# Residual Symmetry Reduction and Consistent Riccati Expansion to a Nonlinear Evolution Equation

Authors: Lamine Thiam; Xi-zhong Liu
Journal: Complexity
 (2019)

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## Abstract

The residual symmetry of a (1 + 1)-dimensional nonlinear evolution equation (NLEE)ut+uxxx−6u2ux+6λux=0 is obtained through Painlevé expansion. By introducing a new dependent variable, the residual symmetry is localized into Lie point symmetry in an enlarged system, and the related symmetry reduction solutions are obtained using the standard Lie symmetry method. Furthermore, the (1 + 1)-dimensional NLEE equation is proved to be integrable in the sense of having a consistent Riccati expansion (CRE), and some new Bäcklund transformations (BTs) are given. In addition, some explicitly expressed solutions including interaction solutions between soliton and cnoidal waves are derived from these BTs.


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## Body

## 1. Introduction

In nonlinear science, the study of nonlinear equations plays an important role in analyzing related complex phenomenon, which exists in the fields of fluid dynamics, plasma, optics, and so on [1]. In the past few decades, many effective methods, including Hirota’s bilinear method [2], Darbux transformation [3, 4] and Bäcklund transformation, Lie symmetry analysis [5, 6], and inverse scattering transformation [7], have been proposed and developed to investigate abundant properties of nonlinear equations. Among these methods, symmetry analysis plays an important role in simplifying and even completely solving nonlinear equations. For many integrable systems, the standard Lie symmetry method can be used to obtain their Lie symmetry group and symmetry reduction solutions. In addition, a finite transformation group related to a symmetry can also be obtained by using Lie’s first theorem.Traditionally, finite transformation related to a nonlocal symmetry cannot be obtained directly in the same way as those of Lie point symmetries. To concur this difficulty, Cheng et al. [8] proposed localizing a nonlocal symmetry in an enlarged nonlinear system by introducing some new dependent variables to the original system, provided all the Lie point symmetries are closed in the enlarged system. Since then, a lot of works have been done on many important nonlinear systems [9–13]. They prove that this localization method is very efficient in obtaining new Bäcklund transformations (BTs) and also new symmetry reduction solutions related to a nonlocal symmetry in an enlarged system. Especially, for many integrable systems, interaction solutions between a soliton and nonlinear periodic waves could be obtained in this way.There exist many different ways to obtain nonlocal symmetries, including potential symmetry [14], Lie–Bäcklund symmetries [15], inverse recursion operators [16, 17], the conformal-invariant form [18], Darboux transformation [19], Bäcklund transformation (BT), and Lax pair. It is found in recent years that, for many nonlinear systems, the coefficient of negative first power of a singular manifold in a truncated Painlevé expansion is readily a nonlocal symmetry of the equation [20], which is called residual symmetry. Compared with other methods for obtaining nonlocal symmetries, the method for obtaining a residual symmetry is very simple, and plenty of studies have been carried out by localizing a residual symmetry into a Lie point symmetry [19, 21, 22]. To obtain more abundant interaction solutions, Lou generalized Riccati expansion method and proposed a new concept of integrability in the sense of having consistent Riccati expansion (CRE) [23]. By applying the CRE method, many new BTs and interaction solutions for various nonlinear systems are obtained [12, 24–29].In this paper, by using residual symmetry localization method and CRE method, we investigate the following (1 + 1)-dimensional nonlinear evolution equation (NLEE):(1)ut+uxxx−6u2ux+6λux=0,which includes the modified KdV equation as a special case by setting λ=0. In Reference [30], NLEE (1) is derived with a Bäcklund transformation relating solutions of KdV equation and solutions of equation (1). It reveals that the vacuum state exists in the KdV equation as well as in NLEE equation (1), and the relationship between vacuum parameter of the set of KdV solutions and the vacuum parameter of equation (1) is found [30, 31]. A series of analytical solutions of NLEE (1) are also given in [32].The paper is organized as follows. In Section1, the residual symmetry of NLEE (1) is derived from the truncated Painlevé expansion and then localized into a Lie point symmetry by introducing a new dependent variable to enlarge the NLEE. On this basis, the finite transformation related to the residual symmetry is also obtained by applying Lie’s first theorem. In Section 2, the general form of Lie point symmetry group as well as symmetry reduction solutions of the enlarged NLEE is obtained by using the standard Lie symmetry method. In Section 3, NLEE (1) is proved to be CRE integrable, and some new BTs are obtained. By applying the CTE method, some concrete explicitly expressed solutions of the NLEE are given, which include the interaction solutions between solitons and background cnoidal waves. The last section contains a summary.


