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Isovectors and similarity solutions for nonlinear reaction-diffusion equations
ISOVECTORS AND SIMILARITY SOLUTIONS FOR NONLINEAR REACTION-SUFFUSION EQUATIONS E. S. $UHUBI?De~rtment
of Mechanical Engineering.
and K. L. CHOWDHURY The University of Calgary, Calgary. Alberta, Canada
Abstract-Nonlinear reaction-diffusion equations are expressed by means of a closed ideal of exterior differential forms over an S-dimensional manifold. The components of the isovector field associated with the system are constructed by using its transport property under Lie derivatives. The similarity solutions which are invariant under the corresponding symmetry group are obtained for fairly general cases and the system is reduced to coupled nonlinear ordinary differential equations.
I. INTRODUCTION Nonlinear reaction-diffusion equations in their diverse forms serve as mathematical models to a number of interesting physical phenomena occurring in various research fields [l-6], e.g. developmental biology, neurodynamics, ecology, plasma physics, chemical reactions etc. Investigations of some of the intricate problems have resulted in pattern formations, stable limit cycles, wave trains, shock fronts, bifurcations and chaotic evolutions. An excellent compilation of the mathematical methods of bifurcation theory and its applications to various scientific disciplines with computer simulations of dynamical systems is given in Refs [7] and f8& Recent search for the invariance groups and special families of exact solutions to nonlinear PDEs, using differential geometric concepts of Lie’s and Cartan’s continuous groups, has been spurred by the works of Vessiot [9, lo], Harrison and Estabrook [ll], Bluman and Cole [ 121 and Ovsjannikov [13]. Walquist and Estabrook [14], Fackdrell [15] and Estabrook [16] used Cartan’s formulations to define and construct prolongation structures of systems of nonlinear PDEs. Edelen’s recent two books [I?‘, IS] have provided extensive theoretical framework of the methods of exterior calculus and further enhanced their applications to various problems of continuum mechanics, thermodynamics and electrodynamics. In this paper, we express quite a genera1 class of nonlinear reaction-diffusion equations by means of a closed idea1 of exterior differential forms over an &dimensional manifold. Using the transport property of the isovectors, i.e. the invariance of the ideal under the Lie dragging by the congruence generated by the isovector field which implies in turn that the Lie derivative of any form in the ideal remains in the ideal, the components of the isovector field are constructed. These components are essentially infinitesimal generators of Lie group which is the symmetry group of the prolongation structure of the reaction-diffusion equations. A similarity soiution of the differential equations is an invariant solution with respect to a particular symmetry group. Therefore they can be obtained by determining the invariants of the group. In this paper the most general symmetry group of one~dimensional reaction-di~usion equations is obtained for arbitrary source functions. Since the latter functions are not specified beforehand the group involves arbitrary functions. These functions are determined for fairly general forms of the source functions and the corresponding similarity solutions and nonlinear ordinary differential equations which they satisfy are provided. 2. BASIC
EQUATIONS
We assume that a one-dimensional reaction-diffusion system of nonlinear differential equations: a, = &a,
process is governed
by the following
+ A+, b, x, t)
b, = D&x,, c B(a, 6, x, f) t Permanent address: Faculty of Science, Department Maslak 80626, Istanbul, Turkey. ES 26:10-A
of Engineering
1027
(2.1) Sciences,
Istanbul Technical
University.
1028
E. S. .tjUHUBI
and K. 1.. C‘flOWDHUR\
where x and t are space and time coordinates,
u and b are the reaction-diffusion variables. functions describing the kinetics of the
A(a, b, x, t) and B(a, b, x, t) are arbitrary nonlinear process and D1 # 0 and D2 # 0 are diffusion constants.
We introduce an S-dimensional manifold u, v, w and _zare given by the relations u=a xv
(x, I, u, b, u, 11.~1,2) where
M with coordinates
v=b x*
w = a,,
(2.2)
= =b,
Let us now define the following exterior differential forms: o1 = D1 du A dt + (A - w) d_x A dt,
w2 = D2 dv A dt + (B - z) dw A dt
(Ye= du - u dx - w dt,
LYE = db - v dx - z dt
da, = -du A dz - dw
A
dt,
da, = -d/f A dx - dz A dt
(2.3)
where A represents the exterior product. If we consider a two-dimensional submanifold H of M on which a, b, u, v, w, z are functions of x and t only, the sectioning of the forms (2.3) by H are CI& = (Dlu, + A - w) dx
A
o&, = (D,v, + B - z) dx
dt,
= (u, - w,) du
A
dt,
dt
CY&= (6, - v) dx + (b, - z) dt
(Y&, = (a, - u) dr + (a, - w) dt, da&
A
da&
= (u, - z,) dx
A
dt
(2.4)
Therefore the solution manifold of eqns (2.1) and (2.2) annuls the exterior forms (2.3). On the other hand one can easily see that dw, = (Aa dx
A
dt) A (Y, + (Ah dx A dt) A a2 - dx A da,
do, = (B, dx
A
dt)
A crl + (&
dx A
dt)
A a2 -
dx A
da,
(2.5)
Hence the forms (ol, 02, aI, a2, da,, da,) generate a closed ideal of the ring of forms and Frobenius theorem can be applied. The isovector field V defined as a vector field with components (VI, V’, V“, Vh, V”, VI’, V”, V”) generates a congruence which leaves the foregoing closed ideal invariant under the corresponding Lie dragging. This implies that the Lie derivatives of the forms (2.3) with respect to the vector field V should remain in the closed ideal generated by them. Therefore we should write
fvoi = Ali da, + A2; da, + B,;wl + B2;w2 + M; A aI+ N, A ~7
(i = 1, 2)
(2.6) , NL where 4, b, PI, ~2, AH, . . . , B22 are arbitrary scalar functions of (x, . . . , z) and M,, are arbitrary l-forms. If we take the exterior derivative of (2.6), and use its commutation property with the Lie derivative, we immediately see that
dfvai = .Ev da, = il; da, + p; da2 + dA, A
al +
dpi
A LYE
Therefore fv dq are already in the ideal. The Lie derivatives of the forms can easily be calculated if we recall the following relation which is valid for all forms f,w=Vidw+d(VJo) where J denotes the contraction
of w with the vector field V. Hence we obtain from (2.3),
V-Ida,=-V”dx+V”du-V”dt+V’dw, VJda,=-V”dx+V”dv-V”dt+V’dz
(2.7)
Nonlinear
reaction-diffusion
If we introduce (2.7) into (2.6), and compare sides we find that &=F,,
PI = Fbt
& = G,,
equations
1029
the coefficients of the same forms from both E;, = F, = 0,
PZ = Gb,
G, = G, = 0
(2.8)
and V” = -F;1 = -G,,,
V’ = -F,
= -G,
V” = F - tit;;, - wF,,
V’=G-UC,,
VU=F,fuF,+vF,,
V”= c;, + UC, + UC,
V”=fi+wF,+zF,,
V” = G, + WC, + zG,
-zG,
(2.9)
Similarly it follows from (2.3), that V_lo,=D,V”dt-D,V’du+(A-w)(V”dt-VW) V _Ido, = (V=A, + VhAh - V”‘) dx A dt - V”A, da A dt + V’A, da A &u -VxA~dbAdt+V*A~dbA~+VxdwAdt-V*dwA~ VJw,=L),V”dt-DzV”dv+(B-y)(V”dt-V’du) V_ZdW2 = (V”B, + VhB, - V’) dx: A dt - VxE, da A dt + V’B, da A dx
-Vx~~dbAdt+VfB~dbA~+VxdzAd~-Vid~A~
(2.10)
We not denote the l-forms M1, N,, M2, N2, by Mi = Eli dt + F,i du + Gli da + H,i db + r,i du + Jli dv + i<,i dw + Lli dz
Ni = Ezi dt + F*; dt- + G*i da + H,i db + I,, du ~ Jzi dV + Kzi dw + L,i dZ Introducing (2.10) and (2.11) into (2.6), and comparing both sides we obtain, for i = 1 DIV,” + (A - w)(Vf + V;) + PA,
(2.11)
coefficients of the same forms from
+ VbA, + VIA, + V’A, - V” = B,I(A - w) + B,,(B
- z) + uEll + vE2, - wF,, - zFzI
D,VU,+(A-w)V”,=-I!?,,,
D,V:+(A-w)V;:=-E,,,
D,(V:+V;)+(A-w)VY,=D,B,,-wZfl-zZz, D, Vu, + (A - w)V:, = -A,1 DIV;
- wKll-
zKz1,
+ (A - w)V: = -AZ1 - wLl, - zLzl,
(A - w)V:, = &,
(A - w)V; = &,
D1 V:: + (A - w)V: = D, Bzl - wJ,, - .zJ21 D,V;
- (A - w)V; = -AtI - uZit - vl,,
(A - w)V:, = uKl, + vK2,, (A - w)V:, = B1, + ul,, + vJzl
{A - w)V: = uLl, + uL21, Gzt =ff,jv
D,Vb=4,,
D,V:,=k,,
D,V:,=DIV:,=D,V;=O (2.12)
J,2 = KII = L,, =Jzl = KZ1 = Lt, =0
where we have defined 4, = E,, + WC,, - Gi, &, = Z& + uH1 1 + vH2,
I?,, = ET?, + wH,, + zH2,,
F,, = F,, + UC,, + vG2,
(2.13)
If we replace Gzl in (2.13) by Hi, in view of the last line of (2.12) we see at once from (2.12)1 that u&l + n&r - wr;;, - zF,, = u& + vi?,, - w& - t& Recalling expressions
(2.8),,, and (2.9) we can easily deduce that the relation (2.12) are to be
satisfied if and only if the isovector field obeys D,(Vu,, - v; - ff V:, - UVi) = 0 D,(V,u+uV:+
UP’;)+ (A - w)(V:
- V::+ uV;:+ uV;)-;(B
- z)V:f 7
+V”A,, + V’A,, + VA,
+ V’A, - V”’ = 0
(2.14)
From the second equation of (2.6)2 we obtain similarly Dz(VY - v: - UV:, - vVb) = 0 &(V;
+ uV:: + uV;> + (B - z)(V;
- V:: + uV: + vV;)
- 2
(A - w)V;; 1
+ VU& + vv&
3. DETERMINATION
OF THE
+ V”B, + V’B, - vz = 0
ISOVECTOR
(2.15)
FIELD
Equations (2.8) imply that F = F(x, t, a, 6, U, W), Introducing
G = G(x, t, a, b, v, t)
(3.1)
(2.9) into (2.14)i and taking into account (2.9), we obtain F,, + uF,, i- uFwh = 0,
F;, = G,
It is easy to see that (2.1~5)~does not provide an independent F,, = G,,v = 0,
(3.2)
F;o= G
equation.
