A nonlocal variable coefficient modified KdV equation derived from two-layer fluid system and its exact solutions
AA nonlocal variable coefficient modified KdV equation derivedfrom two-layer fluid system and its exact solutions
Xi-zhong Liu
Institute of Nonlinear Science, Shaoxing University, Shaoxing 312000, China
Abstract
A nonlocal form of a two-layer fluid system is proposed by a simple symmetry reduction, thenby applying multiple scale method to it a general nonlocal two place variable coefficient modifiedKdV (VCmKdV) equation with shifted space and delayed time reversal is derived. Various exactsolutions of the VCmKdV equation, including elliptic periodic waves, solitary waves and interac-tion solutions between solitons and periodic waves are obtained and analyzed graphically. As anillustration, an approximate solution of the original nonlocal two-layer fluid system is also given.
PACS numbers: 02.30.Jr, 02.30.Ik, 05.45.Yv, 47.35.FgKeywords: nonlocal variable coefficient mKdV equation, periodic waves, interaction solutions a r X i v : . [ n li n . S I] M a r . INTRODUCTION In nonlinear science, to obtain exact solutions of nonlinear equations, including nonlin-ear wave solutions and soliton solutions, etc., is a basic and challenging task. In the pastfew decades, many effective methods have been developed to study a lot of integrable ornon-integrable equations, such as Korteweg-de Vries (KdV) equation along with its multidi-mensional analog Kadomtsev-Petviashvili equation, different type of nonlinear Schr¨odingerequations and so on, to give their various exact solutions. However, these well studied non-linear systems are mostly local ones until Ablowitz and Musslimani [1] in 2013 introduceda PT symmetric nonlocal Schr¨odinger (NNLS) equation iq t ( x, t ) = q xx ( x, t ) ± q ( x, t ) q ∗ ( − x, t ) q ( x, t ) , (1)with ∗ being complex conjugate and q being a complex valued function of the real variables x and t . It is remarkable that, despite of the nonlocal property, Eq. (1) is an integrableinfinite dimensional Hamiltonian equation, which can be solved by inverse scattering trans-form and possesses infinitely number of conservation laws. Although Eq. (1) was derivedwith physical intuition, since then, a wealth of new nonlocal nonlinear integrable infinitedimensional Hamiltonian dynamical systems are introduced from integrability requirement,among which, many of them are constructed by simple symmetry reductions of generalAKNS scattering problems. These include reverse space-time, and in some cases reversetime, nonlocal nonlinear modified Korteweg-deVries equation, nonlocal sine-Gordon equa-tion, derivative NNLS equation and so on [2]. At the same time, many efficient methodshave been developed to obtain abundant wave solutions of nonlocal systems such as solitarywaves, periodic waves, rogue waves, etc. [3–6].In fact, nonlocal phenomena exist commonly in many fields of real nature [7–10]. Toestablish physically meaningful nonlocal systems, Lou proposed to construct two place Alice-Bob system (AB system) by using “AB-BA equivalence principle” and “ ˆ P s - ˆ T d - C principle”[11]. Through this way, many nonlocal version of physically important nonlinear systemsare constructed, such as AB-KdV equation [12, 13], AB-modified KdV equation [14], AB-NLS equation [11], etc., and various exact solutions are given, including some types ofshifted parity and time reversal ( ˆ P s ˆ T d ) symmetry breaking multi-soliton solutions, ˆ P s ˆ T d conserving localized excitation, rogue wave solutions [15, 16] and so