Symbolic computation of exact solutions for the (2 + 1)-dimensional generalized stochastic KP equation

Chaos, Solitons and Fractals 33 (2007) 1552–1557 www.elsevier.com/locate/chaos

Symbolic computation of exact solutions for the (2 + 1)-dimensional generalized stochastic KP equation Xiao-Fei Wu b

a,* ,

Guo-Sheng Hua a, Zheng-Yi Ma

a,b

a College of Information, Lishui University, Lishui Zhejiang 323000, PR China Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, PR China

Accepted 6 March 2006

Abstract In this paper, the F-expansion method is extended and applied to construct the exact solutions of the (2 + 1)-dimensional generalized Wick-type stochastic Kadomtsev–Petviashvili equation by the aid of the symbolic computation system Maple. Some new stochastic exact solutions which include kink-shaped soliton solution, singular soliton solution and triangular periodic solutions are obtained via this method and Hermite transformation. Ó 2006 Elsevier Ltd. All rights reserved.

1. Introduction In recent years, the nonlinear partial differential equations (NPDEs) are widely used to describe many important phenomena and dynamic processes in physics, mechanics, chemistry and biology, etc. Constructing a new approach to derive new exact solutions of NPDEs is one of the central tasks in physics as well as mathematics. With the development of soliton theory, many powerful methods have been presented, such as inverse scattering transform method [1], Hirota’s bilinear method [2], Ba¨cklund transform method [3], homogeneous balance method [4], truncated Painleve´ expansion method [5], similarity reduction method [6], hyperbolic function method [7], F-expansion method [8], and so on. In [9], Wadati first introduced stochastic KdV equation and proved that a soliton under Gaussian noise satisfies a diffusion equation in transformed coordinates. He and Akutsu also studied stochastic KdV equation with and without damping, and discussed the asymptotic form of multi-soliton [10]. In addition, a nonlinear partial differential equation which describes wave propagation in random media was presented by Wadati [11]. Henceforth, many researchers pay more attention to the study of the random waves, which are important subjects of stochastic partial differential equation (SPDE), and have obtained a number of soliton solutions of nonlinear stochastic partial differential equations [12–22]. In [22], Holden et al. used the white noise functional approach to study SPDEs in Wick versions. And Xie [23] studied the exact solutions of the Wick-type stochastic Kadomtsev–Petviashvili (KP) equation by using homogeneous balance method. *

Corresponding author. E-mail address: [email protected] (X.-F. Wu).

0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.03.003

X.-F. Wu et al. / Chaos, Solitons and Fractals 33 (2007) 1552–1557

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In this paper, we will consider the (2 + 1)-dimensional generalized Wick-type stochastic KP equation of the following form 4 2 0 U xt þ 6ðU 2 x þ UU xx Þ þ U xxxx þ H 2 ðtÞU yy þ 6H 1 ðtÞU x  ðH 1 ðtÞ þ 12H 1 ðtÞÞ ¼ 0; d

ð1:1Þ



where e denotes R t the Wick product on the Hida distribution space ðSðR ÞÞ , H1(t) is a white noise function and H 2 ðtÞ ¼ expð6 H 1 ðsÞ dsÞ, which will be discussed in Section 3. And we will give some stochastic exact soliton solutions of Eq. (1.1) by using the extended F-expansion method and the white noise analysis method.

2. SPDEs driven by white noise Let ðSðRd ÞÞ and ðSðRd ÞÞ be the Hida test function space and the Hida distribution space on Rd respectively. And let hn(x) be the n-order Hermite polynomials. Put pffiffiffi 1 1 2 nn ðxÞ ¼ e2x hn ð 2xÞ=ðpðn  1Þ!Þ2 ; n P 1. Then, we obtain that the collection {nn}nP1 constitutes an orthogonal basis for L2 ðRÞ. If we denote a = (a1, . . . , ad) being d-dimensional multi-indices with a1 ; . . . ; ad 2 N, we have that the family of tensor ðiÞ ðiÞ products na ¼ nða1 ;...;ad Þ ¼ na1      nad ða 2 Nd Þ forms an orthogonal basis for L2 ðRd Þ. Let aðiÞ ¼ ða1 ; . . . ; ad Þ be the d ith multi-index number in some fixed ordering of all d-dimensional multi-indices a ¼ ða1 ; . . . ; ad Þ 2 N . We can, and will, assume that this ordering has the property that ðiÞ

