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On the generalized forms of exact solutions to the liouville equation via direct approach
Peqpunon
lnt. J. Engng Sci. Vol. 32, No. 12, pp. 1965-1%9, 1994 Copyright @ 1994 EIscvier Science Ltd Printed in Great Britain. Al1 rights reserved
~7~(~~E~l7-D
o&?o-7225/94 $7.00+ 0.00
ON THE GENERALIZED FORMS OF EXACT SOLUTIONS TO THE LIOUVILLE EQUATION VIA DIRECT APPROACH 0. P. BHUTAN1 Department
and M. H. M. MOUSSA
of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi-l 10 016, India K. Department
VI~AYAKUMAR
of Mathematics, Punjab University, Chandigarh-1~014, (Communicated
India
by D. G. B. EDELEN)
Abstract-Using the direct approach due to Clarkson and Kruskal [I] two generalized forms of exact solutions to the Liouville equation are presented that eventually enable recovering all available exact solutions obtained via group theoretic, Bscklund ~ansfo~ation and PainIevB analysis.
INTRODUCTION
The importance of the Liouville equation in the context of field theoretic model is well known [2] and the equation has been the subject of investigation for solution in [3-51. In [3], Ibragimov has presented an exact solution to the Liouville equation in terms of two arbitrary functions and their derivatives by reducing it to linear form via the BIcklund transformation. Tamizhmani and Lakshmanan [4] have achieved the said linearization through PainleG analysis and the form of the final solution turns out to be the same as reported in 121. In [5], Bhutani and Vijayakumar carried out a detailed analysis of Liouville equation via isovector approach. More specifically, the components of isovector fields tabulated in their complete generality that involve six arbitrary constants al, az-+,, yield, through the solutions of orbital equations, the invariant groups of transformation which reduce the given PDE to NLODE. In order to solve the resulting NLODE in the general case use is made of the variational s~rnet~ [6]. However, in the particular case wherein a, # 0, a4 ZO and a2, a3, a5, a6 are all zero, the resulting NLODE is solved via the standard technique and two new solutions are reported. Herein, using the Clarkson and Kruskal formalism [l] (that does away with the sophisticated tools of functional analysis, Painleve analysis, Bgcklund transformation, variational symmetry, etc.), we have obtained two generalized forms of similarity solutions obtained via the group theoretic technique; the other generalized solution enabled us to obtain Ibragimov’s solution obtained via the BBcklund transformation.
2. LIOUVILLE
EQUATION
AND
The Liouville equation under consideration
METHOD
OF
SOLUTION
can be expressed as
u,t - e” = 0.
(2.1)
u = logv,
(2.2)
Under the transformation
equation (2.1) assumes the following form VU,, - u,u, 1965
u3= 0.
(2.3)
1966
0. P. BHUTAN1 et al.
Following Clarkson and Kruskal, we seek solution of equation (2.3) in the following form v(x, t) = Q(x, t, w(z(x, 0).
(2.4)
It may be remarked here that (2.4) is the most general form for the similarity solution. We now require that substitution of (2.4) in (2.3) yields the ordinary differential equation for w(z). This imposes conditions upon Q and its derivatives that enable us to solve for Q. It turns out that it is sufficient for (2.3) to seek similarity solutions in the special form u(x, 0 = a(x, t) + P(x, Mz(x,
t)),
(2.5)
where Q(X,t), 6(x, t) and z(x, t) are assumed to be suffi~ently differentiable functions and w(z) is two times differentiable. Substituting equation (2.5) into (2.3) and collecting coefficients of like derivatives and powers of w(z), we get P2ZxZPWB + (TPZ*ZrWR + (rrP,z* + Q&Z, + cupz, - PLy,ZI- (w,Pzx)W + (%
+ (/?& - &Pf - 3&)w2 f P&X- P,% - &Ix - 3cu2P)w+ pZzuww’ - p2zXzXw’~ - p2w3 - (ffalx - axat - a”) = 0. (2.6)
In order that equation (2.6) is transformed into an ordinary differential equation for w(z) it is necessary that the ratios for different derivatives and powers of w(z) must be functions of z only. This gives an overdete~ined system of equations for a(x), @(x,t), z(x, t), whose solutions yield the desired similarity solutions. We shall now proceed to determine the general similarity solutions of equation (2.3) using this direct method. Taking the coefficient of ww” (i.e. Pz,zJ as the normalizing coefficient, the coefficient of w3 yields the constraint B’z*zJXz) = P3, where I’(z) is a function to be determined. On making use of the freedom mentioned in remark (A.3.3b) (see Appendix), we get p = Z,ZP
(2.7)
The coefficient of ww’ yields the constraint P2ZXZII(Z)= P2&, where I(z) is to be determined. Hence, using equation (2.7), resealing I(z), and integrating the resulting equation w.r.t. x, we get &I(Z) = s(t)>
(2.8)
where l(t) is a function of integration (recall remark (A.3.2)). Integrating equation (2.8) w.r.t. r, we get r(z) = 0(x) + ff(t),
(2.9)
where e(x) is another function of integration. Using the freedom mentioned in remark (A.3.3c), equation (2.9) yields z = e(x) + a(t).
