of the KP equation. [5] Hu X R, Chen Y and Huang F 2010 Chin. Phys. B 19 080203 It is obvious that equations (40) and (51) can be consid- [6] Liu X Z 2010 Chin. Phys. B 19 080202 [7] Jing J C and Li B 2013 Chin. Phys. B 22 010303 ered as interaction solutions of a soliton with the background [8] Gu C H, Hu H and Zhou Z X 2005 Darboux Transformations in Inte- wave, while equation (60) is corresponding to the single soli- grable Systems Theory and Their Applications to Geometry (Dordrecht: tary wave with nonhomogeneous background for f not being Springer) 3 [9] Rogers C and Schief W K 2002 Backlund¨ and Darboux Transforma- a constant. tions Geometry and Modern Applications in Soliton Theory (Cam- bridge: Cambridge University Press) [10] Jin Y, Jia M and Lou S Y 2013 Chin. Phys. Lett. 30 020203 4. Conclusion and discussion [11] Scott A 2005 Encyclopedia of Nonlinear Science (Taylor and Francis) In summary, the residual symmetry coming from the [12] Lou S Y and Hu X B 1993 Chin. Phys. Lett. 10 577 [13] Lou S Y and Hu X B 1997 J. Math. Phys. 38 6401 Painleve´ truncated expansion of the KP equation is localized [14] Lou S Y and Hu X B 1997 J. Phys. A: Math. Gen. 30 L95 by introducing multiple new dependent variables. For the pro- [15] Lou S Y 1998 Physica Scripta 57 481 longed KP system, the new Backlund¨ transformation is de- [16] Hu X B, Lou S Y and Qian X M 2009 Stud. Appl. Math. 122 305 [17] Li Y Q, Chen J C, Chen Y and Lou S Y 2014 Chin. Phys. Lett. 31 rived by using Lie’s first principle. The general form of Lie 010201 point symmetry for the prolonged system is found, from which [18] Lou S Y, Hu X R and Chen Y 2012 “Nonlocal Symmetries Related to it can be found that the residual symmetry is included as a spe- Backlund¨ Transformation and Their Applications”, arXiv: 1201.3409 [19] Hu X R, Lou S Y and Chen Y 2012 Phys. Rev. E 85 056607 cial case. The Lie point symmetry group for the prolonged KP [20] Cheng X P, Chen C L and Lou S Y 2012 “Interactions Among Different system covers the corresponding one for the original KP equa- Types of Nonlinear Waves Described by the Kadomtsev–Petviashvili Equation”, arXiv: 1208.3259 tion. So many more rich interaction solutions can be found [21] Gao X N, Lou S Y and Tang X Y 2013 JHEP 05 029 by using the classical Lie group approach. From the form of [22] Ma W X 1992 J. Phys. A: Math. Gen. 25 5329 symmetry reduction solutions for the KP equation, we see that [23] Kadomtsev B B and Petviashvili V I 1970 Sov. Phys. Dokl. 15 539 [24] Pelinovsky D E, Stepanyants Yu A and Kivshar Yu A 1995 Phys. Rev. the residual symmetry is to add an additional soliton to the E 51 5016 background wave. In other words, the interaction solutions be- [25] Ma W X and Kaup D 2013 Nonlinear and Modern Mathematical tween soliton and background wave are successfully obtained. Physics (American Institute of Physics, Tampa, Florida, USA) [26] Cao C W and Geng X G 1990 J. Phys. A: Math. Gen. 23 4117 Some special explicit solutions under certain constraints are [27] Cheng Y and Li Y S 1991 Phys. Lett. A 157 22 also given. [28] Zeng Y B, Ma W X and Lin R L 2000 J. Math. Phys. 41 5453 100201-6 Chinese Physics B