ðiÞ

ðjÞ

ðjÞ

i < j ) a1 þ    þ ad 6 a1 þ    þ ad ; i.e., the faðjÞ g1 j¼1 occurs in an increasing order. Now, we define gi ¼ naðiÞ ¼ naðiÞ      naðiÞ ; 1

i P 1.

d

We need to consider multi-indices of arbitrary length. To simplify the notation, we regard multi-indices as elements of the space ðNN 0 Þc of all sequences a = {a1, a2, . . .} with element ai 2 N0 and with compact support, i.e., with only finitely many ai 5 0. We define | ¼ ðNN 0 Þc . For a = (a1, a2, . . .) 2 |, we define 1 Y hai ðhx; gi iÞ; x ¼ ðx1 ; . . . ; xm Þ 2 ðSðRd ÞÞ . H a ðxÞ ¼ i¼1

P For a fixed n 2 N and 8k 2 N, letting the space ðSÞn1 is composed of those f ðxÞ ¼ a ca H a ðxÞ 2 nk¼1 L2 ðlÞ with P P 2 n n 2 ðkÞ 2 ð1Þ ðnÞ ca 2 Rn such that kf k21;k ¼ a c2a ða!Þ2 ð2NÞka < 1, where Q1 ca ¼ jca j ¼ a k¼1 Qðca Þ aj if ca ¼ ðca ; . . . ; ca Þ 2 R , and l is   the white noise measure on ðS ðRÞ; BðS ðRÞÞÞ; a! ¼ k¼1 ak ! and ð2NÞ ¼ j ð2jÞ for a 2 |. P The space ðSÞn1 consists of all formal expansions F ðxÞ ¼ a ba H a ðxÞ with ba 2 Rn such that kF k1;q ¼ P 2 qa < 1 for some q 2 N. The family of seminorms kfk1,k, k 2 N gives rise to a topology on ðSÞn1 , and we a ba ð2NÞ can regard ðSÞn1 as the dual of ðSÞn1 by the action X ðba ; ca Þa!; hF ; f i ¼ a

where (ba, ca) is the usual inner product in Rn . P P The Wick product f e F of two elements f ¼ a aa H a ; F ¼ a ba H a 2 ðSÞn1 with aa ; ba 2 Rn , is defined by X f F ¼ ðaa ; bb ÞH aþb . a;b

We can prove ðSðRd ÞÞ; ðSðRd ÞÞ ; ðSÞ1 and (S)1 are closed under Wick products. Pthat the spaces n For F ¼ a ba H a 2 ðSÞ1 with ba 2 Rn , the Hermite transformation of F, denoted by HðF Þ or Fe ðzÞ, is defined by X HðF Þ ¼ Fe ðzÞ ¼ ba za 2 CN ðwhen convergentÞ; a

where z ¼ ðz1 ; z2 ; . . .Þ 2 CN (the set of all sequences of complex numbers) and za ¼ za11 za22    zann   , if a = (a1, a2, . . .) 2 |, where z0j ¼ 1. For F ; G 2 ðSÞn1 ; by this definition, we have g e FGðzÞ ¼ Fe ðzÞ  GðzÞ

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e for all z such that Fe ðzÞ and GðzÞ exist. The product on the right-hand side of Pthe above formula is the complex bilinear product between two elements of CN defined by ðz11 ; . . . ; z1n Þ  ðz21 ; . . . ; z2n Þ ¼ nk¼1 z1k z2k , where z 2 C. P e ð0Þ 2 Rn is called the generalized expectation of X and is denoted Let X ¼ a aa H a 2 ðSÞn1 . Then the vector c0 ¼ X by E(X). Suppose that g : U ! Cm is an analytic function, where U is a neighborhood of E(X). Assume that the Taylor e Þ 2 ðSÞm . series of g around E(X) has coefficients in Rm . Then the Wick version g ðX Þ ¼ H1 ðf  X 1 Suppose that modelling consideration leads us to consider an SPDE as follows: Aðt; x; @ t ; rx ; U ; xÞ ¼ 0;

ð2:1Þ

where A is some given function, U = U(t, x, x) is the unknown (generalized) stochastic process, and where the operators   @ t ¼ @t@ ; rx ¼ @x@1 ; . . . ; @x@d when x ¼ ðx1 ; . . . ; xd Þ 2 Rd . Firstly, we interpret all products as Wick products and all functions as their Wick versions. We obtain Wick version of Eq. (2.1) as follows: A ðt; x; @ t ; rx ; U ; xÞ ¼ 0.