(2.10)
On combining equations (2.7) and (2.10) we get (2.11)
Exact solutions to the Liouville equation
1967
The coefficient of w” yields P2ZXZ,~(Z) = oazxz,, where I’(z) is to be determined. On making use of equation (2.11) and the freedom in remark (A.3.3a), we get ff(X,t) = 0.
(2.12)
The coefficient of w2 gives PZZXZ,TYZ) = PPUC- PXP,- 3ffp2. On making use of equations (2.11)-(2.12), we get r-(z) = 0.
(2.13)
Further, equation (2.6) simplifies to ww~~-wcw3=~.
(2.14)
Using the substitution (2.15)
w’ =p,
equation (2.14) gets transformed to PP’ +~(w)p2+g(w)=0,
(2.16)
where g(w) = -w2.
f(w)= -;>
The solution to equation (2.16) can be expressed as
P2e2” =C-
2
f
g(w)e2@dw,
(2.17)
where cbtw) = /f(w) dw,
(2.18)
and c is a constant of integration. Using the expressions for f(w) and g(w) and equation (2.18), equation (2.17) can be expressed as p2 = w”(c
+
2w).
(2.19)
For the solution of equation (2.19) two possibilities arise: Case (i): c = 0 Corresponding to this possibility w(z) can be expressed as (2.20) where to is a constant of integration. Combining equations (2-S), (2.10)-(2.11) and (2.20) we arrive at
(2.21)
0. P. BHUTAN1 er 01.
1%8
and hence, the solution to the Liouville equation (2.1) can be expressed as ,E!!z
r+
&dx dr r, = lOg{e(x) + cr(r) + &}* ’
(2.22)
where 0(x) and cr(t) are arbitrary functions of their respective arguments. It may be mentioned that equation (2.22) represents a known general solution to the Liouville equation that coincides with the one obtained by Ibragimov [3] and Tamizhmani and Lakshmanan [4]. Case (ii): c ZO
Corresponding
to this possibility as obtained in equation (2.22) w(z) satisfies W(Z) = - i
On combining equations (2.5) (2.10)-(2.11). the Liouville equation u(x,
t) = log
fi
sech* - 1
(z + zo)
(2.23)
‘.
I
(2.2) and (2.23) we get the following solution to
-f$zsech2
( - G (0(x) + o(t) + ZJ)),
(2.24)
In equation (2.24) e(x), a(t) are arbitary functions and c, z0 arbitrary constants. It may be mentioned here that the solution (2.24) to the Liouville equation (2.1) is completely new and does not seem to have been reported in the literature. Further, choosing c =4, a(t) = log(t + A2)IR, 0(x) = log(x + A,)-‘” and z0 = loge;“‘, where r, A,, A2, are arbitrary constants, we get 1 u(x, t) = log 1: 2 (x + AN +
(t + A2)‘R log(q(x
A21
+
A,))‘”
(2.25) ’
Equation (2.25) represents an exact solution of the Liouville equation reported by Bhutani and Vijayakumar [S] obtained via the isovector approach, for n = 1. Also, it can be seen easily that w(z) =isec*
($(z + Z”))
(2.26)
satisfies equation (2.19). Combining equations (2.5), (2.10) (2.11), (2.2) and (2.26), we obtain cdeda u(x, t) = log 2-d-zsec2
($
(e(x) + ~(0 + G))].
(2.27)
where z0 is a constant of integration. This form of exact solution of the Liouville equation is the same as the one reported in Ibragimov [3], when c = 4.