Since
F,,, = c;,, = G,.,, = 0
El;;,,= G,,,, = 0,
(3.3)
we easily deduce that the function F and G have the following representations: F = 1(I(x, t, a, b)w + tp(x, I, a, b)u + a+, G = v(x,
c, a, b)
t, a, b)z + +(.G r, a, b)v + P(x, t, ~1,b)
(3.4)
Replacing F, in (3.2), by t+?,we obtain t/J7+ U7Jla+ W&11, =0 which implies that qa =
$)h
=
v.r
=
(1
or (3.5)
* = q(t)
Let us introduce the relations (3.4) and (3.5) into (2.9) and resulting expressions into (2.14), and (2.1$. After arranging terms we arrive at the following equations: Dimzx - (a;l + 2#,)A - r%B + &A, + P&z - #A, - VA, - a, + ufD,(&
+ 2%)
- 3&A - r&B - hl + ~~(Qcu,,- &,A) i- ~(24, - yjf) + Z(T - l)cu,, + 2uw& f uz(r - I)& + 2vw#~, + u”(2& + aOu,,)+ u’L~~~+ 2uv(#, + otih) + u’$~*, + 2U%Cpah+ UV2&h = 0
Nonlinear
reaction-diffusion
where we defined
r=- D1 D2
We shall assume that r # 1. Now (3.6) implies that (Yb= Ly,, = 0,
GU= $h = 0,
D,(&,
2& - i/J,= 0,
1031
equations
(3.7)
Pll = Pbb = 0
+ 2~)
Dz($L + 2Phx) - @,= 0
- @r= 0,
aA, + PA,, - $A, - +A, - (aa + 2@,)A + Dla;, aB, + PBh - GBx - YB, - (h
- cu,= 0
+ 2Gx)B + DzPx.r- PI = 0
(3.8)
It follows from the equations in the first line of (3.8) that $ = @(x. t), a@, t, a, b) = y(x, t)u + 6(x, r), 6(x, f, u, b) = o(x, t)b +
E(X,
(3.9)
t)
Then equations in the second line of (3.8) yield
2D,y, = ; +c + ,~5+ y(x, t) = &
[t $x2 + i~x + ol(t)]
2D,o, = ; $x + p 3 o(x, t) = &
[a $x2 + bx + q2(l)]
(3.10)
Finally, the last two equations of (3.8) are simply reduced to (ra + b)Au + (wb + c)Ab - GA, - VA, - (v + 24QA + (D,Y,, - r,)a + D16x.x- 6, = 0 (yu + 6)B, + (ob + E)B, - @Bx - ~JB, - (w + 2&)B + (D2w, - o,)b + D2~,, - E, = 0
(3.11)
If we substitute (3.5) and (3.9) into (2.9) through (3.4) we obtain the components of isovectors as dependent on the still unknown functions q(t), p(t), vi(t), r/*(t), 6(x, t) and E(X, t): V” = -; $(t)x - p(t) V’= -q(t) V” =
& [; T&)X2 + P(t)x + q,(r)]u +6(x,
Vh =
& ; $(t)x2 + P(t)x
+ r/*(t) b + E(X, t) I
2
V” = &
t)
{[f
&)x2
+ P(t)x + r],(t) + Dgj(r)]u
V” = &[ [ i t&)x2 + P(t)x + q2(f) + D,$(l)]u V” = [; $(t)x
+
V’ = [;
+
+ P(t)]u
+ &
[ 4’ &)x2
& [; ij(t)x2 + o(t)x
+ [; $(t)x
+ P(r)]b
+ 20,6(x, + ~D+(x,
t)) I)}
+ P(t)x + r],(t) + 2D,v)(t)]w
& [; ;i;(t)x2+ #a(t)x + ill(r)]u+6,(x, lj$t)x + b(t)]v +&
+ [; vji(t)x + p(i)]u
t)
[$$(t)x2 + @(t)x -I + fj2(f)]b + E,(x, t)
v,(t)
+
2D244z
(3.12)
E. S. SUHUBI
1032
and K. L. CHOWDHUKY
We can exploit eqn (3.11) in two fashions. If A and B are given functions of a, b, x and t. (3.11) will enable us to determine unspecified functions r+~,p, n, , II?, 6 and E. Moreover. components of isovector field are determined through (3.12). Since those components are the infinitesimal generators of the Lie group under which the prolongation structure corresponding to the differential equations (2.1) remains invariant, we can determine the symmetry group of equations (2.1) with given A and B. Conversely, if we specify functions r/j, p, n, , q2, 0 and F. thereby choose a priori the symmetry group which will be admitted by our differential equation then (3.11) constitutes two uncoupled linear partial differential equations with variable coefficients to determine functions A and B compatible with the chosen Lie group of symmetry transformations. In any case the similarity solutions of the reaction-diffusion equations are invariant solution under the appropriate Lie group. Therefore they are obtainable through the invariants of the group which satisfy
the solution of which can be formed by its characteristic dx =---~-~--_--__-__-__ dt da db du vx V’ V” vb V”
fields
dv V”
dw V”
dt V’
(3.14)
A closer scrutiny of the structure of isovector field (3.12) reveals the fact that the first four equations in (3.14) can be solved independently and this is what really concerns us because we will be interested in similarity solutions for a and 6. We have indeed dx
-;
dt =-=-=-
da
db
-q(t)
v”
vb
W(t)x - o(r)
from which we can easily write (3.15) where v(t) is defined by and the similarity variable is (3.16) Then a is determined
from
as “q5) a=e --1\(1;
+ qt; ,ge-w;
51,
6 = WI
t)
(3.17)
where A similar expression can be found for b depending on an unknown function p(c). If these expressions are substituted into (2.1) we obtain a pair of ordinary nonlinear differential equations for o and p in the similarity variable 5. For instance let us consider the case
v(t) = t,
p=q*=q*=~=&=O
which yields V” = -x,
V’ = -2t,
V”=I/b=()
Nonlinear
Therefore
reaction-diffusion
1033
equations
similarity solutions are obtained from dx dt da -_=-=-=X
2t
db
0
0
which gives
a = a(E), Equations
E=$
b = b(E),
(3.18)
(3.16) are reduced now to &Bx+te,+B=o
;xA,+tA,+A=O,
which can be integrated
as A =xe2f(a,
B =x-*g(a,
b, E),
b, E)
(3.19)
where f and g are arbitrary functions of their arguments. If we introduce (3.18) and (3.19) into (2.1) we obtain the ordinary differential equations 2D1a” + i$a’ + $(a,
corresponding
2D,b” + Eb’ + -$g(a,
b, E) = 0,
b, Zj)= 0
(3.20)
to functions A and B given by (3.19)
4. SIMILARITY SOLUTIONS FOR SOME NONLINEAR REACTION-DIFFUSION MODELS In this section we shall try to explore in some detail restrictions imposed by (3.10) on the functions A and B. We shall assume that A and B depend only on a and 6. In this case (3.10) can be rewritten as (ya + 6)A,
+ (ob + E)A~ - (y + 2&)A
(ya + S)B, + (ob + E)B~ - (w + 2&)B
We first consider the equation from
+ (Dlyxx - y,)a + D,S,
+ (Dzwx, - o,)b + D2~,, - E, = 0
satisfied by A. The characteristic
da
db =-= ya + 6 ob + E
- 6, = 0