## 2. Localization of Residual Symmetry

By balancing the dispersion term and nonlinear term, the truncated Painlevé expansion of equation (1) is(2)u=u0ϕ+u1,where ϕ is the singular manifold and u0andu1 are functions of xandt to be determined later. Substituting (2) into (1) and making the coefficients of different powers of ϕ to zero, we have(3)u0=ϕx,(4)u1=−ϕxx2ϕx,where ϕ satisfies the Schwartzian form of (1)(5)C+K+6λ=0,with C=ϕxxx/ϕx−3/2ϕxx2/ϕx2 and K=ϕt/ϕx. By substituting (3) and (4) into (2), the following theorem is readily obtained.Theorem 1.
Ifϕ is a solution of the Schwartzian equation (5), then(6)u=ϕxϕ−ϕxx2ϕx,is a solution of equation (1).As we know, any Schwartzian equation like (5) is form-invariant under Möbious transformation(7)ϕ⟶a1ϕ+b1a2ϕ+b2,a1a2≠b1b2,which means equation (5) have three symmetries:(8)σϕ=d1,σϕ=d2ϕ,σϕ=d3ϕ2,with arbitrary constants d1,d2, and d3. It is interesting that the residue u0 in expansion (2) is just a symmetry of the NLEE, which can be verified by substituting it with equations (4) and (5) into the linearized equation of (1). Obviously, the residual symmetry σu=u0 is linked to the symmetry of (8) by the linearized equation of (4).To get the finite transformation corresponding to the residual symmetryσu=ϕx, we have to localize it into a Lie point symmetry first. To this end, we introduce a new dependent variable as(9)g≡ϕxto enlarge the original equation (1).The linearized equations of the enlarged system (1), (5), and (9) are(10a)σu,t+σu,xxx−12uuxσu−6u2σu,x+6λσu,x,x=0,(10b)ϕxxxσϕ,x+σϕ,xxxϕx−3ϕxxσϕ,xx+σϕ,xϕt+12σϕ,xλ+σϕ,tϕx=0,(10c)σϕ,x−σg=0.When we fixσϕ=−ϕ2, the equations of (10a)–(10c) have a simple solution(11a)σu=g,(11b)σg=−2gϕ,(11c)σϕ=−ϕ2,which means the residual symmetry is localized in the enlarged system.By applying Lie’s first theorem to the initial value problem of the symmetry (11a) and (11b), i.e.,(12)du^εdε=g^ε,u^0=u,dg^εdε=−2ϕ^εg^ε,g^0=g,dϕ^εdε=−ϕ^ε2,ϕ^0=ϕ,we get the following Bäcklund transformation.Theorem 2.
Ifu,g,ϕ is a solution of the prolonged system (1), (5), and (9), then so is u^,g^,ϕ^ with(13a)u^=u+εgεϕ+1,(13b)g^=gεϕ+12,(13c)ϕ^=ϕεϕ+1,with an arbitrary group parameter ε.