Volume 23 Number 10 October 2014

GENERAL

100201 New interaction solutions of the Kadomtsev–Petviashvili equation Liu Xi-Zhong, Yu Jun, Ren Bo and Yang Jian-Rong 100301 Transmission time in the reflectionless complex potential Yin Cheng, Wang Xian-Ping, Shan Ming-Lei, Han Qing-Bang and Zhu Chang-Ping 100302 Relative ordering of square-norm distance correlations in open quantum systems Wu Tao, Song Xue-Ke and Ye Liu 100303 Hybrid double-dot qubit measurement with a quantum point contact Yan Lei, Yin Wen and Wang Fang-Wei 100304 Security of biased BB84 quantum key distribution with finite resource Zhao Liang-Yuan, Li Hong-Wei, Yin Zhen-Qiang, , You Juan and Han Zheng-Fu 100305 Excitations of optomechanically driven Bose Einstein condensates in a cavity: Photodetection measure- ments Neha Aggarwal, Sonam Mahajan, Aranya B. Bhattacherjee and Man Mohan 100306 Relativistic symmetries of Deng Fan and Eckart potentials with Coulomb-like and Yukawa-like tensor interactions Akpan N. Ikot, S. Zarrinkamar, B. H. Yazarloo and H. Hassanabadi 100307 A full quantum network scheme Ma Hai-Qiang, Wei Ke-Jin, Yang Jian-Hui, Li Rui-Xue and Zhu Wu 100401 Thermodynamics of a two-dimensional charged black hole in the geometric framework Han Yi-Wen and Hong Yun 100501 Generalized projective synchronization of the fractional-order chaotic system using adaptive fuzzy slid- ing mode control Wang Li-Ming, Tang Yong-Guang, Chai Yong-Quan and Wu Feng 100502 Periodic solitons in dispersion decreasing fibers with a cosine profile Jia Ren-Xu, Yan Hong-Li, Liu Wen-Jun and Lei Ming 100503 Drift coefficients of motor proteins moving along sidesteps Li Jing-Hui 100504 Finite-time sliding mode synchronization of chaotic systems Ni Jun-Kang, Liu Chong-Xin, Liu Kai and Liu Ling

ATOMIC AND MOLECULAR PHYSICS

103101 Theoretical study of 훾-aminobutyric acid conformers: Intramolecular interactions and ionization ener- gies Wang Ke-Dong, Wang Mei-Ting and Meng Ju

(Continued on the Bookbinding Inside Back Cover) 103201 Nonlinear spectroscopy of barium in parallel electric and magnetic fields Yang Hai-Feng, Gao Wei, Cheng Hong and -Ping 103301 Time-dependent approach to the double-channel dissociation of the NaCs molecule induced by pulsed lasers Zhang Cai-Xia, Niu Yu-Quan and Meng Qing-Tian 103302 Nonsequential double ionization of diatomic molecules by elliptically polarized laser pulses Tong Ai-Hong, Liu Dan and Feng Guo-Qiang 103303 Optical and magneto optical properties of periodic Co double layer film Xia Wen-Bin, Gao Jin-Long, Zhang Shao-Yin, Chen Le-Yi, Tang Yan-Mei, Tang Shao-Long and Du You-Wei 103304 Interferences in photo-detached electron spectra from a non-collinear tri-atomic anion A. Afaq, K. Farooq, M. A. Khan and Yi Xue-Xi

ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID DYNAMICS

104201 Universal form of the power spectrum of the aero-optical aberration caused by the supersonic turbulent boundary layer Gao Qiong, Yi Shi-He and Jiang Zong-Fu 104202 Analysis of detection limit to time-resolved coherent anti-Stokes Raman scattering nanoscopy Liu Wei, Liu Shuang-Long, Chen Dan-Ni and Niu Han-Ben 104203 Scattering properties of polluted dust in 1.6-µm wavelength Fan Meng, Chen Liang-Fu, Li Shen-Shen, Tao Jin-Hua, Su Lin and Zou Ming-Min 104204 Measurement-induced disturbance between two atoms in Tavis Cummings model with dipole dipole interaction Zhang Guo-Feng, Wang Xiao and Lu¨ Guang-Hong 104205 Phase control of group-velocity-based biexciton coherence in a multiple quantum well nanostructure Seyyed Hossein Asadpour and H. Rahimpour Soleimani 104206 227-W output all-fiberized Tm-doped fiber laser at 1908 nm -Yue, Yan Ping, Xiao Qi-Rong, Liu Qiang and Gong Ma-Li 104207 Material growth and device fabrication of terahertz quantum-cascade laser based on bound-to- continuum structure Yin Rong, Wan Wen-Jian, Zhang Zhen-Zhen, Tan Zhi-Yong and Cao Jun-Cheng 104208 Nonlinear polarization rotation-induced pulse shaping in a stretched-pulse ytterbium-doped fiber laser Bai Dong-Bi, Li Wen-Xue, Yang Kang-Wen, Shen Xu-Ling, Chen Xiu-Liang and Zeng He-Ping 104209 Resonant cavity-enhanced quantum dot field-effect transistor as a single-photon detector Dong Yu, Wang Guang-Long, Wang Hong-Pei, Ni Hai-Qiao, Chen Jian-Hui, Gao Feng-Qi, Qiao Zhong-Tao, Yang Xiao-Hong and Niu Zhi-Chuan 104210 Carrier-envelope phase effects on high-harmonic generation driven by mid-infrared laser field Diao Han-Hu, Zheng Ying-Hui, Zhong Yue, Zeng Zhi-Nan, Ge Xiao-Chun, Li Chuang, Li Ru-Xin and Xu Zhi-Zhan