ð2:2Þ

Secondly, we take the Hermite transformation of Eq. (2.2), which turns Wick products into ordinary products (between complex numbers) and the equation takes the form: e x; @ t ; rx ; U e ; z1 ; z2 ; . . .Þ ¼ 0; Aðt;

ð2:3Þ

e ¼ HðU Þ is the Hermite transformation of U and z1, z2, . . . are complex numbers. Suppose that we can find a where U solution u = u(t, x, z) of Eq. (2.3) for each z ¼ ðz1 ; z2 ; . . .Þ 2 Kq ðrÞ, where Kq ðrÞ ¼ fz ¼ ðz1 ; z2 ; . . .Þ 2 CN and P qa a 2 < r2 g for some q, r. Then, under certain conditions, we can take the inverse Hermite transformation a6¼0 jz j ð2NÞ 1 U ¼ H u 2 ðSÞ1 and thereby obtain a solution U of the original Wick equation (2.2). We have the following theorem, which was proved by Holden et al. [22]. Theorem 2.1. Suppose u(t, x, z) is a solution (in the usual strong, pointwise sense) of Eq. (2.3) for (t, x) in some bounded open set G  R Rd ; and for all z 2 Kq ðrÞ, for some q, r. Moreover, suppose that u(t, x, z) and all its partial derivatives, which are involved in Eq. (2.3), are bounded for ðt; x; zÞ 2 G Kq ðrÞ, continuous with respect to (t, x) 2 G for all z 2 Kq ðrÞ e ðt; xÞÞðzÞ and analytic with respect to z 2 Kq ðrÞ, for all (t, x) 2 G. Then there exists U(t, x) 2 (S)1 such that uðt; x; zÞ ¼ ð U for all ðt; x; zÞ 2 G Kq ðrÞ and U(t, x) solves (in the strong sense in (S)1) Eq. (2.2) in (S)1. 3. New exact solutions of Eq. (1.1) Taking the Hermite transformation of Eq. (1.1), we can get the following equation: e xt þ 6ð U e 2x þ U eU e xx Þ þ U e xxxx þ H e 42 ðt; zÞ U e yy þ 6 H e 1 ðt; zÞ U e x  ðH e 1t ðt; zÞ þ 12 H e 21 ðt; zÞÞ ¼ 0; U

ð3:1Þ

N

where zðz1 ; z2 ; . . .Þ 2 ðC Þc is a vector parameter. Now we use the extended F-expansion method to solve Eq. (3.1). For simplicity, we denote e 1 ðt; zÞ; e ðt; x; y; zÞ; H 1 ¼ H 1 ðt; zÞ ¼ H u ¼ uðt; x; y; zÞ ¼ U  Z t  H 2 ðt; zÞ ¼ exp 6 H 1 ðs; zÞ ds .

e 2 ðt; zÞ and H 2 ¼ H 2 ðt; zÞ ¼ H

_ Let W(t) be Gaussian white noise and B(t) be a Brown notion. Then, we have W ðtÞ ¼ BðtÞ. Suppose that  Z t  H 1 ðtÞ ¼ hðtÞ þ cW ðtÞ; H 2 ðtÞ ¼ exp 6 ðhðsÞ þ cW ðsÞÞ ds ;

ð3:2Þ

ð3:3Þ

where c is an arbitrary constant and h(t) is an integrable function on Rþ , we obtain their Hermite transformations e ðt; zÞ; H 1 ðt; zÞ ¼ hðtÞ þ c W