REFERENCES (I] P. A. CLARKSON and M. D. KRUSKAL, /. Math. Phys. 30,220l (1989). [2] F. CALOGERO and A. DEGASPERIS. Spectral Transforms and Solirons. North Holland, Amsterdam (1982). [3] N. H. IBRAGIMOV, Transformurion Groups Applied to A4arhemaricol Physics. Reidel. Boston (1984). [4] K. M. TAMIZHMANI and M. LAKSHMANAN, 1. Marh. Phys. 27,2257 (1986). [S] 0. P. BHUTAN1 and K. VIJAYAKUMAR. Inf. J. Engng Sci. 30, 1049 (1992). (61 0. P. BHUTAN1 and K. VIJAYAKUMAR, J. Ausf. Marh. Sot. Ser. E 32,457 (1991).
APPENDIX In order to put the direct approach into practice we have to follow the remarks recorded below (for details see [I]): Remark (A.3.1): We shall use the coefficient of highest derivatives of w(z) as the normalizing coefficient and require that other
Exact solutions to the Liouville equation
1%9
coefficients are of the form of the normalizing coefficient multiplied by r(z). where I is a function of z to be determined. Remark (A.3.2): Whenever we use on upper case Greek letter to denote a function (e.g. I(z)), then this is a function, to be determined, upon which we can perform any mathematical operation (e.g. differentiation, integration, taking logarithm, exponentiation, taking powers, resealing, etc.) and then also call the resulting function I(z), without loss of generality (e.g. the differential of I(z) will be I(z)). Remark (A.3.3) There are three freedoms in the determination of a, B, z which we can exploit, without loss of generality: (a) If a(x, I) is of the form a@, 1) = a&, t) + /3(x, I)I(z), where a&, t) is specified and I’(z) is any function, then we can assume that I = 0 (make the transformtion w(z) + w(z) - I(z)); (b) If /3(x, r) is of the form /3 = /I&, r)T(z), where & is specified and I(z) is any function, then we can assume that I= 1 (make the transformation w(z)+ w(z)/I(z)); (c) If z(x, t) is defined by an equation of the form T(z) = z&r, I), where b is specified and I’(z) is any invertible function, then we can assume that I = z (make the transformation z + I-‘(z) where I-’ is the inverse of I). (Received 27 July 1993; accepted 15 February 1994)
lnt. J. Engng Sci. Vol. 32, No. 12, pp. 1965-1%9, 1994 Copyright @ 1994 EIscvier Science Ltd Printed in Great Britain. Al1 rights reserved
~7~(~~E~l7-D
o&?o-7225/94 $7.00+ 0.00
ON THE GENERALIZED FORMS OF EXACT SOLUTIONS TO THE LIOUVILLE EQUATION VIA DIRECT APPROACH 0. P. BHUTAN1 Department
and M. H. M. MOUSSA
of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi-l 10 016, India K. Department
VI~AYAKUMAR
of Mathematics, Punjab University, Chandigarh-1~014, (Communicated
India
by D. G. B. EDELEN)
Abstract-Using the direct approach due to Clarkson and Kruskal [I] two generalized forms of exact solutions to the Liouville equation are presented that eventually enable recovering all available exact solutions obtained via group theoretic, Bscklund ~ansfo~ation and PainIevB analysis.
INTRODUCTION
The importance of the Liouville equation in the context of field theoretic model is well known [2] and the equation has been the subject of investigation for solution in [3-51. In [3], Ibragimov has presented an exact solution to the Liouville equation in terms of two arbitrary functions and their derivatives by reducing it to linear form via the BIcklund transformation. Tamizhmani and Lakshmanan [4] have achieved the said linearization through PainleG analysis and the form of the final solution turns out to be the same as reported in 121. In [5], Bhutani and Vijayakumar carried out a detailed analysis of Liouville equation via isovector approach. More specifically, the components of isovector fields tabulated in their complete generality that involve six arbitrary constants al, az-+,, yield, through the solutions of orbital equations, the invariant groups of transformation which reduce the given PDE to NLODE. In order to solve the resulting NLODE in the general case use is made of the variational s~rnet~ [6]. However, in the particular case wherein a, # 0, a4 ZO and a2, a3, a5, a6 are all zero, the resulting NLODE is solved via the standard technique and two new solutions are reported. Herein, using the Clarkson and Kruskal formalism [l] (that does away with the sophisticated tools of functional analysis, Painleve analysis, Bgcklund transformation, variational symmetry, etc.), we have obtained two generalized forms of similarity solutions obtained via the group theoretic technique; the other generalized solution enabled us to obtain Ibragimov’s solution obtained via the BBcklund transformation.