on. Especially, for somecomplicated physically important AB-type models, for example AB-type multiple vortex2nteraction systems in atmosphere system, which are hard to give exact soliton solutionsor periodic wave solutions, the multiple scale expansion method is applied to transformthem into some aimed AB-type nonlinear equations, such as AB-(m)kdV equation [17, 18],AB-NLS equation [19], etc. Through this way, many approximate solutions for the originalcomplicated AB-type systems can be derived from various exact solutions of aimed AB-type equations, such as periodic elliptic wave solutions, soliton solutions and the interactionsolutions between solitons and periodic waves, etc., meanwhile, many physical phenomenalike correlated dipole blocking event can be appropriately explained.In this paper, we use a two-layer fluid system [20] q t + J { ψ , q } + βψ x = 0 , (2) q t + J { ψ , q } + βψ x = 0 , (3)where q = ψ xx + ψ yy + F ( ψ − ψ ) , (4) q = ψ xx + ψ yy + F ( ψ − ψ ) , (5)and J { a, b } = a x b y − b x a y , as a starting point to derive a nonlocal mKdV system and studyits various solutions. In Eqs. (2)-(5), F indicates coupling strength between two layers offluid, which is a small constant; β = β ( L /U ) where β = (2 ω /a ) cos( ϕ ), with a beingthe earth’s radius, ω being the angular frequency of the earth’s rotation and ϕ being thelatitude, U is the characteristic velocity scale while L is characteristic horizontal lengthscale. In deriving Eqs. (2)-(3) in Ref. [20], the constants are fixed as L = 10 m and U = 10 − ms − .The paper is organized as follows. In Sect. II, a general nonlocal variable coefficientmodified KdV (VCmKdV) equation with shifted parity and delayed time reversal is derivedfrom nonlocal version of the two-layer system (2)-(3) by using multiple scale expansionmethod. In Sect. III, various exact solutions of the VCmKdV system are obtained andanalyzed graphically, including elliptic wave solutions, solitary wave solutions, interactionsolutions between elliptic waves and solitons. Especially, a kind of elliptic wave solutionexhibits abundant wave structures for different choices of the coefficient functions of theVCmKdV system, which is analyzed and depicted by 4 cases. In Sect. IV, as a simpleillustration, an approximated solution to the original nonlocal version of the two-layer system3s given by using a known solution of the VCmKdV system. The last section devotes to asummary and discussion. II. DERIVATION OF A NONLOCAL VARIABLE COEFFICIENT MODIFIEDKDV EQUATION
Based on the AB-BA equivalence principle and ˆ P s - ˆ T d - C principle, the nonlocal counter-part of the two-layer system (2)-(3) can be obtained by employing the following symmetryconstraint ψ = ˆ P xs ˆ T d ψ = ψ ( − x + x , y, − t + t ) , (6) q = ˆ P xs ˆ T d q = q ( − x + x , y, − t + t ) . (7)To derive a nonlocal modified KdV type equation with shifted parity and delayed timereversal from it, according to the multiple scale expansion method, the following long waveapproximation assumption is assumed ξ = (cid:15) ( x − c t ) , τ = (cid:15) t, (8)where (cid:15) is a small parameter, and c is an arbitrary constant. The stream function ψ canbe expanded as ψ = c y + U + ψ ( ξ, y, τ ) , (9)where U ≡ U ( y ) is an arbitrary function of y and the last term can be expanded as ψ ( ξ, y, τ ) = (cid:15)φ + (cid:15) φ + (cid:15) φ + O ( (cid:15) ) , (10)with φ i ≡ φ ,i ( ξ, y, τ ) , ( i = 1 , ,