field of this equation follows
dA (Y + 2&P
(4.1)
- (DryXX- vt)a - (Dr&, - 6,)
(4.2)
From the first two equations we obtain G = [a + cVr)l”” b + (E/O) and on a characteristics
defined by c = constant we have
(ra + 6) which can be integrated
(4.3)
2 =(Y +‘WA4 - (Dl~xx
- rda - (@L - 6,)
to yield
where n(x, t) = 2 $
(4.5)
E. S. $UHUBI
1034
and K. L. CHOWDHURY
and f is an arbitrary function of its argument. Although A is assumed to be independent of X, t. many terms in the expression (4.4) depend on x and t. Therefore we should determine y, 6, (r) and 4 in such a way that (4.4) should satisfy A., = 0,
A,=0
(4.6)
If & = 0, i.e. n = 0 then (4.4) changes to A
=f(o(a
+ %, _ DlY;
- Yt ia +$log(a + t, _ 6(DlYxx - YJ-+Y(Dl&* -
A,=A,=O
w (4.7)
Similarly we obtain for the function b
where
with the conditions B, = 0,
B, = 0
(4.10)
which help determine the functions w, E, y and 4. Here, g is an arbitrary argument. If C/Q= 0, i.e. m = 0 then (4.8) should be replaced by
function of its
B, = B, = 0
(4.11)
We shall now consider some special cases as solutions of eqns (4.4)-(4.6)
and (4.8)-(4.10).
(i) All terms in the expressions for A and B are assumed to be independent of x and t (a) We, first, choose that m=n=O
or
@,=f$=O,
(4.12)
Vy=Cl
where c1 is an arbitrary constant. This in turn implies that we have to employ (4.7) and (4.11) as representations of A and B and we should also impose the conditions 6
-_=
Y
k 1,
e-kz,
;-
QY,,
- yr
Y
=m,
0 -=m,,
Y
D2wx
-
w
w,
=mz
(4.13)
ml, m2, are arbitrary constants. On the other hand it follows from the expressions (3.10) for y and o that the relation w = m,y is satisfied if and only if
where kl, k2, m,
ml=:=,
and 2
ql(t)=q2(t)
The relation &Y,, - yr = mY leads, in view of (4.12), to -yt = my or to (ij + mrj)x + q, + mq, = 0 which yields P+mlj=O,
ij,+mq,=O
(4.14)
Nonlinear
reaction-diffusion
1035
equations
or nr = cqeem’
p(t) = c2 + cge-“I, c2,
c3,
c4
are arbitrary constants.
gives
The last condition of (4.13) now becomes -0,
-mlyr
= m,m,y
Hence the relevant isovector components V” = - (c2 + c3e-*‘), 1 V” = 20 (-mc3x 1
(4.15) = m2w which
m, = m
and
(4.16)
are obtained as
V’ = -cl
+ c4)evm’(a + k,),
”
i(_
mc3x + c4)emmr(b + k,)
= 24
(4.17)
and forms of the functions A and B are found from (4.7) and (4.11) as A =f(C)(a
+ k,) - m(a + k,)log(a + k,)
B = g(C)(b
+ k2) - m(b + kJlog(b
(
+ k,)
(a + w
=
(4.18)
b + k2
The similarity solution of (2.1) corresponding system dx - (c2 + c3eAm’) &(-
to this case is obtained
by investigating
the
da mc3x + c4)eFm’(u + k,) db
=
(4.19)
i$-
mc3x + c4)eem’(b + k2)
Defining new constants by a,
2
a22
9 Cl
a3=-,
,
Cl
c4
Cl # 0
Cl
we then obtain dx z
=
a, + a2e --mr
or
x=a,t--e
a2 -mt+ 5 m
which yields the similarity variable as 5=x-a,I+zeWm On a characteristics
(4.20)
a satisfies
1 da 1 - = (ma,x - a3)epm’ = - l ( ma,a,t - uge-mr + ma2E - a3)epm’ a + kl dt 20, 201
which integrates to log@ + k,) = &
a,a,(mt
1
+ 1) - i fz&~-~~+ ma& - a3 + log a(E)
1
which gives upon using (4.20) a + kl = e-“(*sf)a(ij)
(4.21)
where A(x, t) =
&
(ma,, 1
+ i a:epmt +
ala2
-
a3)emmt
(4.22)
Is. s
1036
SUHUBI
and K. L. C’HOWDHURY
Similarly we can obtain for h b + k2 = e-m”/jf;T)
(4.B)
If we introduce (4.21) and (4.23) into (2.1) we see that the functions (u(g) and p(t)
(B )
D,a”+u,d+f
g
lx-mffloga=O
D$” + LQ’ + g~~~~ where primes denote differentiations If the constant m = 0 then
satisfy
-
mplog p
=0
(4.24)
with respect to 5. 11= CJ
p(t) = c2 + c,r,
9 = Cl, and V” = -(cz + CQ),
V’ = -Cl,
v” = &
(c+ “t &+)(a+ ICI), 1
vL$-
(%X -
+
cd@
+
k2)
2
Thus we obtain again b $ k2 = e-‘^~“~‘)~(lj)
a + k, = e-A(X~f)(u(Q, azxt + a$ A(x,r)=&(
1 1 - - alazt2 + - ait3 2
3
(4.25)
>
and
(4.26) (b) We assume that m and n are nonzero constants. Then it follows from (4.4) and (4.8) that we have to write
(4.27) where kI, b, read as A =f(C)(a
Equations
ml, m2, m, n are arbitrary constants. With these equations,
+ kl)l+, + m&a + k,),
B = g(C)@
+ kdftm + m4b + k2),
(4.4) and (4.8) are i; = (a + kP b + kz
(4.27)6,7 clearly show that n=mlm
These relations further imply that v=ny=mu
or
n
r&=20,
l..,
c
4*x
.
+px+rj,
i
=z
m 2
(
1.., . -4+x +pX+r/2
J
and (4.28)
Nonlinear
reaction-diffusion
equations
1037
or l/J = 2c,t + c2,
4ac,
p=c3r
Hence y and w are constants w = m,y gives
71(t) = -
n
and (4.27)4,5 requires
vz(t)
’
that
=
m2 = m3 =
4&c,
-
0. Finally the relation
This last relation is satisfied either by assuming (1) m, = r and q1 = r/2 or (2) These cases correspond of course to (1) n = rm and (2) n = m. The relevant components of isovectors are V”
=
-(c1x
+
c3),
V’
=
=
2
-(2c,t
V” =?(a
+ cz),
+ k,),
(4.29)
m
m 1 = 1, r] , = rq2.
vb=3b+k2)
which leads to the system dt
dx
-(x +
a*) =
da
-(2t + al)
which can be integrated
=
;(a+k,)
db
al=--,
c2 Cl
i(b+k,)’
a2=-,
C3
Cl f0
Cl
to
a+k,= E= (2;t+aq;‘” 9
45) (x + a2)2’n
’
b+k,=
P(5) (x + a2)Um
(4.30)
It is easy to see that (Y(E) and p(g) satisfy
(4.31) Two cases can be obtained from (4.31) by simply taking m, = r, n = rm or m, = 1, n = m. In the last case a family of exact solutions can be found for (4.31). For the sake of clarity let us rewrite (4.31)
(4.32) Let us choose a function U(E), define p = Ua, substitute second equation from the first. We obtain [2$+$(r-l)]n’+[$ 1
this into (4.32) and subtract the
” ’ +y(~-~)]O.+~[R(U-l)U”‘-/(u-‘)]a””=O
However by a change of variable A=&+
(4.33)
this equation modifies to a linear first order equation A’ + V(E)A = W(E)
(4.34)
E. S. $JHUBl
1038
and K. L. CHOWDHURY
whose solution is always reduced to a quadrature.