(Continued on the Bookbinding Inside Back Cover) 104211 Wavelength-dependence of double optical gating for attosecond pulse generation Tian Jia, Li Min, Yu Ji-Zhou, Deng Yong-Kai and Liu Yun-Quan 104212 Code synchronization based on lumped time-delay compensation scheme with a linearly chirped fiber Bragg grating in all-optical analog-to-digital conversion Wang Tao, Kang Zhe, Yuan Jin-Hui, Tian Ye, Yan Bin-Bin, Sang Xin-Zhu and Yu Chong-Xiu 104213 Higher-order solitons in amplitude-disordered waveguide arrays Liu Hai-Dong, Jin Hong-Zhen and Dong Liang-Wei 104214 Nonlinear modes in rotating double well potential with parity time symmetry Pang Wei, Fu Shen-He, Wu Jian-Xiong, Li Yong-Yao and Mai Zhi-Jie 104215 Description and reconstruction of one-dimensional photonic crystal by digital signal processing theory Zhang Juan, Fu Wen-Peng, Zhang Rong-Jun and Wang Yang 104216 Theory study on a photonic-assisted radio frequency phase shifter with direct current voltage control Li Jing, Ning Ti-Gang, Pei Li, Jian Wei, You Hai-Dong, Wen Xiao-Dong, Chen Hong-Yao, Zhang Chan and Zheng Jing-Jing 104217 Improved partial response maximum likelihood method combining modulation code for signal waveform modulation multi-level disc Wang He-Qun, Pei Jing and Pan Long-Fa 104218 Designing of a polarization beam splitter for the wavelength of 1310 nm on dual-core photonic crystal fiber with high birefringence and double-zero dispersion Bao Ya-Jie, Li Shu-Guang, Zhang Wan, An Guo-Wen and Fan Zhen-Kai 104219 Highly sensitive fiber refractive index sensor based on side-core holey structure Han Ya, Xia Li and Liu De-Ming 104220 Passive polarization rotator based on silica photonic crystal fiber for 1.31-µm and 1.55-µm bands via adjusting the fiber length Chen Lei , Zhang Wei-Gang, Wang Li, Bai Zhi-Yong, Zhang Shan-Shan, Wang Biao, Yan Tie-Yi and Jonathan Sieg 104221 Influence of mode competition on beam quality of fiber amplifier Xiao Qi-Rong, Yan Ping, Sun Jun-Yi, Chen Xiao, Ren Hai-Cui and Gong Ma-Li 104222 Laser frequency stabilization and shifting by using modulation transfer spectroscopy Cheng Bing, Wang Zhao-Ying, Wu Bin, Xu Ao-Peng, Wang Qi-Yu, Xu Yun-Fei and Lin Qiang 104223 Er3+ ion concentration effect on transient and steady-state behavior in Er3+:YAG crystal Asadpour Seyyed Hossein and Rahimpour Soleimani H

3+ 104224 Preparation and characterization of nanosized Gd푥Bi0.95−푥VO4:0.05Eu solid solution as red phos- phor Yi Juan, Qiu Jian-Bei, Wang Yu-An and Zhou Da-Cheng 104301 Multi-crack imaging using nonclassical nonlinear acoustic method Zhang Lue, Zhang Ying, Liu Xiao-Zhou and Gong Xiu-Fen