3) being functions of indicated variables. The modelconstants F and β can be reasonable assumed as (cid:15) and (cid:15) order, respectively, i.e., F = F (cid:15), β = β (cid:15) , (11)which means that coupling strength between two layers is small and the effect of the rotationof the earth is even smaller than that.Substituting Eq. (9) with Eqs. (8), (10) and (11) into Eqs. (2) and (3) with (6), (7) andvanishing coefficients of O ( (cid:15) ), we obtain( U y − c ) φ yyξ − U yyy φ ξ = 0 , (12)4nd ( U y − c ) φ yyξ − U yyy φ ξ = 0 , (13)where φ = ˆ P ξs ˆ T τd φ = φ ( − ξ + ξ , y, − τ + τ ).Since Eqs. (12) and (13) are linear with respect to φ and φ , we assume them in thevariable separation form as φ = G ( y, τ ) A ( ξ, τ ) ≡ G A, (14)and φ = P ( y, τ ) B ( ξ, τ ) ≡ P B, (15)with B ( ξ, τ ) = ˆ P ξs ˆ T τd A = A ( − ξ + ξ , − τ + τ ) and P ( y, τ ) = ˆ T τd G = G ( y, − τ + τ ). Nowsubstitute Eqs. (14) and (15) into Eqs. (12) and (13) to get( U y + 2 c ) G yy − U yyy G = 0 , (16)and ( U y + 2 c ) P yy − U yyy P = 0 , (17)which have a general form of solution for G and P G = ( U y + 2 c ) F ( τ ) (cid:90) U y + 2 c ) dy, (18)and P = ( U y + 2 c ) H ( τ ) (cid:90) U y + 2 c ) dy, (19)with H ( τ ) = ˆ T τd F ( τ ) = F ( − τ + τ ) being an arbitrary function.Now vanishing the coefficients of O ( (cid:15) ) leads to( U y + 2 c ) φ yyξ − U yyy φ ξ = ( F U y + 2 F c + φ yyy ) φ ξ + F ( U y + 2 c ) φ ξ − φ y φ yyξ , (20)and( U y + 2 c ) φ yyξ − U yy φ ξ = ( F U y + 2 F c + φ yyy ) φ ξ + F ( U y + 2 c ) φ ξ − φ y φ yyξ , (21)where φ = ˆ P ξs ˆ T τd φ = φ ( − ξ + ξ , y, − τ + τ ).5o solve the Eqs. (20) and (21), it is readily verified that φ and φ can be taken in theform φ = ( G B + G A + G A ) G (22)and φ = ˆ P ξs ˆ T τd φ = ( P B + P A + P A ) P , (23)where P i ( y, τ ) = ˆ T τd G i ( y, τ ) = G i ( y, − τ + τ ) ( i = 1 , , G = − QG (cid:20) F + (cid:90) F − (cid:82) F P QdyQ dy (cid:21) , (24) G = − QG (cid:20) F + (cid:90) F − (cid:82) F G QdyQ dy (cid:21) , (25) G = − QG (cid:20) F + (cid:90) F − (cid:82) ( G G yyy − G yy G y ) dy Q dy (cid:21) , (26)where Q = U y + 2 c and F i , ( i = 3 · · ·
8) are arbitrary functions of τ .To take one step further by vanishing coefficients O ( (cid:15) ) we get( U y + 2 c ) φ yyξ − U yyy φ ξ = ( F φ y + φ yyy + β ) φ ξ + ( F U y + 2 F c + φ yyy ) φ ξ + F ( U y + 2 c ) φ ξ − ( U y + 2 c ) φ ξξξ + ( F φ ξ − φ yyξ ) φ y − φ y φ yyξ + φ yyτ , (27)( U y + 2 c ) φ yyξ − U yyy φ ξ = ( F φ y + φ yyy + β ) φ ξ + ( F U y + 2 F c + φ yyy ) φ ξ + F ( U y + 2 c ) φ ξ − ( U y + 2 c ) φ ξξξ + ( F φ ξ − φ yyξ ) φ y − φ y φ yyξ + φ yyτ , (28)where φ = ˆ P ξs ˆ T τd φ = φ ( − ξ + ξ , y, − τ + τ ).Without using y − average trick as in many literatures, we assume the general form ofvariable separation solutions of Eqs. (27) and (28) as φ = g (cid:90) A ξ Bdξ + g A + g A + g A + g AB + g B + g B + g A ξξ , (29) φ = ˆ P ξs ˆ T τd φ = p (cid:90) B ξ Adξ + p B + p B + p B + p AB + p A + p A + p B ξξ , (30)where p i ( y, τ ) = ˆ T τd g i ( y, τ ) = g i ( y, − τ + τ ) ( i = 1 · · ·
8) are functions to be determined later.Substituting Eqs. (29) and (30) into Eqs. (27) and (28) and requiring coefficients of differentderivatives of A and B be proportional to each other up to a constant level, we arrive at ageneral nonlocal modified KdV system: A τ + ( e A + e A + e B + e ) A ξ + ( e A + e B + e ) B ξ + e A ξξξ + e A = 0 , (31)6 τ + ( s B + s B + s A + s ) B ξ + ( s B + s A + s ) A ξ + s B ξξξ + s B = 0 , (32)where e = F τ F , e i ≡ e i ( τ ) ( i = 1 · · ·
8) and s i ( τ ) = ˆ T τd e i ( τ ) = e i ( − τ + τ ) ( i = 1 · · ·
9) arearbitrary functions, while g i ( i = 1 · · ·