Here we have defined
(4.35)
W(5)=~[2~+~(r-l)]-‘~g(u-‘)ll”-f(u-~)] 1
For every choice of U we can generate in this way an exact solution by solving (4.34) for the function A. Then we have * = A-I’m, P = U-1I” It is interesting to note that some cases which are quite important as far as the application is concerned fall naturally into this category. For instance A = &ub, B = &zb [19], and A = ~,a(~* + ,qb*), B = IZ2b(a2 + p2b2) [20] correspond, respectively, to functions
(4.36) (ii) Coefficients of A and B may depend on x and t We shall only consider a special case in which we shall assume that 6 = E = 0. Let us define (4.37) Let us recall that n(x, t) and m(x, t) were defined by (4.5) and (4.9). Hence A and B are given bY w, A =f ; u1+” + D,Y,, - or Bzg :A blfm$ D2wx b (
i
244
(
uJ
J
wx
which lead to the relations A,,,, =f ‘(5)&w,u’+”
+f (&w,u’+”
log u + (Dlg;y~)&z=o D2wxx
B,,,,,
=
g’(~)&x,,,b’+”
+
&%cx,rjb’+m
log
b
where the subscript (x, t) denotes partial differentiation
+
(
-
w, )
wx
b=O
(4.38)
(x.1)
with respect to x or t. We also defined
f = *-1/n_ bllh U
Equations (4.38) clearly show that their last terms should vanish implying that D,Y,, - yr 2$x
D20, =
-PL,
-
2C&
w,
(4.39)
=-v
where ,Uand Y are arbitrary constants. Employing (3.10) in (4.39) we immediately ... V = 0,
p =o,
4’=2D&j+/~j),
find that
Q2=2D2($$+&)
or r/J(t) = cqt2 + a,t + ag, v(qt*
dt)
= /At +
+ a,t + 6,) +;
(ult
I
B2>
w(t)=W[d
cult2
+
cr2t
+
6,)
1 +P
1 (4.40)
Nonlinear
On the other
reaction-diffusion
1039
equations
hand
and (4.38) becomes (4.41) Equations
(4.41) will have a solution U’(C)
where
p
and
q
are arbitrary
if and only if f(c)
=A%
Then
satisfy
&CC) = 4(x, M)
t)f(C-)J
functions.
and g(c)
A,
and
n
m
should
satisfy
+m(,,,)
=
(x,1) For every choice of the pair integrability conditions
p
and
q
there
corresponds
an isovector
0
(4.42)
field provided
that
the
pA2mCx.,j + qn(x,r, = 0 are met. In the sequel
we shall assume
that
p
f(5)
and
q
are constants.
Then
we have (4.43)
g(E) = 5”
= A,Z’,
and P&,,,
+ q,,,)
0,
=
0
+ m(x,r) =
4 i
0
(x.1)
so that 4
pA+n=c,, where
c, and c2 are arbitrary
constants.
Equation
(4.44) can now be written
pw + 2$x = Cl y, which implies
(4.44)
i+m=c2 4Y + 2& =
as
c2w
that cl =pr,
pr(rlr-
c2=-,
r/2)=2@+,
4 -q(%-v2)=2a4 r
which yields immediately (4.45)
q=-rp
Then
pr
the equation
~L(cY,~~+ a2t + 6,) + i curt
1- [
2D2 v&t2
+ au,t + 6,) +;
n,t
II=
20,(2a,t
+ (Ye)
gives v=pr,
a,-
pz-4
(y2Jr
;;)a,
r-l'
and due to (4.45) we have
q_kr-l
(4.47)
E. S. $UHUBI
1040 so
and K. L. CHOWDHURY
that p and q satisfies the relation p+q=-4
Therefore
(3.48)
A and B become A = A,,a iP+n+Lh-_p
_
B
pa
=
&-Y@//~)+“~+l
_
,.@
or if we write -p=l
/Ip+n+l=k,
so that r-l
we have A = A&b’
4
l=--
k=5r-1 -
1=-p,
k=pr+l,
B = B &lb’+’
- pa,
k+l=5
r-l’
’
_
0
(4.49)
(4.50)
vb
We shall now consider a special case in which all the arbitrary parameters vanish in isovector components except for crl = a,, pi = -a2. The similarity solution corresponding to this case can easily be determined from the system db
da
dx
=-= -(tx - yt) -“:2 [
(4.51)
-&-(x2-2yx)+ptz+~t]a=[$-(x2-2yx)+rpt2+~t]b 1
1
where the constant y is defined as azlaI. We thus obtain from the integration equations the similarity variable g=- X-Y t
x = y - Et,
of the first two
(4.52)
Then 1 da --_= a dt
_-t: [-&‘-2yx)+w2+;]
= -(p
+-g)
-;+A
1
or (4.53)
where (4.54)
Similarly we obtain for b b = ?- e-r’+J)/-j(Q ti
If we introduce and /3:
(4.55)
(4.53) and (4.55) into (2.1) we find the following differential
pll
_
&‘p
+
$
&-l/j’+’ 2
where
&yz
40:
=
0
equations for (Y
(4.56)
Nonlinear
reaction-diffusion
equations
1041
As we have done before we can look for some solutions of (4.56) satisfying the condition /? = U(g)(r for a chosen function U(E). We can easily find that a = A-“-+,
p = uA-“4
(4.57)
and A satisfies the first order linear equation A’-tV;3.=W
(4.58)
where 3 v=-$Y-(r*-l)c*U],
7 WY----D ;, 1
(r& - A,,) Cl’+’
(4.59)
Acknowledgements-This paper was written during E. S. Suhubi’s visit to the Department of Mechanical Engineering. He gratefully acknowledges the University of Calgary and the Killam Foundation for electing him as a Killam Visiting Scholar.
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [13] [14] [15]
[ 161 [17] (181
[19] [20]
R. ARIS, The Mathematical Theory of Reaction-Diffusion in Porous Catalysts. Oxford University Press (1975). A. TURING, Phil. Trans. R. Sot. B237, 37 (1952). P. C. FIFE, BUN. Am. Math. Sot. 84, 693 (1978). N. KOPELL and L. N. HOWARD, Stud. appl. Math. 52, 291 (1973). L. N. HOWARD and N. KOPELL, SIAM AMS Proc. 8, 1 (1974). A. T. WINFREE, SIAM AMS Proc. 8, 13 (1984). 0. GUREL and 0. E. ROSSLER, Ann. N. Y. Acad. Sci. 316, 1 (1979). J. GUCKENHEIMER and P. HOLMES, Nonlinear Oscillutions, Dynamical Systems, and Bifurcation of Vector Fields. Springer, New York (1983). E. VESSIOT, Bull. Sot. Math. France 42, 336 (1924). E. VESSIOT, /. Math. 18, 1 (1939); 21, 1 (1942). B. K. HARRISON and F. B. ESTABROOK, J. math. Phys. 12, 653 (1971). G. W. BLUMAN and J. D. COLE, Similarity Methods for Differential Equations. Springer, New York (1974). L. V. OVSJANNIKOV, Group Analysts of Differential Equations. Academic Press, New York (1982). H. D. WALQUIST and F. B. ESTABROOK, J. math. Phys. 16, 1 (1975). E. D. FACKDRELL, in Geometric Aspects of the Einstein Equations and Integrable Systems (Edited by R. MARTINI). Lecture Notes in Physics No. 239, Springer, Berlin (1985). F. B. ESTABROOK, Geometrical Approaches to Differential Equations (Edited by R. MARTINI). Lecture Notes in Mathematics No. 810, Springer, Berlin (1980). D. G. B. EDELEN, Applied Exterior Culcul~. Wiley, New York (1985). D. G. B. EDELEN, Isovectors Methods for Equations of Balance with Programs for Computer Assistance in Operator Calculations and Exposition of Practical Topics of Exterior Calculus. Sijthoff & Noordhoff, Alphen aan den Rijn, The Netherlands (1980). W. F. AMES, Nonlinear Partial Diflerential Equations in Engineering, Vol. I, p. 13. Academic Press. New York (1965). I. M. FRIEDLIN, SIAM 1. uppl. Math. 46, 222 (1986).
of Mechanical Engineering.
and K. L. CHOWDHURY The University of Calgary, Calgary. Alberta, Canada
Abstract-Nonlinear reaction-diffusion equations are expressed by means of a closed ideal of exterior differential forms over an S-dimensional manifold. The components of the isovector field associated with the system are constructed by using its transport property under Lie derivatives. The similarity solutions which are invariant under the corresponding symmetry group are obtained for fairly general cases and the system is reduced to coupled nonlinear ordinary differential equations.
