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Diffusion equations and Lie and Lie-Bäcklund transformation groups
Physica 114A(1982)95-99 North-Holland Publishing Co.
DIFFUSION LIE-BACKLUND
EQUATIONS AND LIE AND TRANSFORMATION GROUPS W. H. STEEB
Fachbereich Theoretische Physik der Uniuersitiit Paderbom, Pohlweg 55, D-4790 Paderborn, Germany and W. STRAMPP Fachbereich Mathematik der Gesamthochschule Kassel, Wilhelmshdherallee 73, D-3500 Kassel, Germany
Linear and nonlinear diffusion equations are studied with the help of Lie and Lie-BIcklund transformation groups.
1. Introduction
In recent time much attention has been focussed on non-linear diffusion equations and symmetry groups’“). The symmetry groups have been used for deriving similarity solutions and conserved currents of the non-linear diffusion equations, where Lie and Lie-Bticklund transformation groups have been studied. Since the diffusion equations cannot be derived from a Lagrangian we are not able to apply Noether’s theorem for deriving conserved currents and conservation laws of diffusion equations. Hence, approaches are desirable for obtaining conserved currents of field equations when there is no Lagrangian or the Lagrangian is unknown. We derive a theorem where, if at least one conservation law is known, other conservation laws can be found with the help of the symmetry groups of the field equation under consideration.
2. Theory
First of all let us introduce the notation. We study the system of partial differential equations within the jet bundle formalism. A triple (N, 7~,M) is called a fibred manifold, if M and N are differentiable manifolds and 0378-4371/82/O
DO2.75 @ 1982 North-Holland
96
W.H. STEEB AND W. STRAMPP
7~:N + A4 is a surjective
submersion.
For example, let
N = R”+” = {(x,, . . . ) x,, Ill,. . . ) Id,)}, M = R” = {(Xl, 1. . ) x,)} and GT(X,,. . . ,x,, ul,. . . , u,) = (xl,. . . ,x,). The so-called base manifold A4 represents the independent variables. In most cases in physics A4 = W4or an open subset of R4. The manifold N represents both the dependent variables (the fields) and the independent variables. In most cases in physics N will be an open subset of the Euclidean space R4 x R”. Now let dimM=m and dimN=n+m and let (xi,ui) (l
* SS
1,
S
Fv(xip Uj, dUj/aXi,.
where v=l,..., manifold
me
Uji,.,.i,
. . , a’Uj/aX,aX,. . . axi,) = 0,
q. Within the jet bundle formalism
Fy(xi) uj, uji, * . * 9 Uji,
i,) = 0
(1) we consider
the sub(2)
and the contact forms 0, = dnj - Ujidxi, . . . , Oji,. i, = duji, _. i,_, - Uji, _. ip dxk. Throughout we use a summation convention. Consider now the vector field Di defined on J(N)
The summation on the right hand side is restricted to 1~ ii G i26 . . . s i, < m. Di is sometimes called the operator of total differentiation. Together with eq. (2) we consider all differential consequences DiFp = 0,. . . , Di,Db . . . Q,F, = 0. Let R = dx, A dx2 A . . . A dx, be the volume form on M. x, will play the role of the time coordinate. Definition. The (m - I)-form w = fa(xi, Ui,Uji,. . .)(f~/ax, JC4) defined on J(N) is called a conservation law of eq. (1) if (js)*(dw) = 0 whenever s: A4 + N is a solution to eq. (1). js is the jet extension of s up to infinite order. a/ax, JO denotes the contraction.
LIE AND LIE-BACKLUND
TRANSFORMATION
GROUPS
97
Another possibility for defining conserved currents is: The (m - I)-form o given above is called a conservation law if dw E J, where J is the differential ideal generated by F,, DiF,, . . . , and the contact forms. We mention that the first definition is the more general one. For deriving conserved currents we consider the vector field Z = aia/axi + b$/a~j, where ai and bi depends upon (Xi, Uj,Uji,. . .). The prolongation up to infinite order of the vector field Z is given by
a
Z=Z+Cjic+
*"+Cji,.-.i,&+*'*~ Jt,
i,
where Cji = .