 Z t  e ðs; zÞÞ ds ; H 2 ðt; zÞ ¼ exp 6 ðhðsÞ þ c W

P R e ðt; zÞ ¼ 1 t gk ðsÞ dszk . where W k¼1 0 We assume that Eq. (3.1) has the following formal solution: n X ai ðt; y; zÞF i ðnÞ; n ¼ nðt; x; y; zÞ ¼ ax þ gðt; y; zÞ; uðt; x; y; zÞ ¼ i¼0

ð3:4Þ

ð3:5Þ

X.-F. Wu et al. / Chaos, Solitons and Fractals 33 (2007) 1552–1557

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where ai(t, y, z)(i = 0, 1, . . . , n), g(t, y, z) are functions to be determined later, a(50) is a constant, n is a positive integer that will be determined soon, and F(n) is only a function of n and satisfies first-order ODE F 0 ðnÞ ¼ P þ QF ðnÞ2 ; where P, Q are constants to be determined 8 qffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi >

 QP tanhð PQn þ CÞ > > > > qffiffiffiffiffiffiffi > pffiffiffiffiffiffiffiffiffiffi > > <  QP cothð PQn þ CÞ F ðnÞ ¼ qffiffiffi pffiffiffiffiffiffi > >

QP tanð PQn þ CÞ > > > > qffiffiffi > > : P cotðpffiffiffiffiffiffi PQn þ CÞ Q

ð3:6Þ later. And Eq. (3.6) admits the following general solutions: ðPQ < 0Þ; ðPQ < 0Þ; ð3:7Þ ðPQ > 0Þ; ðPQ > 0Þ;

where C is an arbitrary constant. By balancing the highest-order partial derivative uxxxx term with the nonlinear uuxx term in Eq. (3.1), we obtain n = 2. Therefore, we may choose the following ansatz: uðt; x; y; zÞ ¼ a0 ðt; y; zÞ þ a1 ðt; y; zÞF ðnÞ þ a2 ðt; y; zÞF ðnÞ2 .

ð3:8Þ

With the aid of Maple, substituting Eq. (3.8) into Eq. (3.1) with Eq. (3.6), collecting all terms with the same power in Fi(n) (i = 0, 1, . . . , 6), and setting the coefficients of these terms Fi(n) to zero yields a set of equations with respect to unknowns a0(t, y, z), a1(t, y, z), a2(t, y, z) and g(t, y, z), namely 120a2 Q4 a4 þ 60a22 a2 Q2 ¼ 0; 72a1 a2 a2 Q2 þ 24a1 Q4 a4 ¼ 0; 6a2 agt Q2 þ 96a22 a2 PQ þ 36a0 a2 a2 Q2 þ 240a2 a4 Q3 P þ 18a21 a2 Q2 þ 6H 42 a2 g2y Q2 ¼ 0; 2a1 Q2 agt þ 12a0 a1 Q2 a2 þ 2H 42 a2 gyy Q þ 4H 42 a2y gy Q þ 108a1 a2 a2 PQ þ 12H 1 a2 aQ þ 2H 42 a1 Q2 g2y þ 2a2t aQ þ 40a1 Q3 a4 P ¼ 0; 8H 42 a2 g2y PQ þ H 42 a1 gyy Q þ 36a22 a2 P 2 þ a1t aQ þ 2H 42 a1y gy Q þ 48a0 a2 a2 PQ 4

2

ð3:9Þ

2

þ 136a2 a Q P þ 24a21 a2 PQ þ H 42 a2yy þ 8a2 agt PQ þ 6H 1 a1 aQ ¼ 0; 2H 42 a2 gyy P þ H 42 a1yy þ 2H 42 a1 Qg2y P þ 2a2t aP þ 4H 42 a2y gy P þ 2a1 Qagt P þ 12a0 a1 Qa2 P þ 16a1 Q2 a4 P 2 þ 12H 1 a2 aP þ 36a1 a2 a2 P 2 ¼ 0; H 42 a0yy þ a1t aP þ 2H 42 a2 g2y P 2 þ H 42 a1 gyy P  H 1t þ 6a21 a2 P 2 þ 6H 1 a1 aP  12H 21 þ 16a2 a4 QP 3 þ 12a0 a2 a2 P 2 þ 2a2 agt P 2 þ 2H 42 a1y gy P ¼ 0. Under the condition (3.2), we can derive the following solutions from Eq. (3.9): a1 ðt; y; zÞ ¼ 0; a2 ðt; y; zÞ ¼ 2a2 Q2 ;  Z t  gðt; y; zÞ ¼ 3ay 2 H 1 ðt; zÞ exp 24 H 1 ðs; zÞds þ C 0 y þ C 1 ;    Z t    Z t 1 2 C2 2 4 a0 ðt; y; zÞ ¼ y H 1t ðt; zÞ þ 6y 2 H 21 ðt; zÞ exp 24 H 1 ðs; zÞds  02 exp 24 H 1 ðs; zÞds þ yC 0 H 1 ðt; zÞ  a2 PQ; 6a 2 a 3 ð3:10Þ