## 3. Residual Symmetry Reduction Solutions of Equation (1)

The general form of a Lie point symmetry of the enlarged NLEE system (1), (5), and (9) can be written in the form as follows:(14)V=X∂∂x+T∂∂t+U∂∂u+G∂∂g+Φ∂∂ϕ,which means the system is invariant under the following transformation:(15)x,t,u,g,ϕ⟶x+εX,t+εT,u+εU,g+εG,ϕ+εΦ,with the infinitesimal parameter ε. Equivalently, the symmetry in the form (13a)-(13b) can be written as a function form as follows:(16a)σu=Xux+Tut−U,(16b)σg=Xgx+Tgt−G,(16c)σϕ=Xϕx+Tϕt−Φ.Substituting equation (16a)–(16c) with the enlarged NLEE system into equation (10a)–(10c) and vanishing all the derivatives of dependent variables u,g, and ϕ, over-determined linear equations for the infinitesimals X,T,U,G,andΦ are obtained. After calculation by computer using the software Maple, we get the result(17)X=4c1λt+c13x+c4,T=c1t+c2,U=−c13u+c3g,G=−13g6c3ϕ+c1−3c5,Φ=−c3ϕ2+c5ϕ+c6,with arbitrary constants cii=1,…,6. It is obvious that the symmetry for u,g,andϕ in equation (17) contains the symmetry of (11a)–(11c) as a special case.In consideration of equation (17), the symmetries in (16a)–(16c) can be written as(18)σu=4c1λt+c13x+c4ux+c1t+c2ut+c13u−c3g,σg=4c1λt+c13x+c4gx+c1t+c2gt+13g6c3ϕ+c1−3c5,σϕ=4c1λt+c13x+c4ϕx+c1t+c2ϕt+c3ϕ2−c5ϕ−c6.The group invariant solutions of the enlarged NLEE system can be obtained by applying the symmetry constraintsσu=σg=σϕ=0 in equation (19), which is equivalent to solving the characteristic equation(19)dx4c1λt+c1/3x+c4=dtc1t+c2=du−c1/3u+c3g=dg−1/3g6c3ϕ+c1−3c5=dϕ−c3ϕ2+c5ϕ+c6.Without loss of generality, we consider symmetry reduction solutions of the enlarged NLEE system in the following two cases.Case 1.
(ci≠0i=1,…,6). After solving equation (19), we get the symmetry reduction solutions of the enlarged NLEE system (1), (5), and (9) as(20)ϕ=tanhΔ1c1Φ+lnΔ/2c1Δ1+c52c3,Δ=c1t+c2,Δ1=4c3c6+c52,(21)g=−2GΔ1/3coshΔ1c1Φ+lnΔ/c1+1,(22)u=UΔ1/3+4c3GΔ1expΔ1ΦΔΔ1/c1+1Δ1/3,where U,G, and Φ are group invariant functions of a group invariant variable ξ=−6c1λt−c1x+18c2λ−3c4/c1t+c21/3c1.
The symmetry reduction equations forU,G, and Φ can be obtained by substituting equations (20)–(22) into the enlarged NLEE system (1), (5), and (9). The results are(23)G=−Δ12Φξ4c3,(24)U=Δ1Φξ2−Φξξ2Φξ,where Φ satisfies the following symmetry reduction equation:(25)−3Δ12Φξ4−2c1ξΦξ2+61+ΦξξξΦξ−9Φξξ2=0.
It is obvious that once any solution of equation (25) is given, the solution of the NLEE can be obtained by substituting it with equations (23) and (24) into equation (22). To give a concrete example, we take a simple solution for (25) under the condition Δ1=c1/3 as(26)Φ=d2c1+3lnd1ξc1,which leads to a simple solution for NLEE (1)(27)u=exp1/3d2c1d1exp1/3d2c1d1ξΔ1/3+1,with arbitrary constants d1 and d2.Case 2.
(c1=0 and ci≠0i=2,…,6).
In this case, similar to case 1, the symmetry reduction solutions of the enlarged NLEE system (1), (5), and (9) are(28)ϕ=tanhΔ1ϕ′+t/2c2Δ1+c52c3,g=−2g′coshΔ1ϕ′+t/c2+1,u=u′−2c3g′tanhΔ1ϕ′+t/2c2Δ1,with u′,g′, and ϕ′ being group invariant functions of group invariant variable x′=x−c4/c2t.
The corresponding symmetry reduction equations foru′,g′, and ϕ′ are(29)g′=−Δ12ϕx′′4c2c3,(30)u′=−ϕx′x′′2ϕx′′,(31)Δ12ϕx′′4−2c26c2λ−c4ϕx′′2−2c22ϕx′x′x′′+1ϕx′′+3c22ϕx′x′′2=0.


## 4. CRE Solvability and Interaction Solutions

### 4.1. CRE Integrable

By leading order analysis, the Riccati expansion of NLEE (1) is(32)u=v0+v1Rw,w=wx,y,t,where v0andv1 are functions of x,t and Rw is a solution of the Riccati equation(33)Rw=a0+a1R+a2R2.Substituting equations (32) with (33) into equation (1) and vanishing all the coefficients of different powers of Rw, we get(34)v0=a1wx2+wxx2wx,v1=a2wx,and three different equations of w. Fortunately, these three equations are consistent with each other, one of which is(35)δwx2+2K′+C′+6λ=0,with C′=wxxx/wx−3/2wxx2/wx2, K′=wt/wx, and δ=4a0a2−a12.The consistency of different equations ofw means that NLEE (1) is integrable under the meaning of having a consistent Riccati expansion. In summary, we have the following theorem.Theorem 3.
Ifw is a solution of(36)δwx2+2K′+C′+6λ=0,then(37)u=a2wxRw+a1wx2+wxx2wx,is a solution of NLEE (1), with R=Rw being a solution of the Riccati equation (33).


