Physica 3D (1981) 637-643 North-Holland Publishing Company

SYMMETRIES OF NON-LINEAR REACTION-DIFFUSION EQUATIONS AND THEIR

SOLUTIONS

W.-H. STEEB Uniuersitiit Paderborn, Theoretische Physik, D-4790 Paderborn, Fed. Rep. Germany

and

W. STRAMPP

Gesamthochschule Kassel, Fachbereich Mathematik, D-3500 Kassel, Fed. Rep. Germany

Received 10 June, 1980 Revised 27 January 1981

This paper is concerned with the symmetries of a certain class of non-linear reaction-diffusion equations. The symmetries are used for deriving solutions of these equations. Subsequently, we compare the solutions with those given by other authors.

1. Introduction

Non-linear reaction-diffusion equations have been considered a great deal in the literature [l-lo]. The evolution of the system will be described by the following non-linear coupled system of diffusion equations:

g = DV*c + F(c). (1.1)

D denotes an n x II matrix of diffusion coefficients and c(x, t) is an n-column vector of concentrations. c(x, t) depends on time t and the spatial position vector (x1, x2, x3). F(s) is a (-1 smooth map from R” into W”, and V2 is the n-dimensional Laplace operator. The com- ponents of c are the concentration of certain chemical species which react with each other at each point and also diffuse through the medium. The equation &z/at = F(c) governs the chemical kinetics of the system of reactions. Thus F(c) is a polynomial of the n-tuple c = (cl,. . . , c,). We assume that eq. (1) does not depend explicitly on time and spatial coordinate.

In most of the papers [l-lo] the reaction- diffusion equations have been investigated for certain boundary and initial conditions. The main object was to find dissipative structures [l-5]. Another aim was to develop perturbation techniques [6]. Plane wave solutions to reaction diffusion equations have been studied by Kopell and Howard 171. Obviously, in most cases ex- plicit solutions of eq. (1) cannot be found.

So far, the application of the theory of trans- formation groups to non-linear dissipative par- tial differential equations, which play an im- portant role in biomathematics, has been over- looked. The present paper concerns the Lie theory of transformation groups and solutions of non-linear reaction-diffusion equations. We note that in mathematical physics this method of solving partial differential equations has been applied a great deal [ll-161. Recently, one of the authors developed a method within this theory for obtaining limit cycles in non-linear autonomous systems of differential equations [17, 181.

There are two possibilities for describing

0 167-2789/8 1 /ooocroooO /$02.50 @ North-Holland Publishing Company

638 W.-H. Steeb and W. Stramppl Non-linear reaction-ditusion equations

similarity methods for differential equations. The traditional approach, working with vector fields only, has been established by Ovsjannikov [II], Bluman and Cole [12], and Ames [13]. Ovsjannikov proceeds from the fact that a differential equation has a symmetry group or invariance group when solutions of the differential equation are carried into solutions by the transformation group. Classical examples of symmetry groups are rotations. A differential equation with rotational symmetry has rota- tional symmetric solutions. For arbitrary sym- metries the notation similarity solution has become convenient. In the modern approach, the partial differential equations are cast into an equivalent set of differential forms, and the in- variance is investigated in a more systematic way using the Lie derivative of the differential forms with respect to a vector field [14-161. The vector field is an infinitesimal generator which yields via a Lie series the associated trans- formation group. With the aid of the Lie derivative we are able to express the invarance of the partial differential equations, under the transformation group.

The model under investigation is that of Dreitlein and Smoes [8]. This model is given by

ac, 2 -=~+fdc,, CA at

ac2 -=~+f*(CI.c2), at

where

fdc,, c2) = CdcL - d - c:> + 2%

f2(CI, c2) = c2(1* - d - 4 - 2Cl.

cr(x, t) and c2(x, t) denote the concentration of certain chemical species. The quantities cr(x, t), c2(x, t), x and t are given so that they are dimensionless (compare Rosen [93). F is a real parameter.