(Continued on the Bookbinding Inside Back Cover) 104302 Investigation on the relationship between overpressure and sub-harmonic response from encapsulated microbubbles , Fan Ting-Bo, Xu Di and Zhang Dong 104303 Conductivity reconstruction algorithms and numerical simulations for magneto–acousto–electrical to- mography with piston transducer in scan mode Guo Liang, Liu Guo-Qiang, Xia Hui, and Lu Min-Hua 104304 Acoustic anechoic layers with singly periodic array of scatterers: Computational methods, absorption mechanisms, and optimal design Yang Hai-Bin, Li Yue, Zhao Hong-Gang, Wen Ji-Hong and Wen Xi-Sen 104501 Higher-order differential variational principle and differential equations of motion for mechanical sys- tems in event space Zhang Xiang-Wu, Li Yuan-Yuan, Zhao Xiao-Xia and Luo Wen-Feng 104502 Hopf bifurcation control for a coupled nonlinear relative rotation system with time-delay feedbacks Liu Shuang, Li Xue, Tan Shu-Xian and Li Hai-Bin 104701 MHD boundary layer flow of Casson fluid passing through an exponentially stretching permeable surface with thermal radiation Swati Mukhopadhyay, Iswar Chandra Moindal and Tasawar Hayat 104702 Experimental study on supersonic film cooling on the surface of a blunt body in hypersonic flow Fu Jia, Yi Shi-He, Wang Xiao-Hu, He Lin and Ge Yong 104703 Convection and correlation of coherent structure in turbulent boundary layer using tomographic particle image velocimetry , Guan Xin-Lei and Jiang Nan

PHYSICS OF GASES, PLASMAS, AND ELECTRIC DISCHARGES

105201 Numerical study of fast ion behavior in the presence of magnetic islands and toroidal field ripple in the EAST tokamak Zhang Wei, Hu Li-Qun, Sun You-Wen, Ding Si-Ye, Zhang Zi-Jun and Liu Song-Lin 105202 A radial non-uniform helicon equilibrium discharge model -Guo, Cheng Mou-Sen, Wang Mo-Ge and Li Xiao-Kang 105203 Nitriding molybdenum: Effects of duration and fill gas pressure when using 100-Hz pulse DC discharge technique U. Ikhlaq, R. Ahmad, M. Shafiq, S. Saleem, M. S. Shah, T. Hussain, I. A. Khan, K. Abbas and M. S. Abbas

CONDENSED MATTER: STRUCTURAL, MECHANICAL, AND THERMAL PROPERTIES

106101 Oxygen vacancy-induced room-temperature ferromagnetism in D–D neutron irradiated single-crystal

TiO2 (001) rutile Xu Nan-Nan, Li Gong-Ping, Pan Xiao-Dong, -Bo, Chen Jing-Sheng and Bao Liang-Man 106102 Pattern imprinting in deep sub-micron static random access memories induced by total dose irradiation Zheng Qi-Wen, Yu Xue-Feng, Cui Jiang-Wei, Guo Qi, Ren Di-Yuan, Cong Zhong-Chao and Zhou Hang

(Continued on the Bookbinding Inside Back Cover) 106103 A model of crack based on dislocations in smectic A liquid crystals Fan Tian-You and Tang Zhi-Yi 106104 Elastic fields around a nanosized elliptic hole in decagonal quasicrystals Li Lian-He and Yun Guo-Hong 106105 Molecular dynamics simulation of an argon cluster filled inside carbon nanotubes Cui Shu-Wen, Zhu Ru-Zeng, Wang Xiao-Song and Yang Hong-Xiu 106106 Epitaxial evolution on buried cracks in a strain-controlled AlN/GaN superlattice interlayer between Al- GaN/GaN multiple quantum wells and a GaN template Huang Cheng-Cheng, Zhang Xia, Xu Fu-Jun, Xu Zheng-Yu, Chen Guang, Yang Zhi-Jian, Tang Ning, Wang Xin-Qiang and Shen Bo 106107 Segregation of alloying atoms at a tilt symmetric grain boundary in tungsten and their strengthening and embrittling effects -Wu, Kong Xiang-Shan, Liu-Wei, Liu Chang-Song and Fang Qian-Feng

106108 First-principles calculations of structural, electronic, and thermodynamic properties of ZnO1−푥S푥 al- loys Muhammad Zafar, Shabbir Ahmed, M. Shakil and M. A. Choudhary

106201 Enhancement of ferromagnetic resonance in Al2O3-doped Co2FeAl Heusler alloy film prepared by oblique sputtering Li Shan-Dong, Cai Zhi-Yi, Xu Jie, Cao Xiao-Qin, Du Hong-Lei, Xue Qian, Gao Xiao-Yang and Xie Shi-Ming 106501 Thermal transport properties of defective graphene: A molecular dynamics investigation Yang Yu-Lin and Lu Yu