where P, Q and a are arbitrary constants, C0 and C1 are integrable constants. From Eqs. (3.5) and (3.10), we have  Z t  nðt; x; y; zÞ ¼ ax þ gðt; y; zÞ ¼ ax  3ay 2 H 1 ðt; zÞ exp 24 H 1 ðs; zÞ ds þ C 0 y þ C 1 .

ð3:11Þ

Substituting Eqs. (3.10) and (3.11) into Eq. (3.8) with Eq. (3.7), we can obtain the exact solutions of Eq. (3.1) as follows: Case 1: For PQ < 0, Eq. (3.1) has the following kink-shaped soliton solution: pffiffiffiffiffiffiffiffiffiffi u1 ðt; x; y; zÞ ¼ a0 ðt; y; zÞ þ 2a2 PQtanh2 ð PQnðt; x; y; zÞ þ CÞ;

ð3:12Þ

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and singular soliton solution pffiffiffiffiffiffiffiffiffiffi u2 ðt; x; y; zÞ ¼ a0 ðt; y; zÞ þ 2a2 PQcoth2 ð PQnðt; x; y; zÞ þ CÞ; where a0(t, y, z) and n(t, x, y, z) are determined by (3.10) and (3.11), respectively. Case 2: For PQ > 0, Eq. (3.1) has the following triangular periodic solutions pffiffiffiffiffiffi u3 ðt; x; y; zÞ ¼ a0 ðt; y; zÞ  2a2 PQ tan2 ð PQnðt; x; y; zÞ þ CÞ;

ð3:13Þ

ð3:14Þ

and pffiffiffiffiffiffi u4 ðt; x; y; zÞ ¼ a0 ðt; y; zÞ  2a2 PQcot2 ð PQnðt; x; y; zÞ þ CÞ;

ð3:15Þ

where a0(t, y, z) and n(t, x, y, z) are determined by (3.10) and (3.11), respectively. In order to get exact solutions of Eq. (1.1), we give the following condition: (a) Suppose that for (t, x, y) in a bounded open set G  Rþ R2 , for all z 2 Kq ðrÞ for some q > 0 and r > 0 such that Hi(t, z) (i = 1, 2) satisfy that u(t, x, y, z) and all its partial derivatives, which are involved in Eq. (3.1), are uniformly bounded for ðt; x; y; zÞ 2 G Kq ðrÞ, continuous with respect to (t, x, y) 2 G for all z 2 Kq ðrÞ and analytic with respect to z 2 Kq ðrÞ for all (t, x, y) 2 G. From (a), Theorem 2.1 implies that there exists U(t, x, y) 2 (S)1 such that uðt; x; y; zÞ ¼ ðHU ðt; x; yÞÞðzÞ for all ðt; x; y; zÞ 2 G Kq ðrÞ, and U(t, x, y) solves Eq. (1.1). Consequently, U(t, x, y) is the inverse Hermite transformation of u(t, x, y, z). Since exp fBðtÞg ¼ expfBðtÞ  12 t2 g (see Lemma 2.6.16 in [22]), hence, for ui(t, x, y, z) (i = 1, 2, 3, 4), we attain that some stochastic exact solutions of Eq. (1.1) as follows: Case 1: For PQ < 0, Eq. (1.1) has the following stochastic kink-shaped soliton solution: pffiffiffiffiffiffiffiffiffiffi U 1 ðt; x; yÞ ¼ A0 ðt; yÞ þ 2a2 PQtanh2 ð PQhðt; x; yÞ þ CÞ; and stochastic singular soliton solution pffiffiffiffiffiffiffiffiffiffi U 2 ðt; x; yÞ ¼ A0 ðt; yÞ þ 2a2 PQcoth2 ð PQhðt; x; yÞ þ CÞ;