### 4.2. Consistent tanh-Function Expansion

When we take a special solution of the Riccati equation (33) as Rw=tanhw, the consistent Riccati expansion (31) reduces to(38)u=u1′tanhw+u0′,which is called consistent tanh expansion (CTE).Following the same logic as in the CRE case, we get the following nonauto BT.Theorem 4.
Ifw satisfies the following equation:(39)2wx2−K−C−6λ=0,then(40)u=wxtanhw−wxx2wxis a solution of NLEE (1).To give some concrete exact solutions of NLEE (1), we first take the form of w in (39) as(41)w=k0x+ω0t+g,where g is a arbitrary function of x and t, k0, and ω0 are arbitrary constants. Three special cases of (41) are listed as follows.Case 3.
We take a simple solution of equation (39) as(42)w=2k3t−6kλt+kx+d,with kandd being arbitrary constants. By using Theorem 4, we obtain a kink soliton solution for NLEE (1):(43)u=ktanh2k3t−6kλt+kx+d.Case 4.
We further constrain the form of (41) as(44)w=k0x+ω0t+WX,X=k1x+ω1t,where k0,ω0,k1, andω1 are arbitrary constants. By substituting (44) into (39), it is interesting to find that W1X≡WXX satisfies the following elliptic function equation:(45)W1X2=C0+C1W1+C2W12+C3W13+C4W14,with(46)C0=C1k13−C2k0k12+C3k02k1−4k03k0k14,C4=4,ω0=C1k13−32C2k0k12+32C3k02k1−4k03−6k0λ,ω1=k1C2k12−3C3k0k1+24k02−12λ2,with arbitrary constants C1, C2, and C3. By Theorem 4, equation (44) leads to a solution of NLEE (1) as(47)u=k0+W1k1tanhk0x+ω0t+W−k122k0+W1k1W1,X,which describes an interaction mode between soliton and elliptic waves. To illustrate this point more clearly, we take a special Jacobi elliptic solution of equation (45) as(48)W1=μ0+μ1snb1X,p,with arbitrary constants μ0andμ1. By substituting equation (48) into equation (45) and vanishing coefficients of different powers of sn, we get(49)C1=C312816C2−C32,C3=8m+16k0k1,μ0=−C316,μ1=5C32k12−64C2k12+32C3k0k1−256k0216k1,b1=−C3k1−16k08k1,p=5C32k12−64C2k12+32C3k0k1−256k02C3k1−16k0.
Figure1 displays the interaction solution of (47) with equations (46), (48), and (49) and the parameters are fixed by(50)C0=11607102400,C1=−866951200,C2=−82696400,C3=1,b1=78,μ0=−116,μ1=−63160,k0=12,k1=1,λ=2,ω0=−5453110240,ω1=−13306912800,p=910.
Figures1(a) and 1(b) describe the interaction structure between a kink soliton and background cnoidal waves in one-dimensional variables of x and t, respectively, which have similar structures.Figure 1
The interaction solution (46) with parameters being fixed by (49): (a) two-dimensional plot with t=0; (b) two-dimensional plot with x=0.
(a)
(b)Case 5.
We take the form of equation (41) as(51)w=k1x+l1t+a1Eπsnk2x+l2t,p,q,p,where Eπ is the third type of incomplete elliptic integral and k1,l1,a1,k2,l2,p,andq are all arbitrary constants. Substituting equation (51) into equation (39) and vanishing coefficients of different powers of sn function, we get several types of solutions, two of which are(52)a1=−p2q−p2−q2+qq,k1=0,l1=−4−p2q−p2−q2+q/qk23p2q,l2=−2k22p2q−3k22p2+k22q+3λqk2q,(53)l1=2k13+6a1k12k2+6a12k22−6λk1+a12a12k23−6k2λ−l2,q=0.
Figures2 and 3 display the solution (40) with equations (51) and (52) in three dimensions and two dimensions, respectively, and the parameters are fixed by(54)p=910,q=12,k2=1,λ=12,a1=3110,k1=0,l1=−8131125,l2=3110.
Figure2 shows that the kink soliton can be seen as being composed by cnoidal waves, which can be seen more clearly by the density plot of Figure 2(b). As for Figures 3(a) and 3(b), they describe this solution one dimensionally with t=1/2 and x=0, respectively.
Whenq=0, the third-type incomplete elliptic integral in equation (50) becomes the first type. Figure 4 displays the interaction solution of (40) with equations (51) and (53). The parameters are fixed by(55)a1=1,k1=1,k2=1,l1=9,l2=1,λ=12,p=910,q=0.Figure 2
The interaction solution (40) with equations (51) and (52) and the parameters are fixed by (54): (a) the three-dimensional plot; (b) the corresponding density plot.
(a)
(b)Figure 3
The interaction solution (39) with equations (51) and (52) and the parameters are fixed by (54): (a) the two-dimensional plot with t=1/2; (b) the two-dimensional plot with x=0.
(a)
(b)Figure 4
The interaction solution (39) with equations (50) (in Section 4.2) and the parameters are fixed by (54): (a) the three-dimensional plot; (b) the two-dimensional plot with t=1/2.
(a)
(b)