I. INTRODUCTION Nonlinear reaction-diffusion equations in their diverse forms serve as mathematical models to a number of interesting physical phenomena occurring in various research fields [l-6], e.g. developmental biology, neurodynamics, ecology, plasma physics, chemical reactions etc. Investigations of some of the intricate problems have resulted in pattern formations, stable limit cycles, wave trains, shock fronts, bifurcations and chaotic evolutions. An excellent compilation of the mathematical methods of bifurcation theory and its applications to various scientific disciplines with computer simulations of dynamical systems is given in Refs [7] and f8& Recent search for the invariance groups and special families of exact solutions to nonlinear PDEs, using differential geometric concepts of Lie’s and Cartan’s continuous groups, has been spurred by the works of Vessiot [9, lo], Harrison and Estabrook [ll], Bluman and Cole [ 121 and Ovsjannikov [13]. Walquist and Estabrook [14], Fackdrell [15] and Estabrook [16] used Cartan’s formulations to define and construct prolongation structures of systems of nonlinear PDEs. Edelen’s recent two books [I?‘, IS] have provided extensive theoretical framework of the methods of exterior calculus and further enhanced their applications to various problems of continuum mechanics, thermodynamics and electrodynamics. In this paper, we express quite a genera1 class of nonlinear reaction-diffusion equations by means of a closed idea1 of exterior differential forms over an &dimensional manifold. Using the transport property of the isovectors, i.e. the invariance of the ideal under the Lie dragging by the congruence generated by the isovector field which implies in turn that the Lie derivative of any form in the ideal remains in the ideal, the components of the isovector field are constructed. These components are essentially infinitesimal generators of Lie group which is the symmetry group of the prolongation structure of the reaction-diffusion equations. A similarity soiution of the differential equations is an invariant solution with respect to a particular symmetry group. Therefore they can be obtained by determining the invariants of the group. In this paper the most general symmetry group of one~dimensional reaction-di~usion equations is obtained for arbitrary source functions. Since the latter functions are not specified beforehand the group involves arbitrary functions. These functions are determined for fairly general forms of the source functions and the corresponding similarity solutions and nonlinear ordinary differential equations which they satisfy are provided. 2. BASIC
EQUATIONS
We assume that a one-dimensional reaction-diffusion system of nonlinear differential equations: a, = &a,
process is governed
by the following
+ A+, b, x, t)
b, = D&x,, c B(a, 6, x, f) t Permanent address: Faculty of Science, Department Maslak 80626, Istanbul, Turkey. ES 26:10-A
of Engineering
1027
(2.1) Sciences,
Istanbul Technical
University.
1028
E. S. .tjUHUBI
and K. 1.. C‘flOWDHUR\
where x and t are space and time coordinates,
u and b are the reaction-diffusion variables. functions describing the kinetics of the
A(a, b, x, t) and B(a, b, x, t) are arbitrary nonlinear process and D1 # 0 and D2 # 0 are diffusion constants.
We introduce an S-dimensional manifold u, v, w and _zare given by the relations u=a xv
(x, I, u, b, u, 11.~1,2) where
M with coordinates
v=b x*
w = a,,
(2.2)
= =b,
Let us now define the following exterior differential forms: o1 = D1 du A dt + (A - w) d_x A dt,
w2 = D2 dv A dt + (B - z) dw A dt
(Ye= du - u dx - w dt,
LYE = db - v dx - z dt
da, = -du A dz - dw
A
dt,
da, = -d/f A dx - dz A dt
(2.3)
where A represents the exterior product. If we consider a two-dimensional submanifold H of M on which a, b, u, v, w, z are functions of x and t only, the sectioning of the forms (2.3) by H are CI& = (Dlu, + A - w) dx
A
o&, = (D,v, + B - z) dx
dt,
= (u, - w,) du
A
dt,
dt
CY&= (6, - v) dx + (b, - z) dt
(Y&, = (a, - u) dr + (a, - w) dt, da&
A
da&
= (u, - z,) dx
A
dt
(2.4)
Therefore the solution manifold of eqns (2.1) and (2.2) annuls the exterior forms (2.3). On the other hand one can easily see that dw, = (Aa dx
A
dt) A (Y, + (Ah dx A dt) A a2 - dx A da,
do, = (B, dx
A
dt)
A crl + (&
dx A
dt)
A a2 -
dx A
da,
(2.5)
Hence the forms (ol, 02, aI, a2, da,, da,) generate a closed ideal of the ring of forms and Frobenius theorem can be applied. The isovector field V defined as a vector field with components (VI, V’, V“, Vh, V”, VI’, V”, V”) generates a congruence which leaves the foregoing closed ideal invariant under the corresponding Lie dragging. This implies that the Lie derivatives of the forms (2.3) with respect to the vector field V should remain in the closed ideal generated by them. Therefore we should write
fvoi = Ali da, + A2; da, + B,;wl + B2;w2 + M; A aI+ N, A ~7
(i = 1, 2)
(2.6) , NL where 4, b, PI, ~2, AH, . . . , B22 are arbitrary scalar functions of (x, . . . , z) and M,, are arbitrary l-forms. If we take the exterior derivative of (2.6), and use its commutation property with the Lie derivative, we immediately see that
dfvai = .Ev da, = il; da, + p; da2 + dA, A
al +
dpi
A LYE
Therefore fv dq are already in the ideal. The Lie derivatives of the forms can easily be calculated if we recall the following relation which is valid for all forms f,w=Vidw+d(VJo) where J denotes the contraction
of w with the vector field V. Hence we obtain from (2.3),
V-Ida,=-V”dx+V”du-V”dt+V’dw, VJda,=-V”dx+V”dv-V”dt+V’dz
(2.7)
Nonlinear
reaction-diffusion
If we introduce (2.7) into (2.6), and compare sides we find that &=F,,
PI = Fbt
& = G,,
equations
1029
the coefficients of the same forms from both E;, = F, = 0,
PZ = Gb,
G, = G, = 0
(2.8)
and V” = -F;1 = -G,,,
V’ = -F,
= -G,
V” = F - tit;;, - wF,,
V’=G-UC,,
VU=F,fuF,+vF,,
V”= c;, + UC, + UC,
V”=fi+wF,+zF,,
V” = G, + WC, + zG,
-zG,
(2.9)
Similarly it follows from (2.3), that V_lo,=D,V”dt-D,V’du+(A-w)(V”dt-VW) V _Ido, = (V=A, + VhAh - V”‘) dx A dt - V”A, da A dt + V’A, da A &u -VxA~dbAdt+V*A~dbA~+VxdwAdt-V*dwA~ VJw,=L),V”dt-DzV”dv+(B-y)(V”dt-V’du) V_ZdW2 = (V”B, + VhB, - V’) dx: A dt - VxE, da A dt + V’B, da A dx
-Vx~~dbAdt+VfB~dbA~+VxdzAd~-Vid~A~
(2.10)
We not denote the l-forms M1, N,, M2, N2, by Mi = Eli dt + F,i du + Gli da + H,i db + r,i du + Jli dv + i<,i dw + Lli dz
Ni = Ezi dt + F*; dt- + G*i da + H,i db + I,, du ~ Jzi dV + Kzi dw + L,i dZ Introducing (2.10) and (2.11) into (2.6), and comparing both sides we obtain, for i = 1 DIV,” + (A - w)(Vf + V;) + PA,
(2.11)
coefficients of the same forms from
+ VbA, + VIA, + V’A, - V” = B,I(A - w) + B,,(B
- z) + uEll + vE2, - wF,, - zFzI
D,VU,+(A-w)V”,=-I!?,,,
D,V:+(A-w)V;:=-E,,,
D,(V:+V;)+(A-w)VY,=D,B,,-wZfl-zZz, D, Vu, + (A - w)V:, = -A,1 DIV;
- wKll-
zKz1,
+ (A - w)V: = -AZ1 - wLl, - zLzl,
(A - w)V:, = &,
(A - w)V; = &,
D1 V:: + (A - w)V: = D, Bzl - wJ,, - .zJ21 D,V;
- (A - w)V; = -AtI - uZit - vl,,
(A - w)V:, = uKl, + vK2,, (A - w)V:, = B1, + ul,, + vJzl
{A - w)V: = uLl, + uL21, Gzt =ff,jv
D,Vb=4,,
D,V:,=k,,
D,V:,=DIV:,=D,V;=O (2.12)
J,2 = KII = L,, =Jzl = KZ1 = Lt, =0
where we have defined 4, = E,, + WC,, - Gi, &, = Z& + uH1 1 + vH2,
I?,, = ET?, + wH,, + zH2,,
F,, = F,, + UC,, + vG2,
(2.13)
If we replace Gzl in (2.13) by Hi, in view of the last line of (2.12) we see at once from (2.12)1 that u&l + n&r - wr;;, - zF,, = u& + vi?,, - w& - t& Recalling expressions
(2.8),,, and (2.9) we can easily deduce that the relation (2.12) are to be
satisfied if and only if the isovector field obeys D,(Vu,, - v; - ff V:, - UVi) = 0 D,(V,u+uV:+
UP’;)+ (A - w)(V:
- V::+ uV;:+ uV;)-;(B
- z)V:f 7
+V”A,, + V’A,, + VA,
+ V’A, - V”’ = 0
(2.14)
From the second equation of (2.6)2 we obtain similarly Dz(VY - v: - UV:, - vVb) = 0 &(V;
+ uV:: + uV;> + (B - z)(V;
- V:: + uV: + vV;)
- 2
(A - w)V;; 1
+ VU& + vv&
3. DETERMINATION
OF THE
+ V”B, + V’B, - vz = 0
ISOVECTOR
(2.15)
FIELD
Equations (2.8) imply that F = F(x, t, a, 6, U, W), Introducing
G = G(x, t, a, b, v, t)
(3.1)
(2.9) into (2.14)i and taking into account (2.9), we obtain F,, + uF,, i- uFwh = 0,
F;, = G,
It is easy to see that (2.1~5)~does not provide an independent F,, = G,,v = 0,
(3.2)
F;o= G
equation.