In section 2 we briefly describe the properties of the system given above without diffusion.

In section 3 the mathematical notion of symmetry or group invariance is developed in the language of vector fields, differential forms, and intrinsic operators on forms, such as the Lie derivative.

Solutions of the Dreitlein-Smoes model within the developed approach are derived in section 4. Moreover, we give a discussion of the solutions.

2. System without diffusion

Without diffusion we have the autonomous system of differential equations

dcl- dt - C,(F - c: - c;, + 2cz,

(2.1)

dC? dt = c*(p - c: - c:> - 2c,.

This system has one critical point (time-in- dependent solution or steady state solution), namely (0, 0) for all values of F. If p < 0, then this critical point is stable. In other words the origin is an attractor. This can be proved using the Liapunov function V = (CT+ ~$2. It fol- lows that V = r2(p - r2). If I_L < 0, then V < 0. The origin becomes unstable when p > 0. The system (2.1) has a stable limit cycle, which is given by c: + c: = CL. In the present case all trajectories outside and inside the limit cycle tend to the limit cycle.

Other non-linear systems with two degrees of freedom and limit cycle behaviour can be cast in the form given above [17].

For the system given above a chemical reac- tion scheme has been proposed by Dreitlein and Smoes [8]. The chemical reaction scheme con- tains autocatalytic steps.

W.-H. Steeb and W. StrampplNon-linear reaction-diffusion equations 639

3. Mathematical approach

In this section we study the reaction-diffusion equations given by eq. (1.1) within the geometric approach, where we apply the modern for- mulation [14-161.

The first step is to cast the system of partial differential equations into a system of partial differential equations of first order, and the second is to find an equivalent set of differential forms. With the help of this equivalent set of differential forms we formulate the concept of invariance for the partial differential equations. For the sake of simplicity we consider the case with one spatial coordinate and where c = (cl, cJ. Moreover, we put Q1 = Dz2 = 1 and D12 = Dzl = 0. The extension to more than two species and more than one spatial coordinate is straightforward. Hence, we study the following’ partial differential equations:

v = “‘$’ t)+ F,(c& t), c&z, t)),

v = “2c;($ t, + Fz(c,(x, f), cz(x, f)).

Introducing the abbreviations (i = 1,2)

&(X, t) = v,

(3.1)

(3.2) aci(X, t)

we find that (i = 1,2)

Ui(X, t) = y + Fl(c,(x, t), c~(x, t)). (3.3)

The system of partial differential equations given by eqs. (3.2) and (3.3) is now cast into an equivalent set of differential forms [14]. In the eight-dimensional space of the variables {x, t, cl, c2, Us, ~2, ulr u2} we adopt the basis one-forms

WI,. . . , dv2}. The set of first order partial differential equations given by eq. (3.2) and (3.3) may be associated with the following differential forms (i = 1, 2):

cti = dci - Ui dt - Vi dx,

(3.4) pi = Fi(cl, ~2) dx A dt - Uidx A dt - dui A dt,

where A denotes the exterior product of differential forms. Any regular two-dimensional solution manifold {Ci(X, t), Ui(X, t), v~(x, t)} (i = 1, 2) in the eight-dimensional space satisfying eqs. (3.2) and (3.3) will annul this set of formsiffkw, as may be verified by sectioning the forms into the solution manifold. We have

dci(X, t) = 2 dx + 2 dt (3.5)

and similarly for Ui and ui. Let the sectioned forms be denoted by a tilde. Then we have

(3.6)

Thus C$ = 0 and pi = 0 (i = 1, 2) yield the partial differential equations given by eqs. (3.2) and (3.3) since dx and dt are basic one-forms and dx A dt is a basic two-form. For investigating the symmetries of the partial differential equa- tions (3.1) (or 3.2 and (3.3)) we consider now the oneforms ai and two-forms &(i = 1, 2) given by eq. (3.4). The condition of invariance of the partial differential equations (3.2) and (3.3) can now be described with the help of the differen- tial forms ai and pi and with an infinitesimal generator V (sometimes called a vector field) which generates via a Lie series a transfor- mation group. The invariance condition is now formulated with the help of the Lie derivative. For a differential form y the Lie derivative Lzy