Di(bj)
-
UjkDi(&)
.
Cji,...i,
=
Di,(Cji,...i,_,)-
Ujil...i,-,rDi,(4)*
(5)
Definition. The system of partial differential equations (1) which is described within the jet bundle formalism by eq. (2) and the contact forms is called invariant under the vector field Z if LzF, a 0, where P stands for the restriction to solutions of eq. (1). Again we can give a definition which is not so general, but frequently used. Here the system of partial differential equations is called invariant if LzFy E J, Lzej E .I,. . . ) where J is the differential ideal generated by F, DiF”, and the contact forms. If the vector field Z leaves invariant the system of partial differential equations, then the vector field
Q=
Uj&+Di(Uj)$+*
I
II
does SO, where Vi = bj - air+. In the following we consider the vector field v instead of Z for investigating the symmetries. Assume that the vector field v is integrable to the corresponding group action u +exp(ev)u. Then, owing to invariance, a solution s: M+N is carried into a new solution exp(ev)s. Theorem. Assume that the system of partial differential equations (1) is invariant under the vector field v. Let w be a conservation law of eq. (1). Then LVW is also a conservation law of eq. (1). Proof.
Let w be a conservation
law. Since solutions
s : A4 + N are mapped
W.H. STEEB AND W. STRAMPP
98
into solutions by the transformation group which is generated by V we obtain (j exp(eV)s)* do = 0. Owing to the identity
$ (j exp(eV)s)*
0 = (j exp(ee)s)*lpw
we find 0 = g (j exp(ee)s)*
do = (j exp(EV)s)* d&o).
(8)
In the last step we have used the fact that the Lie derivative and the exterior derivative commute. Setting E = 0, it follows that (js)* d&o) = 0 which completes the proof. We mention that recently Steinberg and Wolf3) have derived a less comprehensive version of the theorem given above.
3. Examples Example 1. Consider the diffusion equation au/at = #u/ax*. We put x1 = x, x2 = t and u1 = u. Consequently, F = u12- ulll = 0. The heat equation is invariant under the infinite extension of the vertical vector field V = (x,~~,+2x~u~~)a/au,. The vector field V is associated with the (x, t)-scale change. With the help of eq. (5) we calculate the prolongation of the vector field V up to infinite order. With this vector field we find LcF = 2F + x,(D,F) + x2(D2F). A conservation law of the heat equation can be given at once, namely o = u1 dx, + uIl dx2, since (js)* do = 0 is nothing more than the diffusion equation. Straightforward calculation shows that L,o
= (xru,, + 2x2~12)dx, + (u,l+ xrurrr + 2x214112) dx2.
(9)
According to the theorem given above, the right-hand side is a conservation law. Repeated application of v leads to a hierarchy of conservation laws. Example 2. Consider the non-linear equation au/at = (au/ax)* + a2u/axz. We use the same notation as above. Consequently, F = u12- uil - ulll = 0. A conservation law is given by w = x1 exp(ur) dxr + (x,urr exp(ur) - exp(ur)) dx2.
(IO)
The non-linear equation is invariant under the infinite extension of the vertical vector field V = (xru,r + 2x2u&/au1. Again Loo is a conservation law. Since the linear heat equation can be transformed into the non-linear heat
LIE AND LIE-BACKLUND
TRANSFORMATION
GROUPS
99
equation via the transformation ul +exp(uJ it is obvious that conservation laws of the linear heat equation can be transformed into conservation laws of the non-linear heat equation, where the field transformation must be extended up to infinite order.
References 1) G. Bluman and S. Kumei, J. Math. Phys. 21 (1980) 1019. 2) N.Kh. Ibragimov and A.B. Shabat, Funktsional’nyi Analiz i ego Prilozheniya 14 (1980) 25; Functional Analysis and its Applications 14 (1980). 3) S. Steinberg and K.B. Wolf, J. Math. Anal. Appl. 80 (1981) 36. 4) A. Munier, J.R. Burgan, J. Gutitrrez, E. Fijalkows and M.R. Feix, SIAM J. Appl. Math. 40 (1981) 191. 5) W.-H. Steeb and W. Strampp, Physica 3D (1981) 637.