2. LIOUVILLE
EQUATION
AND
The Liouville equation under consideration
METHOD
OF
SOLUTION
can be expressed as
u,t - e” = 0.
(2.1)
u = logv,
(2.2)
Under the transformation
equation (2.1) assumes the following form VU,, - u,u, 1965
u3= 0.
(2.3)
1966
0. P. BHUTAN1 et al.
Following Clarkson and Kruskal, we seek solution of equation (2.3) in the following form v(x, t) = Q(x, t, w(z(x, 0).
(2.4)
It may be remarked here that (2.4) is the most general form for the similarity solution. We now require that substitution of (2.4) in (2.3) yields the ordinary differential equation for w(z). This imposes conditions upon Q and its derivatives that enable us to solve for Q. It turns out that it is sufficient for (2.3) to seek similarity solutions in the special form u(x, 0 = a(x, t) + P(x, Mz(x,
t)),
(2.5)
where Q(X,t), 6(x, t) and z(x, t) are assumed to be suffi~ently differentiable functions and w(z) is two times differentiable. Substituting equation (2.5) into (2.3) and collecting coefficients of like derivatives and powers of w(z), we get P2ZxZPWB + (TPZ*ZrWR + (rrP,z* + Q&Z, + cupz, - PLy,ZI- (w,Pzx)W + (%
+ (/?& - &Pf - 3&)w2 f P&X- P,% - &Ix - 3cu2P)w+ pZzuww’ - p2zXzXw’~ - p2w3 - (ffalx - axat - a”) = 0. (2.6)
In order that equation (2.6) is transformed into an ordinary differential equation for w(z) it is necessary that the ratios for different derivatives and powers of w(z) must be functions of z only. This gives an overdete~ined system of equations for a(x), @(x,t), z(x, t), whose solutions yield the desired similarity solutions. We shall now proceed to determine the general similarity solutions of equation (2.3) using this direct method. Taking the coefficient of ww” (i.e. Pz,zJ as the normalizing coefficient, the coefficient of w3 yields the constraint B’z*zJXz) = P3, where I’(z) is a function to be determined. On making use of the freedom mentioned in remark (A.3.3b) (see Appendix), we get p = Z,ZP
(2.7)
The coefficient of ww’ yields the constraint P2ZXZII(Z)= P2&, where I(z) is to be determined. Hence, using equation (2.7), resealing I(z), and integrating the resulting equation w.r.t. x, we get &I(Z) = s(t)>
(2.8)
where l(t) is a function of integration (recall remark (A.3.2)). Integrating equation (2.8) w.r.t. r, we get r(z) = 0(x) + ff(t),
(2.9)
where e(x) is another function of integration. Using the freedom mentioned in remark (A.3.3c), equation (2.9) yields z = e(x) + a(t).
(2.10)
On combining equations (2.7) and (2.10) we get (2.11)
Exact solutions to the Liouville equation
1967
The coefficient of w” yields P2ZXZ,~(Z) = oazxz,, where I’(z) is to be determined. On making use of equation (2.11) and the freedom in remark (A.3.3a), we get ff(X,t) = 0.
(2.12)
The coefficient of w2 gives PZZXZ,TYZ) = PPUC- PXP,- 3ffp2. On making use of equations (2.11)-(2.12), we get r-(z) = 0.
(2.13)
Further, equation (2.6) simplifies to ww~~-wcw3=~.
(2.14)
Using the substitution (2.15)
w’ =p,
equation (2.14) gets transformed to PP’ +~(w)p2+g(w)=0,
(2.16)
where g(w) = -w2.
f(w)= -;>
The solution to equation (2.16) can be expressed as
P2e2” =C-
2
f
g(w)e2@dw,
(2.17)
where cbtw) = /f(w) dw,
(2.18)
and c is a constant of integration. Using the expressions for f(w) and g(w) and equation (2.18), equation (2.17) can be expressed as p2 = w”(c
+
2w).