8) in Eq. (29) are determined by g = − Q ( m + (cid:90) ( m − (cid:90) G y G y + (4 G G yy − F P ) G y + 2 G G yy G y +( e − e ) G yy + F G P y + G G yyy dy ) Q − dy ) ,g = − Q ( m + (cid:90) (cid:0) m − (cid:90) ( G G yy − G yy G ) G y − G y G y + G G yyy +2 G G yy G y + 3 G G G yyy − e G yy dy (cid:1) (3 Q ) − dy ) ,g = − Q ( m + (cid:90) ( m − (cid:90) G G yy − G yy G ) G y − G y G y + 2 F G U y G +4 F G G c + G G yyy + 2 G G yy G y + 2 G G G yyy − e G yy dy )(2 Q ) − dy ) ,g = − Q ( m + (cid:90) ( m − (cid:90) F ( G G + P P ) U y + 2 F G G c + 2 F P P c + G β − e G yy dy ) Q − dy ) ,g = − Q ( m + (cid:90) ( m − (cid:90) ( F P − G G yy − G yy G ) G y − G y G y + G G G yyy − e G yy dy ) Q − dy ) ,g = − Q ( m + (cid:90) ( m − (cid:90) F U y P P + 4 F P P c − e G yy dy )(2 Q ) − dy ) ,g = − Q ( m + (cid:90) ( m − (cid:90) F ( G G + P P ) U y − e G yy + 2 F c ( G G + P P ) dy ) Q − dy ) ,g = − Q ( m + (cid:90) ( m + (cid:90) U y G + 2 G c + e G yy dy ) Q − dy ) , (33)with m i ≡ m i ( τ ) ( i = 1 · · ·
16) being arbitrary functions.
III. EXACT SOLUTIONS OF THE NONLOCAL VCMKDV EQUATIONA. exact solutions of the VCmKdV system with constant coefficients
To give more interesting solutions of the VCmKdV equation (31), for simplicity, we firstassume the coefficient of it are all constants. Under this assumption as well as e = 0 inEq. (31), i.e. F ( τ ) is a nonzero constant, we use elliptic function expansion method andgeneralized tanh expansion method to give some periodic wave solutions and interactionsolutions between soliton and periodic waves, respectively.7 . periodic wave solutions Using elliptic function expansion method, after some routine work, it can be verified Eq.(31) admits two kinds of elliptic wave solutions, the first one is A = − e − e + e − e e ± a m (cid:114) − e e sn (cid:0) a ( ξ − ξ ) + b ( τ − τ ) , m (cid:1) , (34)with b = a (cid:2) e e ( m + 1) a − e − e ) e + (3 e + e − e − e )( e − e + e − e ) (cid:3) e , (35)and the other one is A = − e + e + e + e e ± a m (cid:114) e e cn (cid:0) a ( ξ − ξ ) + b ( τ − τ ) , m (cid:1) , (36)with b = − a (cid:2) e (2 m − e a + (4 e + 4 e ) e − ( e + e + e + e ) (cid:3) e , (37)and the others being arbitrary constants. It’s interesting to see that the solution of (34) isˆ P ξs ˆ T τd symmetry breaking while the solution of (36) is ˆ P ξs ˆ T τd symmetry conserving, whichare shown in Figs. 1(a) and 1(b), respectively, with upper sign and the parameters are fixedas a = e = e = e = e = e = e = ξ = τ = 1 , e = − , e = 2 , b = − . , m = 0 . , (38)for Fig. 1(a), and a = e = e = e = e = e = e = e = 1 , ξ = τ = 0 , e = 2 , b = − . , m = 0 . , (39)for Fig. 1(b), respectively, at a specific time τ = 1. When m in Eqs. (34) and (36)approaches to unity, it reduces to tanh and sech functions, respectively, which are shown inFigs. 2(a) and 2(b), with upper sign and the parameters are fixed as a = e = e = e = e = e = e = ξ = τ = m = 1 , e = − , e = 2 , b = − . , (40)for Fig. 2(a), and a = e = e = e = e = e = e = e = m = 1 , ξ = τ = 0 , e = 2 , b = − . , (41)for Fig. 2(b), respectively, at a specific time τ = 1.8 (a) (b) FIG. 1: Profiles of elliptic periodic wave solution at a specific time τ = 1 with upper sign: (a) snsolution of (34) with (35); (b) cn solution (36) with (37). (a) (b) FIG. 2: Profiles of soliton solution at a specific time τ = 1 with upper sign: (a) kink soliton solutionof (34) with (35); (b) bright soliton solution of (36) with (37). . interaction solutions between solitons and periodic waves To obtain interaction solutions between solitons and periodic waves, a simple but effectiveway is using generalized tanh expansion method. To this end, we assume the solution of Eq.