106801 Stacking stability of MoS2 bilayer: An 푎푏 푖푛푖푡푖표 study Tao Peng, -Hong, Yang Teng and Zhang Zhi-Dong 106802 First-principles calculations on Si (220) located 6H–SiC (101¯0) surface with different stacking sites He Xiao-Min, Chen Zhi-Ming, Pu Hong-Bin, Li Lian-Bi and Huang Lei 106803 Influence of roughness on the detection of mechanical characteristics of low-푘 film by the surface acoustic waves Xiao Xia, Tao Ye and Sun Yuan 106804 Analysis of phase shift of surface plasmon polaritons at metallic subwavelength hole arrays Li Jiang-Yan, Qiu Kang-Sheng and Ma Hai-Qiang

CONDENSED MATTER: ELECTRONIC STRUCTURE, ELECTRICAL, MAGNETIC, AND OPTI- CAL PROPERTIES 107101 Low-resistance Ohmic contact on polarization-doped AlGaN/GaN heterojunction Li Shi-Bin, Yu Hong-Ping, Zhang Ting, Chen Zhi and -Ming

107102 Electronic structure and optical properties of Mg푥Zn1−푥S bulk crystal using first-principles calculations Yu Zhi-Qiang, Xu Zhi-Mou and Wu Xing-Hui 107103 Influence of electron–phonon interaction on the properties of transport through double quantum dot with ferromagnetic leads Luo Kan, Wang Fa-Qiang, Liang Rui-Sheng and Ren Zhen-Zhen

(Continued on the Bookbinding Inside Back Cover) 107104 Transverse Zeeman background correction method for air mercury measurement Li Chuan-Xin, Si Fu-Qi, Liu Wen-Qing, Zhou Hai-Jin, Jiang Yu and Hu Ren-Zhi 107301 Ferromagnetic barrier-induced negative differential conductance on the surface of a topological insula- tor An Xing-Tao

107302 Analysis of optoelectronic properties of TiO2 nanowiers/Si heterojunction arrays Saeideh Ramezani Sani 107303 Mechanism of improving forward and reverse blocking voltages in AlGaN/GaN HEMTs by using Schot- tky drain Zhao Sheng-Lei, Mi Min-Han, Hou Bin, Luo Jun, Wang Yi, Dai Yang, Zhang Jin-Cheng, Ma Xiao-Hua and Hao Yue 107304 Electron states scattering off line edges on the surface of topological insulator Shao Huai-Hua, Liu Yi-Man, Zhou Xiao-Ying and Zhou Guang-Hui 107305 Effects of shape of terminus on excitation of surface plasmon modes on metal nanowires Qiao Ya-Nan and Yang Shu

107306 Effects of different dopants on switching behavior of HfO2-based resistive random access memory Deng Ning, Pang Hua and Wu Wei 107307 Interfacial thermal resistance between high-density polyethylene (HDPE) and sapphire Zheng Kun, Zhu Jie, -Mei, Tang Da-Wei and Wang Fo-Song

107401 Optical conductivity as a probe of a hidden Fermi-liquid behavior in BaFe1.904Ni0.096As2 Yang Yan-Xing, Xiong Rui, Fang Zhi-Hao, Xu Bing, Xiao Hong, Qiu Xiang-Gang, Shi Jing and Wang Kai 107402 Precursor evolution and growth mechanism of BTO/YBCO films by TFA–MOD process Wang Hong-Yan, Ding Fa-Zhu, Gu Hong-Wei, Zhang Teng and Peng Xing-Yu 107501 Phase constitution and microstructure of Ce–Fe–B strip-casting alloy Yan Chang-Jiang, Guo Shuai, Chen Ren-Jie, Lee Dong and Yan A-Ru 107502 Asymmetric exchange bias training effect in spin glass (FeAu)/FeNi bilayers Rui Wen-Bin, He Mao-Cheng, You Biao, Shi Zhong, Zhou Shi-Ming, Xiao Ming-Wen, Gao Yuan, Zhang Wei, and Du Jun

107503 Cation distributions estimated using the magnetic moments of the spinel ferrites Co1−푥Cr푥Fe2O4 at 10 K Shang Zhi-Feng, Qi Wei-Hua, Ji Deng-Hui, Xu Jing, Tang Gui-De, Zhang Xiao-Yun, Li Zhuang-Zhi and Lang Li-Li 107801 Temperature dependence of the photothermal laser cooling efficiency for a micro-cantilever Ding Li-Ping, Mao Tian-Hua, Fu Hao and Cao Geng-Yu 107802 Photo-induced intramolecular electron transfer and intramolecular vibrational relaxation of rhodamine 6G in DMSO revealed by multiplex transient grating spectroscopy Jiang Li-Lin, Liu Wei-Long, Song Yun-Fei and Sun Shan-Lin 107803 Mid-gap photoluminescence and magnetic properties of GaMnN films grown by metal–organic chemical vapor deposition Xing Hai-Ying, Xu Zhang-Cheng, Cui Ming-Qi, Xie Yu-Xin and Zhang Guo-Yi