ð3:16Þ

ð3:17Þ

where  Z t   Z t  1  ðtÞexp 24 H ðsÞ ds A0 ðt; yÞ ¼ y 2 H 01 ðtÞexp 24 H 1 ðsÞ ds þ 6y 2 H 2 1 1 2   Z t C2 2 4  02 exp 24 H 1 ðsÞ ds þ yC 0 H 1 ðtÞ  a2 PQ a 3 6a  Z t   1 2 0 1 2 0 ¼ y ðh ðtÞ þ cW ðtÞÞ exp 24 hðsÞ ds þ 24c BðtÞ  t 2 2  Z t   1 2 2 2 þ 6y ðhðtÞ þ cW ðtÞÞ exp 24 hðsÞ ds þ 24c BðtÞ  t 2    Z t 2 C 1 2 4 þ yC 0 ðhðtÞ þ cW ðtÞÞ  a2 PQ;  02 exp 24 hðsÞ ds  24c BðtÞ  t2 6a 2 a 3  Z t  hðt; x; yÞ ¼ ax  3ay 2 H 1 ðtÞexp 24 H 1 ðsÞ ds þ C 0 y þ C 1  Z t  1 2 ¼ ax  3ay ðhðtÞ þ cW ðtÞÞ exp 24 hðsÞ ds þ 24cðBðtÞ  t Þ þ C 0 y þ C 1 . 2 2

Case 2: For PQ > 0, Eq. (1.1) has the following stochastic triangular periodic solutions: pffiffiffiffiffiffi U 3 ðt; x; yÞ ¼ A0 ðt; yÞ  2a2 PQ tan2 ð PQhðt; x; yÞ þ CÞ;

ð3:18Þ

ð3:19Þ

ð3:20Þ

and pffiffiffiffiffiffi U 4 ðt; x; yÞ ¼ A0 ðt; yÞ  2a2 PQcot2 ð PQhðt; x; yÞ þ CÞ; where A0(t, y) and h(t, x, y) are determined by (3.18) and (3.19), respectively.

ð3:21Þ

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4. Conclusion In this paper, we construct some new stochastic exact solutions of the (2 + 1)-dimensional generalized Wick-type stochastic KP equation via the extended F-expansion method and Hermite transformation, which include kink-shaped soliton solution, singular soliton solution and triangular periodic solutions. These solutions may be of great significance for explanation of some practical physical problems. If we let H1(t) = f(t), H2(t) = g(t), Wick product e is an ordinary product ‘‘Æ’’, Eq. (1.1) can be changed into the (2 + 1)-dimensional generalized KP equation with variable coefficients, which is written as ð4:1Þ uxt þ 6ðu2x þ uuxx Þ þ uxxxx þ g4 ðtÞuyy þ 6f ðtÞux  ðf 0 ðtÞ þ 12f 2 ðtÞÞ ¼ 0; Rt where f(t) is an integrable function on Rþ , and gðtÞ ¼ expð6 f ðsÞ dsÞ. Huang and Zhang [24] studied and gave new soliton solutions of Eq. (4.1) via the extended homogeneous balance method. Noting that there exists a unitary mapping between the Wiener white noise space and the Poisson white noise space, we can easily obtain the solution of the Poisson SPDE by applying this mapping to the solution of the corresponding Gaussian SPDE. A nice and concise account of this connection was given by Benth and Gjerde [25]. We can see this in Section 4.9 of [22] as well. Hence, we can attain stochastic soliton solutions as we do in this paper if the coefficient Hi(t) (i = 1, 2) are Poisson white noise functions in Eq. (1.1).

Acknowledgements The project was supported by the National Natural Science Foundation of China (Grant No. 10272071) and the Key Academic Discipline of Zhejiang Province of China (Grant No. 200412).

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