(31) has the form A = a + a tanh( f ) with a , a and f are all undetermined functions of ξ and τ . By carrying out the standard procedure, we have A = − √− e e f ξξ e f ξ − e − e + e − e e + √− e e f ξ e tanh( f ) , (42)with f = ξ + h arctanh( h sn( h ξ , m )), ξ = k ( ξ − ξ ) + ω ( τ − τ ), ξ = k ( ξ − ξ ) + ω ( τ − τ ), and the constants an be classified into 44 cases, here, we just list 2 of them:case (1) h = 1 , h = ±√ m, h = 2 k k ( m + 1) , (43) ω = k m + 1) e (cid:8) [8 k ( m + 14 m + 1) e − m + 1) ( e − e )] e +( m + 1) (3 e + e − e − e )( e − e + e − e ) (cid:9) , (44) ω = k m + 1) e (cid:8) (8 k (5 m + 6 m + 5) e − m + 1) ( e − e )) e +( m + 1) (3 e + e − e − e )( e − e + e − e ) (cid:9) , (45)case (2) h = 12 , h = − m, h = 2 k k m , (46) ω = k m e (cid:8) [8 k ( m + 3) e − m ( e − e )] e + m (3 e + e − e − e )( e − e + e − e ) (cid:9) , (47) ω = k m e (cid:8) [8 k (5 m − e − m ( e − e )] e + m (3 e + e − e − e )( e − e + e − e ) (cid:9) , (48)with the others remain free.To show the special feature of interaction solutions, case (1) of the solution (42) is dis-played in Fig. 3 for A and Fig. 4 for B , with the parameters are fixed as e = e = e = e = e = k = h = ξ = τ = 1 , e = − . , e = 2 , e = 2 , k = 8 ,m = 0 . , h = 0 . , h = 10 , ω = − . , ω = − . (49)10 a) (b) FIG. 3: Profiles for A of interaction solutions (42) in case (1): (a) three dimensional view; (b)density plot. Fig. 3 shows the structure of a kink soliton interacting with elliptic periodic waves, whileFig. 4 shows the structure of a anti-kink soliton interacting with elliptic periodic waves,both of which are with a nonzero phase change.
B. exact solutions of the nonlocal VCmKdV equation with variable coefficients
For the general case, i.e. e i ( τ ) ( i = 1 · · ·
9) in Eq. (31) are not constants, we give thefollowing two kinds of periodic wave solutions, one is A = 12 k m cn (cid:0) k ( ξ − ξ ) + ω ( τ )( τ − τ ) , m (cid:1) , (50)with ω ( τ ) = − τ − τ (cid:90) τ − τ k (cid:0) k (2 m − e ( τ + 12 τ ) + e ( τ + 12 τ ) + e ( τ + 12 τ ) (cid:1) dτ, (51)and the other one is A = − k m sn (cid:0) k ( ξ − ξ ) + ω ( τ )( τ − τ ) , m (cid:1) , (52)with ω ( τ ) = 22 τ − τ (cid:90) τ − τ k (cid:0) k ( m + 1) e ( τ + 12 τ ) − e ( τ + 12 τ ) − e ( τ + 12 τ ) (cid:1) dτ, (53)11 a) (b) FIG. 4: Profiles for B = ˆ P ξs ˆ T τd A of interaction solutions (42) in case (1): (a) three dimensionalview; (b) density plot. and the others are arbitrary constants. In Eqs. (50) and (52), sn and cn are elliptic sinefunction and cosine functions, respectively, with modulus m . In addition, both of thesesolutions are under the common condition e ( τ ) = e ( τ ) − e ( τ ) − e ( τ ) − e ( τ ) , e ( τ ) = e ( τ ) = 0 . (54)Choosing different functions of e i ( τ ) , ( i = 1 · · ·
8) results in rich structure of the solution(50), here we show it graphically by fixing the parameters as e = e = e = e = e = e = cos( τ ) , e = − τ ) , k = 0 . , ξ = τ = 0 , (55)for Fig. 5, e = e = e = e = e = e = sech( τ ) , e = − τ ) , k = 0 . , ξ = τ = 0 , (56)for Fig. 6, e = e = e = e = cos( τ ) , e = e = cos( τ ) , e = − τ ) , k = 0 . , ξ = τ = 0 , (57)for Fig. 7, e = e = e = e = τ , e = e = τ cosh( τ ) , e = − τ , k = 0 . , ξ = τ = 0 , (58)12 a) (b) FIG. 5: Profiles of the solution (50) with the parameters being fixed by (55) and (a)m=0.9; (b)m=1. for Fig. 8, respectively. In each of the figures, the right wave is the limiting case of the leftwaves by taking m → P ξs ˆ T τd symmetry, so the field B have exactly the same dynamic behavior as A . IV. AN ILLUSTRATION: APPROXIMATE SOLUTIONS OF THE NONLOCALTWO-LAYER FLUID SYSTEM