640 W.-H. Steeb and W. Strampp/Non-linear reaction-difusion equutions

may be viewed as the propagator of the form y down the trajectories of the vector field 2. Lz( *) maps a p-form into a p-form. Lz( *) is a linear operator and satisfies the product rule with res- pect to the exterior product A. The definition [la] of the Lie derivative of a differential form with respect to a vector field is not useful for practical calculations rather we use the indentity

Lzy = d(Z_Jy) + ZAdy. (3.7)

dy denotes the exterior derivative of the differential form yd( - ) is a linear operator and maps a p-form into a (p + 1)-form. Z_l y denotes the inner-product (sometimes called contraction) of the vector field Z with the form y. Z _I (s) is a linear operator and maps a p- form into a (p-1)-form. The invariance of the partial differential equation can now be for- mulated as follows: We wish to find infinitesimal generators (sometimes called isogroups) that the Lie derivative of ai and pi with respect to the infinitesimal generators is in the idea1 of (Y~, da;, pi. In this case the partial differential equations (3.2) and (3.3) are invariant under the trans- formation group generated by V. The trans- formation group generated by the vector field V is a symmetry group which carries solutions into solutions. The knowledge of such vector fields can now be used for deriving similarity solu- tions of the partial differential equations (3.2) and (3.3). What we do is the following: Let

VI,. . ., V, be vector fields such that the Lie derivative of the differential forms ai and pi with respect to V,, . . . , V, is the ideal of ai, dczi,

pi, and dpi. We annul the forms Zi ~i(Vi J ai) and consider the section. This leads to a system of linear partial differential equations. With the help of these equations we obtain similarity solutions.

4. The reaction-diffusion model

Now we wish to study the reaction-diffusion mode1 given by eq. (2.1) with the help of the

approach described above. We consider the differential forms (i = I, 2)

ai = dci - Ui dt - pi dx,

(4.1) pi = fi dX A d t - Ui dx A dt - dtii A dt,

and the vector fields (infinitesimal generators)

x=&,

T=+, (4.2b)

d -v2au,. (4.2~)

The vector field X is associated with the trans- formation group X-+X-tEl. (t-t, Cl-+

Cl,. . ., v2+ vz). The vector field T is associated with the transformation group t +

t + E?(X + x, . . . ) v2+ VZ). Finally, for the vector field Z we have the transformation group

cl-+clcos~2-cpsin~3,

c2-+c2 cos l 3+ cl sin l 3,

ul-+ulcos~3-u2sin~3,

u2-+ u2cos l 3+ ulsin l 3,

vI+vIcos~3-v2sin~3,

~~+v2cos~3+v~sin~~,

(4.3)

and x+x, t+t. We mention that the vector fields defined on

Iw’ from a basis of an Abelian Lie algebra. We now show that the partial differential equations given by eq. (3.1) and (3.2) are invariant under the transformation groups given above. For this purpose we have to calculate the Lie derivative of the differential forms ai and &(i = 1, 2) with respect to the vector fields given by (4.2). We find [15]

W.-H. Steeb and W. Stramppl Non-linear reaction-diffusion equations 641

Lxa, = 0, Lxaz = 0;

LTCY, = 0, l&a* = 0;

LXP, = 0, Lx/32 = 0; (4.4)

LTPl = 0, LTh! = 0;

Lza, = ff2, LZCQ = -(Y1;

LzP1= -P2, LzP2 = PI.