DIFFUSION LIE-BACKLUND
EQUATIONS AND LIE AND TRANSFORMATION GROUPS W. H. STEEB
Fachbereich Theoretische Physik der Uniuersitiit Paderbom, Pohlweg 55, D-4790 Paderborn, Germany and W. STRAMPP Fachbereich Mathematik der Gesamthochschule Kassel, Wilhelmshdherallee 73, D-3500 Kassel, Germany
Linear and nonlinear diffusion equations are studied with the help of Lie and Lie-BIcklund transformation groups.
1. Introduction
In recent time much attention has been focussed on non-linear diffusion equations and symmetry groups’“). The symmetry groups have been used for deriving similarity solutions and conserved currents of the non-linear diffusion equations, where Lie and Lie-Bticklund transformation groups have been studied. Since the diffusion equations cannot be derived from a Lagrangian we are not able to apply Noether’s theorem for deriving conserved currents and conservation laws of diffusion equations. Hence, approaches are desirable for obtaining conserved currents of field equations when there is no Lagrangian or the Lagrangian is unknown. We derive a theorem where, if at least one conservation law is known, other conservation laws can be found with the help of the symmetry groups of the field equation under consideration.
2. Theory
First of all let us introduce the notation. We study the system of partial differential equations within the jet bundle formalism. A triple (N, 7~,M) is called a fibred manifold, if M and N are differentiable manifolds and 0378-4371/82/O
DO2.75 @ 1982 North-Holland
96
W.H. STEEB AND W. STRAMPP
7~:N + A4 is a surjective
submersion.
For example, let
N = R”+” = {(x,, . . . ) x,, Ill,. . . ) Id,)}, M = R” = {(Xl, 1. . ) x,)} and GT(X,,. . . ,x,, ul,. . . , u,) = (xl,. . . ,x,). The so-called base manifold A4 represents the independent variables. In most cases in physics A4 = W4or an open subset of R4. The manifold N represents both the dependent variables (the fields) and the independent variables. In most cases in physics N will be an open subset of the Euclidean space R4 x R”. Now let dimM=m and dimN=n+m and let (xi,ui) (l
* SS
1,
S
Fv(xip Uj, dUj/aXi,.
where v=l,..., manifold
me
Uji,.,.i,
. . , a’Uj/aX,aX,. . . axi,) = 0,
q. Within the jet bundle formalism
Fy(xi) uj, uji, * . * 9 Uji,
i,) = 0
(1) we consider
the sub(2)
and the contact forms 0, = dnj - Ujidxi, . . . , Oji,. i, = duji, _. i,_, - Uji, _. ip dxk. Throughout we use a summation convention. Consider now the vector field Di defined on J(N)
The summation on the right hand side is restricted to 1~ ii G i26 . . . s i, < m. Di is sometimes called the operator of total differentiation. Together with eq. (2) we consider all differential consequences DiFp = 0,. . . , Di,Db . . . Q,F, = 0. Let R = dx, A dx2 A . . . A dx, be the volume form on M. x, will play the role of the time coordinate. Definition. The (m - I)-form w = fa(xi, Ui,Uji,. . .)(f~/ax, JC4) defined on J(N) is called a conservation law of eq. (1) if (js)*(dw) = 0 whenever s: A4 + N is a solution to eq. (1). js is the jet extension of s up to infinite order. a/ax, JO denotes the contraction.
LIE AND LIE-BACKLUND
TRANSFORMATION
GROUPS
97
Another possibility for defining conserved currents is: The (m - I)-form o given above is called a conservation law if dw E J, where J is the differential ideal generated by F,, DiF,, . . . , and the contact forms. We mention that the first definition is the more general one. For deriving conserved currents we consider the vector field Z = aia/axi + b$/a~j, where ai and bi depends upon (Xi, Uj,Uji,. . .). The prolongation up to infinite order of the vector field Z is given by
a
Z=Z+Cjic+
*"+Cji,.-.i,&+*'*~ Jt,
i,
where Cji = .
Di(bj)
-
UjkDi(&)
.