Since
F,,, = c;,, = G,.,, = 0
El;;,,= G,,,, = 0,
(3.3)
we easily deduce that the function F and G have the following representations: F = 1(I(x, t, a, b)w + tp(x, I, a, b)u + a+, G = v(x,
c, a, b)
t, a, b)z + +(.G r, a, b)v + P(x, t, ~1,b)
(3.4)
Replacing F, in (3.2), by t+?,we obtain t/J7+ U7Jla+ W&11, =0 which implies that qa =
$)h
=
v.r
=
(1
or (3.5)
* = q(t)
Let us introduce the relations (3.4) and (3.5) into (2.9) and resulting expressions into (2.14), and (2.1$. After arranging terms we arrive at the following equations: Dimzx - (a;l + 2#,)A - r%B + &A, + P&z - #A, - VA, - a, + ufD,(&
+ 2%)
- 3&A - r&B - hl + ~~(Qcu,,- &,A) i- ~(24, - yjf) + Z(T - l)cu,, + 2uw& f uz(r - I)& + 2vw#~, + u”(2& + aOu,,)+ u’L~~~+ 2uv(#, + otih) + u’$~*, + 2U%Cpah+ UV2&h = 0
Nonlinear
reaction-diffusion
where we defined
r=- D1 D2
We shall assume that r # 1. Now (3.6) implies that (Yb= Ly,, = 0,
GU= $h = 0,
D,(&,
2& - i/J,= 0,
1031
equations
(3.7)
Pll = Pbb = 0
+ 2~)
Dz($L + 2Phx) - @,= 0
- @r= 0,
aA, + PA,, - $A, - +A, - (aa + 2@,)A + Dla;, aB, + PBh - GBx - YB, - (h
- cu,= 0
+ 2Gx)B + DzPx.r- PI = 0
(3.8)
It follows from the equations in the first line of (3.8) that $ = @(x. t), a@, t, a, b) = y(x, t)u + 6(x, r), 6(x, f, u, b) = o(x, t)b +
E(X,
(3.9)
t)
Then equations in the second line of (3.8) yield
2D,y, = ; +c + ,~5+ y(x, t) = &
[t $x2 + i~x + ol(t)]
2D,o, = ; $x + p 3 o(x, t) = &
[a $x2 + bx + q2(l)]
(3.10)
Finally, the last two equations of (3.8) are simply reduced to (ra + b)Au + (wb + c)Ab - GA, - VA, - (v + 24QA + (D,Y,, - r,)a + D16x.x- 6, = 0 (yu + 6)B, + (ob + E)B, - @Bx - ~JB, - (w + 2&)B + (D2w, - o,)b + D2~,, - E, = 0
(3.11)
If we substitute (3.5) and (3.9) into (2.9) through (3.4) we obtain the components of isovectors as dependent on the still unknown functions q(t), p(t), vi(t), r/*(t), 6(x, t) and E(X, t): V” = -; $(t)x - p(t) V’= -q(t) V” =
& [; T&)X2 + P(t)x + q,(r)]u +6(x,
Vh =
& ; $(t)x2 + P(t)x
+ r/*(t) b + E(X, t) I
2
V” = &
t)
{[f
&)x2
+ P(t)x + r],(t) + Dgj(r)]u
V” = &[ [ i t&)x2 + P(t)x + q2(f) + D,$(l)]u V” = [; $(t)x
+
V’ = [;
+
+ P(t)]u
+ &
[ 4’ &)x2
& [; ij(t)x2 + o(t)x
+ [; $(t)x
+ P(r)]b
+ 20,6(x, + ~D+(x,
t)) I)}
+ P(t)x + r],(t) + 2D,v)(t)]w
& [; ;i;(t)x2+ #a(t)x + ill(r)]u+6,(x, lj$t)x + b(t)]v +&
+ [; vji(t)x + p(i)]u
t)
[$$(t)x2 + @(t)x -I + fj2(f)]b + E,(x, t)
v,(t)
+
2D244z
(3.12)
E. S. SUHUBI
1032
and K. L. CHOWDHUKY
We can exploit eqn (3.11) in two fashions. If A and B are given functions of a, b, x and t. (3.11) will enable us to determine unspecified functions r+~,p, n, , II?, 6 and E. Moreover. components of isovector field are determined through (3.12). Since those components are the infinitesimal generators of the Lie group under which the prolongation structure corresponding to the differential equations (2.1) remains invariant, we can determine the symmetry group of equations (2.1) with given A and B. Conversely, if we specify functions r/j, p, n, , q2, 0 and F. thereby choose a priori the symmetry group which will be admitted by our differential equation then (3.11) constitutes two uncoupled linear partial differential equations with variable coefficients to determine functions A and B compatible with the chosen Lie group of symmetry transformations. In any case the similarity solutions of the reaction-diffusion equations are invariant solution under the appropriate Lie group. Therefore they are obtainable through the invariants of the group which satisfy
the solution of which can be formed by its characteristic dx =---~-~--_--__-__-__ dt da db du vx V’ V” vb V”
fields
dv V”
dw V”
dt V’
(3.14)
A closer scrutiny of the structure of isovector field (3.12) reveals the fact that the first four equations in (3.14) can be solved independently and this is what really concerns us because we will be interested in similarity solutions for a and 6. We have indeed dx
-;
dt =-=-=-
da
db
-q(t)
v”
vb
W(t)x - o(r)
from which we can easily write (3.15) where v(t) is defined by and the similarity variable is (3.16) Then a is determined
from
as “q5) a=e --1\(1;
+ qt; ,ge-w;
51,
6 = WI
t)
(3.17)
where A similar expression can be found for b depending on an unknown function p(c). If these expressions are substituted into (2.1) we obtain a pair of ordinary nonlinear differential equations for o and p in the similarity variable 5. For instance let us consider the case
v(t) = t,
p=q*=q*=~=&=O
which yields V” = -x,
V’ = -2t,
V”=I/b=()
Nonlinear
Therefore
reaction-diffusion
1033
equations
similarity solutions are obtained from dx dt da -_=-=-=X
2t
db
0
0
which gives
a = a(E), Equations
E=$
b = b(E),
(3.18)
(3.16) are reduced now to &Bx+te,+B=o
;xA,+tA,+A=O,
which can be integrated
as A =xe2f(a,
B =x-*g(a,
b, E),
b, E)
(3.19)
where f and g are arbitrary functions of their arguments. If we introduce (3.18) and (3.19) into (2.1) we obtain the ordinary differential equations 2D1a” + i$a’ + $(a,
corresponding
2D,b” + Eb’ + -$g(a,
b, E) = 0,
b, Zj)= 0
(3.20)
to functions A and B given by (3.19)
4. SIMILARITY SOLUTIONS FOR SOME NONLINEAR REACTION-DIFFUSION MODELS In this section we shall try to explore in some detail restrictions imposed by (3.10) on the functions A and B. We shall assume that A and B depend only on a and 6. In this case (3.10) can be rewritten as (ya + 6)A,
+ (ob + E)A~ - (y + 2&)A
(ya + S)B, + (ob + E)B~ - (w + 2&)B
We first consider the equation from
+ (Dlyxx - y,)a + D,S,
+ (Dzwx, - o,)b + D2~,, - E, = 0
satisfied by A. The characteristic
da
db =-= ya + 6 ob + E
- 6, = 0
field of this equation follows
dA (Y + 2&P
(4.1)
- (DryXX- vt)a - (Dr&, - 6,)
(4.2)
From the first two equations we obtain G = [a + cVr)l”” b + (E/O) and on a characteristics
defined by c = constant we have
(ra + 6) which can be integrated
(4.3)
2 =(Y +‘WA4 - (Dl~xx
- rda - (@L - 6,)
to yield
where n(x, t) = 2 $
(4.5)
E. S. $UHUBI
1034
and K. L. CHOWDHURY
and f is an arbitrary function of its argument. Although A is assumed to be independent of X, t. many terms in the expression (4.4) depend on x and t. Therefore we should determine y, 6, (r) and 4 in such a way that (4.4) should satisfy A., = 0,
A,=0
(4.6)
If & = 0, i.e. n = 0 then (4.4) changes to A
=f(o(a
+ %, _ DlY;
- Yt ia +$log(a + t, _ 6(DlYxx - YJ-+Y(Dl&* -
A,=A,=O
w (4.7)
Similarly we obtain for the function b
where
with the conditions B, = 0,
B, = 0
(4.10)
which help determine the functions w, E, y and 4. Here, g is an arbitrary argument. If C/Q= 0, i.e. m = 0 then (4.8) should be replaced by
function of its
B, = B, = 0
(4.11)
We shall now consider some special cases as solutions of eqns (4.4)-(4.6)
and (4.8)-(4.10).