According to the discussion given in section 3 the partial differential equations given by eq. (3.1) and (3.2) are invariant under the following transformation groups: x +x + cl; t + t + e2 and the transformation group given by (4.3). The invariance under the space and time translation is due to the fact that the differential equations (3.1) and (3.2) do not depend explicitly on x and t. Z represents the rotational symmetry of the partial differential eqs.

In order to find similarity solutions we con- sider a linear combination of the infinitesimal generators X, T and Z, i.e.

V = a,X + a2T + a3Z, (4.5)

where al, a2, a3 E R. The contraction (inner product) of the infinitesimal generator V with the one-forms (Y~ and CQ, respectively, yields the O-forms (functions)

V _I lyI = -alul - a2ul - a3c2,

(4.6)

V J (~2 = -alu2 - a2u2 + a3cl.

To obtain a similarity solution, we annul the forms V Jai and V Ja2 and obtain a linear system of partial differential equations, namely

at acdx, t) acdx, t) _

at +a2 ax - a3c2,

(4.7)

aI aczcx, t) ) a

2 3c2(x, t) a3c, ---=

at ax

Let al 3 0. Then we find as the general solution

to eq. (4.7)

cdx, t) = G(T) cos(a3tlaJ - ?2(77) sh(a3tlaA

(4.8)

c2(x, t) = Cd77) sin(a3tlaJ + C2(77) cos(a3t/a&

where 7 = arx - a2t. Now let a2 f 0. Then we find as the general solution to eq. (4.7)

cdx, t) = FdO cos(a3xla2) - E2(5) sin(a3xla2),

(4.9) c2k t) = G(5) sin(a3xla2) + C2(f) cos(a3xla21,

where 5 = a2t - alx. Inserting eq. (4.8) into eq. (1.2) and after some algebraic manipulation we obtain (’ = d/dq)

(4.10)

-a2Ei+zCl - C’i-f2(C,, 152) = 0.

In order to find a solution of the autonomous system (4.10) which can be given explicitly we

Put

El =o, a2 = 0, al= 1, a3=2.

Then eq. (4.10) reduces to

(4.11)

CI + Cz(/_& - C:) = 0. (4.12)

This non-linear ordinary equation can be exactly solved [19] and we find the solution given by Rosen [9] for the model (1.2). The solution takes the form

cj(x, t) = g(x) cos(2t); ct(x, t) = -g(x) sin(2t), (4.13)

where

g(x) = k (k : l -->“‘sn [ (j$$2(x - x0), k].

642 W.-H. Steeb and W. Strampp/ Non-linear reaction-diflusion equations

sn is the Jacobi elliptic function of first kind

[191. Let us briefly discuss this solution. x0 and k

are disposable real constants of integration which must be adjusted to satisfy the boundary condition. In particular, note that sn(x, k = 0) = sin x and sn(x, k = 1) = tanh x. The solution describes dissipative structures (time oscillation and spatial structure). Chemical dissipative space structures have been observed experi- mentally in the case of the Zhabotinski reaction [20]. It is possible to impose boundary con- ditions of Dirichlet type. This means, let R be a one-dimensional finite region, then CI(X, t) = cz(x, t) = 0 when x E aR. aR is the boundary of R. F = 0 is a bifurcation point. In this case the solution vanishes.

In order to find further solutions of eqs. (4.10) we must solve these equations numerically. The equations can be cast into an autonomous sys- tem of four ordinary differential equations of first order. It seems that chaotic behaviour can appear in these equations.

Inserting eq. (4.9) into eq. (1.2) and after some algebraic manipulation we obtain (’ = d/dl)

2

a*ci - a:c; + 2 F2 + 2ala, _ -cc;-f*(C,,C*)=O.

a2

Now let us put a3 = 0 and a2 = 2. We obtain

2l?; - a:?;, - f,(El, l?z) = 0, (4.15)

2Ei - a& - f2(& E2) = 0.

As a particular solution to (4.15) we find

c,(x, t) = (CL - k2)“* cos(kx + 2t); c2(x, t)

= -(p - k2)“* sin(kx + 2t). (4.16)

This solution has also been described by Dreit- lein and Smoes [7].