Cji,...i,
=
Di,(Cji,...i,_,)-
Ujil...i,-,rDi,(4)*
(5)
Definition. The system of partial differential equations (1) which is described within the jet bundle formalism by eq. (2) and the contact forms is called invariant under the vector field Z if LzF, a 0, where P stands for the restriction to solutions of eq. (1). Again we can give a definition which is not so general, but frequently used. Here the system of partial differential equations is called invariant if LzFy E J, Lzej E .I,. . . ) where J is the differential ideal generated by F, DiF”, and the contact forms. If the vector field Z leaves invariant the system of partial differential equations, then the vector field
Q=
Uj&+Di(Uj)$+*
I
II
does SO, where Vi = bj - air+. In the following we consider the vector field v instead of Z for investigating the symmetries. Assume that the vector field v is integrable to the corresponding group action u +exp(ev)u. Then, owing to invariance, a solution s: M+N is carried into a new solution exp(ev)s. Theorem. Assume that the system of partial differential equations (1) is invariant under the vector field v. Let w be a conservation law of eq. (1). Then LVW is also a conservation law of eq. (1). Proof.
Let w be a conservation
law. Since solutions
s : A4 + N are mapped
W.H. STEEB AND W. STRAMPP
98
into solutions by the transformation group which is generated by V we obtain (j exp(eV)s)* do = 0. Owing to the identity
$ (j exp(eV)s)*
0 = (j exp(ee)s)*lpw
we find 0 = g (j exp(ee)s)*
do = (j exp(EV)s)* d&o).
(8)
In the last step we have used the fact that the Lie derivative and the exterior derivative commute. Setting E = 0, it follows that (js)* d&o) = 0 which completes the proof. We mention that recently Steinberg and Wolf3) have derived a less comprehensive version of the theorem given above.
3. Examples Example 1. Consider the diffusion equation au/at = #u/ax*. We put x1 = x, x2 = t and u1 = u. Consequently, F = u12- ulll = 0. The heat equation is invariant under the infinite extension of the vertical vector field V = (x,~~,+2x~u~~)a/au,. The vector field V is associated with the (x, t)-scale change. With the help of eq. (5) we calculate the prolongation of the vector field V up to infinite order. With this vector field we find LcF = 2F + x,(D,F) + x2(D2F). A conservation law of the heat equation can be given at once, namely o = u1 dx, + uIl dx2, since (js)* do = 0 is nothing more than the diffusion equation. Straightforward calculation shows that L,o
= (xru,, + 2x2~12)dx, + (u,l+ xrurrr + 2x214112) dx2.
(9)
According to the theorem given above, the right-hand side is a conservation law. Repeated application of v leads to a hierarchy of conservation laws. Example 2. Consider the non-linear equation au/at = (au/ax)* + a2u/axz. We use the same notation as above. Consequently, F = u12- uil - ulll = 0. A conservation law is given by w = x1 exp(ur) dxr + (x,urr exp(ur) - exp(ur)) dx2.
(IO)
The non-linear equation is invariant under the infinite extension of the vertical vector field V = (xru,r + 2x2u&/au1. Again Loo is a conservation law. Since the linear heat equation can be transformed into the non-linear heat
LIE AND LIE-BACKLUND
TRANSFORMATION
GROUPS
99
equation via the transformation ul +exp(uJ it is obvious that conservation laws of the linear heat equation can be transformed into conservation laws of the non-linear heat equation, where the field transformation must be extended up to infinite order.
References 1) G. Bluman and S. Kumei, J. Math. Phys. 21 (1980) 1019. 2) N.Kh. Ibragimov and A.B. Shabat, Funktsional’nyi Analiz i ego Prilozheniya 14 (1980) 25; Functional Analysis and its Applications 14 (1980). 3) S. Steinberg and K.B. Wolf, J. Math. Anal. Appl. 80 (1981) 36. 4) A. Munier, J.R. Burgan, J. Gutitrrez, E. Fijalkows and M.R. Feix, SIAM J. Appl. Math. 40 (1981) 191. 5) W.-H. Steeb and W. Strampp, Physica 3D (1981) 637.