Given the abundant known exact solutions of the nonlocal VCmKdV equation, it ismeaningful to use them to generate approximate solutions of the original two-layer fluidsystem and give them appropriate physical explanations. To this end, up to O ( (cid:15) ), we firstobtain the approximate solution of Eqs. (2) and (3) from Eqs. (10) and (14) ψ = c y + U ( y ) + (cid:15)G ( y ) A ( ξ, τ ) , ψ = ˆ P ξs ˆ T τd ψ (59)with ξ = (cid:15) ( x − c t ) , τ = (cid:15) t , and G is given by (18). When taking A ( ξ, τ ) in Eq. (59) asthe solution of Eq. (36), i.e. the solution of (31) with constant coefficients, and fixing the13 a) (b) FIG. 6: Profiles of the solution (50) with the parameters being fixed by (56) and (a) m =0.9;(b) m =1. (a) (b) FIG. 7: Profiles of the solution (50) with the parameters being fixed by (57) and (a) m =0.9;(b) m =1. arbitrary functions and parameters as U = sin y a) (b) FIG. 8: Profiles of the solution (50) with the parameters being fixed by (58) and (a) m =0.9;(b) m =1. and C = 0 , C = 200 , a = 2 , b = − , e = e = e = e = e = e = e = c = 1 ,e = 2 , m = 1 , t = 10 , x = 160 , (cid:15) = 0 . , (61)and by substituting them into Eq. (59), a special stream function is obtained, of which, thedensity distribution and streamlines are shown in Fig. 9. Observing from Figs. 9(a) and9(b), it’s obvious that the vertices have the property of congestion both in space and time,which are mainly localized around the line of x = 75 and t = −
71, respectively. It should benoted that the existence of an arbitrary function U in Eq. (59) could generate more kindsof approximate solutions for the original nonlocal two-layer fluid system. Furthermore, ifwe take the A in Eq. (59) as the solution of Eq. (31) with variable coefficients (e.g. thesolution of (52)) more complex can be generated by choosing different ω ( τ ), which havemore potentiality of physical applications. V. CONCLUSION AND DISCUSSION
In summary, a nonlocal VCmKdV system with shifted parity and delayed time reversal isderived from a two-layer liquid system by applying AB-BA equivalence principle and multiple15 a) (b)
FIG. 9: Density plot with stream lines of the approximate solution (59) with (36) at:(a) t =1;(b) x =1. scale expansion method. Various exact solutions of the VCmKdV system are obtained,including elliptic periodic waves, solitary waves and interaction solutions between solitonsand periodic waves. As an illustration, a simple approximate solution of the original nonlocaltwo-layer liquid system are given and analyzed.From the derivation process of nonlocal VCmKdV system, it’s clear that multiple scalemethod is powerful in obtaining aimed nonlocal nonlinear systems. In this sense, their existsa lot of work to derive other types of nonlocal equations from many physically importantmodel and use their approximate solutions to analyze related phenomena, which needs tobe explored in our future work. Acknowledgments
This work was supported by the National Natural Science Foundation of China underGrant Nos. 11405110 and the Natural Science Foundation of Zhejiang Province of Chinaunder Grant No. LY18A050001. 16 ompliance with ethical standardsConflict of interest statement
The authors declare that they have no conflicts of interest to this work. There is noprofessional or other personal interest of any nature or kind in any product that could beconstrued as influencing the position presented in the manuscript entitled.
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