Let us now discuss the solutions given by eq. (4.16). Solution (4.16) describes a propagating wave [8]. In contrast to the linear wave, the amplitude of the wave depends on the wave- number k and the parameter CL. If I_L < 0, then the solution ceases to exist. Moreover k2 < p(p >O). Since A =2r/k we find A >2rriq/c*-. Consequently, there is a cutoff wave length

A, = -&.

The phase velocity of the wave, namely u = lw]/k has a lower limit u, = 2/d/cL. All waves must propagate with at least this velocity. Here it is worthwhile to mention a theorem due to Howard and Kopell [7].

Theorem: If c = F(c) be an autonomous system of ordinary differential equations. If c = F(c) has a stable limit cycle, then there is a one-parameter family of plane waves of the diffusion equation &/at = a2c/ax2 + F(c). i.e. c;(x, t) = ?i(kx - wt), where Fi is a 2a-periodic function of its arguments.

Hence the solution given by Dreitlein and Smoes is an application of the theorem des- cribed above.

In order to find further solutions of the sys- tem (4.15) again we are forced to solve these equations numerically.

A comment about the ordinary differential equation (4.15) is in order. The equation can be cast into an autonomous system of differential equations. Now the right-hand side of these differential equations is associated with a vector field. Here we can use symmetries in order to obtain first integrals of this autonomous sys- tem. Let V be the vector field which is asso- ciated with the right-hand side. Let Z be an- other vector field such that LzV = 0, where LzV = [Z, V] ([ , ] commutator). The know- ledge that the Lie derivative of the vector field V with respect to the vector field Z vanishes can be used to find first integrals [13, 211. We notice that the same approach can be performed for the eq. (4.10).

W.-H. Steeb and W. StrampplNon-linear reaction-diffusion equations 643

To sum up, we have shown that the solutions given by Rosen [9] and Dreitlein and Smoes [8] are special cases of the solution obtained by applying the technique described above.

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J.F. Auchmuty and G. Nicolis, Bull. Math. Biol. 37 (1975) 323. G. Rosen, Bull. Math. Biol. 37 (1975) 277. G. Rosen, Bull. Math. Biol. 40 (1978) 853. M. Ashkenazi and H.G. Othmer, J. Math. Biol. 5 (1978) 305. J.D. Murray, Lectures on Non-linear Differential Equation Models in Biology (Clarendon Press, Oxford, 1977). Y. Kuramoto and T. Yamada, Progr. Theor. Phys. 55 (1976) 412. L.N. Howard and N. Kopell, SIAM-AMS-Proc. Vol. 8 (1974) 1.

[8] J. Dreitlein and M.-L. Smoes, J. Theor. Biol. 46 (1974) 559.

[9] G. Rosen, J. Theor. Biol. 54 (1975) 391. [IO] M. Herschkowitz-Kaufmann and G. Nicolis, J. Chem.

Phys. 56 (1972) 1890. [ 1 I] L. Ovsjannikov, Gruppovye Svostva Differentsialng

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[13] G.W. Bluman and J.D. Cole, Similarity Methods for Differential Equations (Springer, New York, 1974).

[14] B.K. Harrison and Estabrook, J. Math. Phys. 12 (1971) 653.

[15] W.-H. Steeb, Lett. Nuovo Cimento 22 (1978) 159. [16] W.-H. Steeb, Phys. Lett. 69A (1978) 159. [17] W.-H. Steeb, J. Phys. A 10 (1977) L221. [18] W.-H. Steeb, Physica 95A (1979) 181. [19] H.T. Davis, Introduction to Non-linear Differential and

Integral Equations (Dover, New York, 1962) p. 207. [20] H. Busse, J. Phys. Chem. 73 (1969) 750. [21] W.-H. Steeb, Int. J. Theor. Phys. 16 (1977) 671.