## 4.1. CRE Integrable

By leading order analysis, the Riccati expansion of NLEE (1) is(32)u=v0+v1Rw,w=wx,y,t,where v0andv1 are functions of x,t and Rw is a solution of the Riccati equation(33)Rw=a0+a1R+a2R2.Substituting equations (32) with (33) into equation (1) and vanishing all the coefficients of different powers of Rw, we get(34)v0=a1wx2+wxx2wx,v1=a2wx,and three different equations of w. Fortunately, these three equations are consistent with each other, one of which is(35)δwx2+2K′+C′+6λ=0,with C′=wxxx/wx−3/2wxx2/wx2, K′=wt/wx, and δ=4a0a2−a12.The consistency of different equations ofw means that NLEE (1) is integrable under the meaning of having a consistent Riccati expansion. In summary, we have the following theorem.Theorem 3.
Ifw is a solution of(36)δwx2+2K′+C′+6λ=0,then(37)u=a2wxRw+a1wx2+wxx2wx,is a solution of NLEE (1), with R=Rw being a solution of the Riccati equation (33).


## 4.2. Consistent tanh-Function Expansion

When we take a special solution of the Riccati equation (33) as Rw=tanhw, the consistent Riccati expansion (31) reduces to(38)u=u1′tanhw+u0′,which is called consistent tanh expansion (CTE).Following the same logic as in the CRE case, we get the following nonauto BT.Theorem 4.
Ifw satisfies the following equation:(39)2wx2−K−C−6λ=0,then(40)u=wxtanhw−wxx2wxis a solution of NLEE (1).To give some concrete exact solutions of NLEE (1), we first take the form of w in (39) as(41)w=k0x+ω0t+g,where g is a arbitrary function of x and t, k0, and ω0 are arbitrary constants. Three special cases of (41) are listed as follows.Case 3.
We take a simple solution of equation (39) as(42)w=2k3t−6kλt+kx+d,with kandd being arbitrary constants. By using Theorem 4, we obtain a kink soliton solution for NLEE (1):(43)u=ktanh2k3t−6kλt+kx+d.Case 4.
We further constrain the form of (41) as(44)w=k0x+ω0t+WX,X=k1x+ω1t,where k0,ω0,k1, andω1 are arbitrary constants. By substituting (44) into (39), it is interesting to find that W1X≡WXX satisfies the following elliptic function equation:(45)W1X2=C0+C1W1+C2W12+C3W13+C4W14,with(46)C0=C1k13−C2k0k12+C3k02k1−4k03k0k14,C4=4,ω0=C1k13−32C2k0k12+32C3k02k1−4k03−6k0λ,ω1=k1C2k12−3C3k0k1+24k02−12λ2,with arbitrary constants C1, C2, and C3. By Theorem 4, equation (44) leads to a solution of NLEE (1) as(47)u=k0+W1k1tanhk0x+ω0t+W−k122k0+W1k1W1,X,which describes an interaction mode between soliton and elliptic waves. To illustrate this point more clearly, we take a special Jacobi elliptic solution of equation (45) as(48)W1=μ0+μ1snb1X,p,with arbitrary constants μ0andμ1. By substituting equation (48) into equation (45) and vanishing coefficients of different powers of sn, we get(49)C1=C312816C2−C32,C3=8m+16k0k1,μ0=−C316,μ1=5C32k12−64C2k12+32C3k0k1−256k0216k1,b1=−C3k1−16k08k1,p=5C32k12−64C2k12+32C3k0k1−256k02C3k1−16k0.
Figure1 displays the interaction solution of (47) with equations (46), (48), and (49) and the parameters are fixed by(50)C0=11607102400,C1=−866951200,C2=−82696400,C3=1,b1=78,μ0=−116,μ1=−63160,k0=12,k1=1,λ=2,ω0=−5453110240,ω1=−13306912800,p=910.
Figures1(a) and 1(b) describe the interaction structure between a kink soliton and background cnoidal waves in one-dimensional variables of x and t, respectively, which have similar structures.Figure 1