(2.19)
For the solution of equation (2.19) two possibilities arise: Case (i): c = 0 Corresponding to this possibility w(z) can be expressed as (2.20) where to is a constant of integration. Combining equations (2-S), (2.10)-(2.11) and (2.20) we arrive at
(2.21)
0. P. BHUTAN1 er 01.
1%8
and hence, the solution to the Liouville equation (2.1) can be expressed as ,E!!z
r+
&dx dr r, = lOg{e(x) + cr(r) + &}* ’
(2.22)
where 0(x) and cr(t) are arbitrary functions of their respective arguments. It may be mentioned that equation (2.22) represents a known general solution to the Liouville equation that coincides with the one obtained by Ibragimov [3] and Tamizhmani and Lakshmanan [4]. Case (ii): c ZO
Corresponding
to this possibility as obtained in equation (2.22) w(z) satisfies W(Z) = - i
On combining equations (2.5) (2.10)-(2.11). the Liouville equation u(x,
t) = log
fi
sech* - 1
(z + zo)
(2.23)
‘.
I
(2.2) and (2.23) we get the following solution to
-f$zsech2
( - G (0(x) + o(t) + ZJ)),
(2.24)
In equation (2.24) e(x), a(t) are arbitary functions and c, z0 arbitrary constants. It may be mentioned here that the solution (2.24) to the Liouville equation (2.1) is completely new and does not seem to have been reported in the literature. Further, choosing c =4, a(t) = log(t + A2)IR, 0(x) = log(x + A,)-‘” and z0 = loge;“‘, where r, A,, A2, are arbitrary constants, we get 1 u(x, t) = log 1: 2 (x + AN +
(t + A2)‘R log(q(x
A21
+
A,))‘”
(2.25) ’
Equation (2.25) represents an exact solution of the Liouville equation reported by Bhutani and Vijayakumar [S] obtained via the isovector approach, for n = 1. Also, it can be seen easily that w(z) =isec*
($(z + Z”))
(2.26)
satisfies equation (2.19). Combining equations (2.5), (2.10) (2.11), (2.2) and (2.26), we obtain cdeda u(x, t) = log 2-d-zsec2
($
(e(x) + ~(0 + G))].
(2.27)
where z0 is a constant of integration. This form of exact solution of the Liouville equation is the same as the one reported in Ibragimov [3], when c = 4.
REFERENCES (I] P. A. CLARKSON and M. D. KRUSKAL, /. Math. Phys. 30,220l (1989). [2] F. CALOGERO and A. DEGASPERIS. Spectral Transforms and Solirons. North Holland, Amsterdam (1982). [3] N. H. IBRAGIMOV, Transformurion Groups Applied to A4arhemaricol Physics. Reidel. Boston (1984). [4] K. M. TAMIZHMANI and M. LAKSHMANAN, 1. Marh. Phys. 27,2257 (1986). [S] 0. P. BHUTAN1 and K. VIJAYAKUMAR. Inf. J. Engng Sci. 30, 1049 (1992). (61 0. P. BHUTAN1 and K. VIJAYAKUMAR, J. Ausf. Marh. Sot. Ser. E 32,457 (1991).
APPENDIX In order to put the direct approach into practice we have to follow the remarks recorded below (for details see [I]): Remark (A.3.1): We shall use the coefficient of highest derivatives of w(z) as the normalizing coefficient and require that other
Exact solutions to the Liouville equation
1%9
coefficients are of the form of the normalizing coefficient multiplied by r(z). where I is a function of z to be determined. Remark (A.3.2): Whenever we use on upper case Greek letter to denote a function (e.g. I(z)), then this is a function, to be determined, upon which we can perform any mathematical operation (e.g. differentiation, integration, taking logarithm, exponentiation, taking powers, resealing, etc.) and then also call the resulting function I(z), without loss of generality (e.g. the differential of I(z) will be I(z)). Remark (A.3.3) There are three freedoms in the determination of a, B, z which we can exploit, without loss of generality: (a) If a(x, I) is of the form a@, 1) = a&, t) + /3(x, I)I(z), where a&, t) is specified and I’(z) is any function, then we can assume that I = 0 (make the transformtion w(z) + w(z) - I(z)); (b) If /3(x, r) is of the form /3 = /I&, r)T(z), where & is specified and I(z) is any function, then we can assume that I= 1 (make the transformation w(z)+ w(z)/I(z)); (c) If z(x, t) is defined by an equation of the form T(z) = z&r, I), where b is specified and I’(z) is any invertible function, then we can assume that I = z (make the transformation z + I-‘(z) where I-’ is the inverse of I). (Received 27 July 1993; accepted 15 February 1994)