(i) All terms in the expressions for A and B are assumed to be independent of x and t (a) We, first, choose that m=n=O
or
@,=f$=O,
(4.12)
Vy=Cl
where c1 is an arbitrary constant. This in turn implies that we have to employ (4.7) and (4.11) as representations of A and B and we should also impose the conditions 6
-_=
Y
k 1,
e-kz,
;-
QY,,
- yr
Y
=m,
0 -=m,,
Y
D2wx
-
w
w,
=mz
(4.13)
ml, m2, are arbitrary constants. On the other hand it follows from the expressions (3.10) for y and o that the relation w = m,y is satisfied if and only if
where kl, k2, m,
ml=:=,
and 2
ql(t)=q2(t)
The relation &Y,, - yr = mY leads, in view of (4.12), to -yt = my or to (ij + mrj)x + q, + mq, = 0 which yields P+mlj=O,
ij,+mq,=O
(4.14)
Nonlinear
reaction-diffusion
1035
equations
or nr = cqeem’
p(t) = c2 + cge-“I, c2,
c3,
c4
are arbitrary constants.
gives
The last condition of (4.13) now becomes -0,
-mlyr
= m,m,y
Hence the relevant isovector components V” = - (c2 + c3e-*‘), 1 V” = 20 (-mc3x 1
(4.15) = m2w which
m, = m
and
(4.16)
are obtained as
V’ = -cl
+ c4)evm’(a + k,),
”
i(_
mc3x + c4)emmr(b + k,)
= 24
(4.17)
and forms of the functions A and B are found from (4.7) and (4.11) as A =f(C)(a
+ k,) - m(a + k,)log(a + k,)
B = g(C)(b
+ k2) - m(b + kJlog(b
(
+ k,)
(a + w
=
(4.18)
b + k2
The similarity solution of (2.1) corresponding system dx - (c2 + c3eAm’) &(-
to this case is obtained
by investigating
the
da mc3x + c4)eFm’(u + k,) db
=
(4.19)
i$-
mc3x + c4)eem’(b + k2)
Defining new constants by a,
2
a22
9 Cl
a3=-,
,
Cl
c4
Cl # 0
Cl
we then obtain dx z
=
a, + a2e --mr
or
x=a,t--e
a2 -mt+ 5 m
which yields the similarity variable as 5=x-a,I+zeWm On a characteristics
(4.20)
a satisfies
1 da 1 - = (ma,x - a3)epm’ = - l ( ma,a,t - uge-mr + ma2E - a3)epm’ a + kl dt 20, 201
which integrates to log@ + k,) = &
a,a,(mt
1
+ 1) - i fz&~-~~+ ma& - a3 + log a(E)
1
which gives upon using (4.20) a + kl = e-“(*sf)a(ij)
(4.21)
where A(x, t) =
&
(ma,, 1
+ i a:epmt +
ala2
-
a3)emmt
(4.22)
Is. s
1036
SUHUBI
and K. L. C’HOWDHURY
Similarly we can obtain for h b + k2 = e-m”/jf;T)
(4.B)
If we introduce (4.21) and (4.23) into (2.1) we see that the functions (u(g) and p(t)
(B )
D,a”+u,d+f
g
lx-mffloga=O
D$” + LQ’ + g~~~~ where primes denote differentiations If the constant m = 0 then
satisfy
-
mplog p
=0
(4.24)
with respect to 5. 11= CJ
p(t) = c2 + c,r,
9 = Cl, and V” = -(cz + CQ),
V’ = -Cl,
v” = &
(c+ “t &+)(a+ ICI), 1
vL$-
(%X -
+
cd@
+
k2)
2
Thus we obtain again b $ k2 = e-‘^~“~‘)~(lj)
a + k, = e-A(X~f)(u(Q, azxt + a$ A(x,r)=&(
1 1 - - alazt2 + - ait3 2
3
(4.25)
>
and
(4.26) (b) We assume that m and n are nonzero constants. Then it follows from (4.4) and (4.8) that we have to write
(4.27) where kI, b, read as A =f(C)(a
Equations
ml, m2, m, n are arbitrary constants. With these equations,
+ kl)l+, + m&a + k,),
B = g(C)@
+ kdftm + m4b + k2),
(4.4) and (4.8) are i; = (a + kP b + kz
(4.27)6,7 clearly show that n=mlm
These relations further imply that v=ny=mu
or
n
r&=20,
l..,
c
4*x
.
+px+rj,
i
=z
m 2
(
1.., . -4+x +pX+r/2
J
and (4.28)
Nonlinear
reaction-diffusion
equations
1037
or l/J = 2c,t + c2,
4ac,
p=c3r
Hence y and w are constants w = m,y gives
71(t) = -
n
and (4.27)4,5 requires
vz(t)
’
that
=
m2 = m3 =
4&c,
-
0. Finally the relation
This last relation is satisfied either by assuming (1) m, = r and q1 = r/2 or (2) These cases correspond of course to (1) n = rm and (2) n = m. The relevant components of isovectors are V”
=
-(c1x
+
c3),
V’
=
=
2
-(2c,t
V” =?(a
+ cz),
+ k,),
(4.29)
m
m 1 = 1, r] , = rq2.
vb=3b+k2)
which leads to the system dt
dx
-(x +
a*) =
da
-(2t + al)
which can be integrated
=
;(a+k,)
db
al=--,
c2 Cl
i(b+k,)’
a2=-,
C3
Cl f0
Cl
to
a+k,= E= (2;t+aq;‘” 9
45) (x + a2)2’n
’
b+k,=
P(5) (x + a2)Um
(4.30)
It is easy to see that (Y(E) and p(g) satisfy
(4.31) Two cases can be obtained from (4.31) by simply taking m, = r, n = rm or m, = 1, n = m. In the last case a family of exact solutions can be found for (4.31). For the sake of clarity let us rewrite (4.31)
(4.32) Let us choose a function U(E), define p = Ua, substitute second equation from the first. We obtain [2$+$(r-l)]n’+[$ 1
this into (4.32) and subtract the
” ’ +y(~-~)]O.+~[R(U-l)U”‘-/(u-‘)]a””=O
However by a change of variable A=&+
(4.33)
this equation modifies to a linear first order equation A’ + V(E)A = W(E)
(4.34)
E. S. $JHUBl
1038
and K. L. CHOWDHURY
whose solution is always reduced to a quadrature.