The interaction solution (46) with parameters being fixed by (49): (a) two-dimensional plot with t=0; (b) two-dimensional plot with x=0.
(a)
(b)Case 5.
We take the form of equation (41) as(51)w=k1x+l1t+a1Eπsnk2x+l2t,p,q,p,where Eπ is the third type of incomplete elliptic integral and k1,l1,a1,k2,l2,p,andq are all arbitrary constants. Substituting equation (51) into equation (39) and vanishing coefficients of different powers of sn function, we get several types of solutions, two of which are(52)a1=−p2q−p2−q2+qq,k1=0,l1=−4−p2q−p2−q2+q/qk23p2q,l2=−2k22p2q−3k22p2+k22q+3λqk2q,(53)l1=2k13+6a1k12k2+6a12k22−6λk1+a12a12k23−6k2λ−l2,q=0.
Figures2 and 3 display the solution (40) with equations (51) and (52) in three dimensions and two dimensions, respectively, and the parameters are fixed by(54)p=910,q=12,k2=1,λ=12,a1=3110,k1=0,l1=−8131125,l2=3110.
Figure2 shows that the kink soliton can be seen as being composed by cnoidal waves, which can be seen more clearly by the density plot of Figure 2(b). As for Figures 3(a) and 3(b), they describe this solution one dimensionally with t=1/2 and x=0, respectively.
Whenq=0, the third-type incomplete elliptic integral in equation (50) becomes the first type. Figure 4 displays the interaction solution of (40) with equations (51) and (53). The parameters are fixed by(55)a1=1,k1=1,k2=1,l1=9,l2=1,λ=12,p=910,q=0.Figure 2
The interaction solution (40) with equations (51) and (52) and the parameters are fixed by (54): (a) the three-dimensional plot; (b) the corresponding density plot.
(a)
(b)Figure 3
The interaction solution (39) with equations (51) and (52) and the parameters are fixed by (54): (a) the two-dimensional plot with t=1/2; (b) the two-dimensional plot with x=0.
(a)
(b)Figure 4
The interaction solution (39) with equations (50) (in Section 4.2) and the parameters are fixed by (54): (a) the three-dimensional plot; (b) the two-dimensional plot with t=1/2.
(a)
(b)


## 5. Conclusion

In summary, the NLEE is investigated through residual symmetry and the CRE method, respectively. The residual symmetry is obtained through the truncated Painlevé expansion and localized into a Lie point symmetry by enlarging the original NLEE into a new system. By applying the standard symmetry method, the general form of a Lie point symmetry of the enlarged NLEE as well as the corresponding symmetry reduction solutions is obtained, which could be used to describe the interaction mode between soliton and nonlinear waves. The NLEE is proved to be CRE integrable, and some new BTs are derived from this property. As far as we know, the literatures giving interaction solutions between soliton and cnoidal waves are mainly focused on (2 + 1)-dimensional systems. As for (1 + 1)-dimensional case, this kind of soliton-cnoidal wave interaction solutions are hard to give in an explicitly expression form (see, e.g. [10]). Fortunately, in this paper, two kinds of soliton-cnoidal wave solutions for equation (1) are obtained with a detailed analysis.



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