(Continued on the Bookbinding Inside Back Cover) 107804 Pressure-dependent terahertz optical characterization of heptafluoropropane Leng Wen-Xiu, Ge Li-Na, Xu Shan-Sen, Zhan Hong-Lei and Zhao Kun

3+ 3+ 3+ 107805 Energy transfer relation of a novel Ce /Pr /Eu co-doped Sr2.975−푥La푥AlO4+푥F1−푥 solid solu- tion phosphor Wang Yan-Ze, , Sun Liang, Li Rui and Wang Da-Jian

3+ 107806 Optimum fluorescence emission around 1.8 µm for LiYF4 single crystals of various Tm -doping con- centrations Li Shan-Shan, Xia Hai-Ping, Fu Li, Dong Yan-Ming, Gu Xue-Mei, Zhang Jian-Li, Wang Dong-Jie, Zhang Yue-Pin, Jiang Hao-Chuan and Chen Bao-Jiu 107807 Terahertz plasmon and surface-plasmon modes in cylindrical metallic nanowires Wu Ping, Xu Wen, Li Long-Long, Lu Tie-Cheng and Wu Wei-Dong 107808 Plasmon-induced absorption in stacked metamaterials based on phase retardation Wan Ming-Li, -Qing, Dai Ke-Jie, Song Yue-Li, Zhou Feng-Qun and -Na

INTERDISCIPLINARY PHYSICS AND RELATED AREAS OF SCIENCE AND TECHNOLOGY

108101 Influences of anionic and cationic dopants on the morphology and optical properties of PbS nanostruc- tures Ramin Yousefi, Mohsen Cheragizade, Farid Jamali-Sheini, M. R. Mahmoudian, Abdolhossein Saaedi´ and Nay Ming Huang

108201 K2S-activated carbons developed from coal and their methane adsorption behaviors Feng Yan-Yan, Yang Wen and Chu Wei 108401 Theoretical and numerical studies on a planar gyrotron with transverse energy extraction Chen Zai-Gao, Wang Jian-Guo and Wang Yue 108402 Modified GIT model for predicting wind-speed behavior of low-grazing-angle radar sea clutter -Shi, Zhang Jin-Peng, Li Xin, Wu Zhen-Sen 108501 Investigation of strain effect on the hole mobility in GOI tri-gate pFETs including quantum confinement Qin Jie-Yu, Du Gang and Liu Xiao-Yan 108701 Initial conformation of kinesin’s neck linker Geng Yi-Zhao, Ji Qing, Liu Shu-Xia and Yan Shi-Wei 108702 Reconstruction of a 6-MeV bremsstrahlung spectrum by multi-layer absorption based on LiF:Mg, Cu, P Huang Jian-Wei and Wang Nai-Yan 108703 Plasticity-induced characteristic changes of pattern dynamics and the related phase transitions in small- world neuronal networks Huang Xu-Hui and Hu Gang 108901 Consensus of heterogeneous multi-agent systems based on sampled data with a small sampling delay Wang Na, Wu Zhi-Hai and Peng Li

(Continued on the Bookbinding Inside Back Cover) 108902 Network dynamics and its relationships to topology and coupling structure in excitable complex net- works Zhang Li-Sheng, Gu Wei-Feng, Hu Gang and Mi Yuan-Yuan

GEOPHYSICS, ASTRONOMY, AND ASTROPHYSICS

109101 Physical analysis on improving the recovery accuracy of the Earth’s gravity field by a combination of satellite observations in along-track and cross-track directions Zheng Wei, Hsu Hou-Tse, Zhong Min and Yun Mei-Juan 109401 Study of typical space wave–particle coupling events possibly related with seismic activity Zhang Zhen-Xia, Wang Chen-Yu, Shen Xu-Hui, Li Xin-Qiao and Wu Shu-Gui 109402 Variational regularization method of solving the Cauchy problem for Laplace’s equation: Innovation of the Grad–Shafranov (GS) reconstruction Yan Bing and Huang Si-Xun