Here we have defined
(4.35)
W(5)=~[2~+~(r-l)]-‘~g(u-‘)ll”-f(u-~)] 1
For every choice of U we can generate in this way an exact solution by solving (4.34) for the function A. Then we have * = A-I’m, P = U-1I” It is interesting to note that some cases which are quite important as far as the application is concerned fall naturally into this category. For instance A = &ub, B = &zb [19], and A = ~,a(~* + ,qb*), B = IZ2b(a2 + p2b2) [20] correspond, respectively, to functions
(4.36) (ii) Coefficients of A and B may depend on x and t We shall only consider a special case in which we shall assume that 6 = E = 0. Let us define (4.37) Let us recall that n(x, t) and m(x, t) were defined by (4.5) and (4.9). Hence A and B are given bY w, A =f ; u1+” + D,Y,, - or Bzg :A blfm$ D2wx b (
i
244
(
uJ
J
wx
which lead to the relations A,,,, =f ‘(5)&w,u’+”
+f (&w,u’+”
log u + (Dlg;y~)&z=o D2wxx
B,,,,,
=
g’(~)&x,,,b’+”
+
&%cx,rjb’+m
log
b
where the subscript (x, t) denotes partial differentiation
+
(
-
w, )
wx
b=O
(4.38)
(x.1)
with respect to x or t. We also defined
f = *-1/n_ bllh U
Equations (4.38) clearly show that their last terms should vanish implying that D,Y,, - yr 2$x
D20, =
-PL,
-
2C&
w,
(4.39)
=-v
where ,Uand Y are arbitrary constants. Employing (3.10) in (4.39) we immediately ... V = 0,
p =o,
4’=2D&j+/~j),
find that
Q2=2D2($$+&)
or r/J(t) = cqt2 + a,t + ag, v(qt*
dt)
= /At +
+ a,t + 6,) +;
(ult
I
B2>
w(t)=W[d
cult2
+
cr2t
+
6,)
1 +P
1 (4.40)
Nonlinear
On the other
reaction-diffusion
1039
equations
hand
and (4.38) becomes (4.41) Equations
(4.41) will have a solution U’(C)
where
p
and
q
are arbitrary
if and only if f(c)
=A%
Then
satisfy
&CC) = 4(x, M)
t)f(C-)J
functions.
and g(c)
A,
and
n
m
should
satisfy
+m(,,,)
=
(x,1) For every choice of the pair integrability conditions
p
and
q
there
corresponds
an isovector
0
(4.42)
field provided
that
the
pA2mCx.,j + qn(x,r, = 0 are met. In the sequel
we shall assume
that
p
f(5)
and
q
are constants.
Then
we have (4.43)
g(E) = 5”
= A,Z’,
and P&,,,
+ q,,,)
0,
=
0
+ m(x,r) =
4 i
0
(x.1)
so that 4
pA+n=c,, where
c, and c2 are arbitrary
constants.
Equation
(4.44) can now be written
pw + 2$x = Cl y, which implies
(4.44)
i+m=c2 4Y + 2& =
as
c2w
that cl =pr,
pr(rlr-
c2=-,
r/2)=2@+,
4 -q(%-v2)=2a4 r
which yields immediately (4.45)
q=-rp
Then
pr
the equation
~L(cY,~~+ a2t + 6,) + i curt
1- [
2D2 v&t2
+ au,t + 6,) +;
n,t
II=
20,(2a,t
+ (Ye)
gives v=pr,
a,-
pz-4
(y2Jr
;;)a,
r-l'
and due to (4.45) we have
q_kr-l
(4.47)
E. S. $UHUBI
1040 so
and K. L. CHOWDHURY
that p and q satisfies the relation p+q=-4
Therefore
(3.48)
A and B become A = A,,a iP+n+Lh-_p
_
B
pa
=
&-Y@//~)+“~+l
_
,.@
or if we write -p=l
/Ip+n+l=k,
so that r-l
we have A = A&b’
4
l=--
k=5r-1 -
1=-p,
k=pr+l,
B = B &lb’+’
- pa,
k+l=5
r-l’
’
_
0
(4.49)
(4.50)
vb
We shall now consider a special case in which all the arbitrary parameters vanish in isovector components except for crl = a,, pi = -a2. The similarity solution corresponding to this case can easily be determined from the system db
da
dx
=-= -(tx - yt) -“:2 [
(4.51)
-&-(x2-2yx)+ptz+~t]a=[$-(x2-2yx)+rpt2+~t]b 1
1
where the constant y is defined as azlaI. We thus obtain from the integration equations the similarity variable g=- X-Y t
x = y - Et,
of the first two
(4.52)
Then 1 da --_= a dt
_-t: [-&‘-2yx)+w2+;]
= -(p
+-g)
-;+A
1
or (4.53)
where (4.54)
Similarly we obtain for b b = ?- e-r’+J)/-j(Q ti
If we introduce and /3:
(4.55)
(4.53) and (4.55) into (2.1) we find the following differential
pll
_
&‘p
+
$
&-l/j’+’ 2
where
&yz
40:
=
0
equations for (Y
(4.56)
Nonlinear
reaction-diffusion
equations
1041
As we have done before we can look for some solutions of (4.56) satisfying the condition /? = U(g)(r for a chosen function U(E). We can easily find that a = A-“-+,
p = uA-“4
(4.57)
and A satisfies the first order linear equation A’-tV;3.=W
(4.58)
where 3 v=-$Y-(r*-l)c*U],
7 WY----D ;, 1
(r& - A,,) Cl’+’
(4.59)
Acknowledgements-This paper was written during E. S. Suhubi’s visit to the Department of Mechanical Engineering. He gratefully acknowledges the University of Calgary and the Killam Foundation for electing him as a Killam Visiting Scholar.
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [13] [14] [15]
[ 161 [17] (181
[19] [20]
R. ARIS, The Mathematical Theory of Reaction-Diffusion in Porous Catalysts. Oxford University Press (1975). A. TURING, Phil. Trans. R. Sot. B237, 37 (1952). P. C. FIFE, BUN. Am. Math. Sot. 84, 693 (1978). N. KOPELL and L. N. HOWARD, Stud. appl. Math. 52, 291 (1973). L. N. HOWARD and N. KOPELL, SIAM AMS Proc. 8, 1 (1974). A. T. WINFREE, SIAM AMS Proc. 8, 13 (1984). 0. GUREL and 0. E. ROSSLER, Ann. N. Y. Acad. Sci. 316, 1 (1979). J. GUCKENHEIMER and P. HOLMES, Nonlinear Oscillutions, Dynamical Systems, and Bifurcation of Vector Fields. Springer, New York (1983). E. VESSIOT, Bull. Sot. Math. France 42, 336 (1924). E. VESSIOT, /. Math. 18, 1 (1939); 21, 1 (1942). B. K. HARRISON and F. B. ESTABROOK, J. math. Phys. 12, 653 (1971). G. W. BLUMAN and J. D. COLE, Similarity Methods for Differential Equations. Springer, New York (1974). L. V. OVSJANNIKOV, Group Analysts of Differential Equations. Academic Press, New York (1982). H. D. WALQUIST and F. B. ESTABROOK, J. math. Phys. 16, 1 (1975). E. D. FACKDRELL, in Geometric Aspects of the Einstein Equations and Integrable Systems (Edited by R. MARTINI). Lecture Notes in Physics No. 239, Springer, Berlin (1985). F. B. ESTABROOK, Geometrical Approaches to Differential Equations (Edited by R. MARTINI). Lecture Notes in Mathematics No. 810, Springer, Berlin (1980). D. G. B. EDELEN, Applied Exterior Culcul~. Wiley, New York (1985). D. G. B. EDELEN, Isovectors Methods for Equations of Balance with Programs for Computer Assistance in Operator Calculations and Exposition of Practical Topics of Exterior Calculus. Sijthoff & Noordhoff, Alphen aan den Rijn, The Netherlands (1980). W. F. AMES, Nonlinear Partial Diflerential Equations in Engineering, Vol. I, p. 13. Academic Press. New York (1965). I. M. FRIEDLIN, SIAM 1. uppl. Math. 46, 222 (1986).