fulltext

A proposal for the analytical solution of stationary

one-dimensional reaction-diffusion equations

Departamento de Física, Facultad de Ciencias Exactas y Naturales Universidad Nacional de Mar del Plata

Deán Funes 3350, (7600) Mar del Plata, Argentina

(ricevuto il 23 Agosto 1996; approvato il 20 Novembre 1996)

Summary. — A method is proposed which allows to find exact stationary solutions of one-dimensional reaction-diffusion (1DRD) equations, whenever both the reaction term and the diffusion coefficient are known as functions of the spatial variable. The method relies on mapping the 1DRD equation onto a Schrödinger one.

PACS 02.30.Jr – Partial differential equations.

PACS 82.40 – Chemical kinetics and reactions: special regimes and techniques.

1. – Introduction

The dynamics of physical systems whose components diffuse and interact in a usually nonlinear but local fashion, has been the subject of a large amount of work during the two last decades [1]. Besides their obvious interest in many chemical applications [2, 3], such systems are paradigmatic of complex behaviour in

non-equilibrium processes [4]; therefore their study transcends the limits of physics and

chemistry, and applies equally well to areas such as biology [5] or economics [6]. It is well known that chemical systems which are locally in thermodynamic equilibrium, but hold far from chemical equilibrium, can undergo phase transitions to new stable states with striking behaviour. The emerging states may be either steady states in which the relative concentrations of the chemical components vary in space (Turing structures), uniform states in which the concentrations of some constituents vary in time (chemical clocks), or may even be nonlinear travelling waves in the concentration of some component. These states, which are metastable near equilibrium but become stable far enough from it, have been named dissipative structures by Prigogine. Their existence is maintained by entropy production and by a related flow of matter and energy from the surrounding [7]. The classic example of such a behaviour in chemical systems (and the best known realistic example of an excitable system) is the

Belousov-Zhabotinskii (BZ) reaction, whose overall effect is the catalytic oxidation of

malonic acid in an acidic bromate solution [8]. Spatial patterns occurring in the BZ 767

reaction are very convenient for experimental study [9]. A typical reaction-diffusion equation has the following form:

¯f(x , t)

¯t 4 ˜ Q

[

]

(1)

where g1is some constant, and the nonlinear function F(f) is called the reaction term.

D(f) is the diffusion coefficient which is usually taken as constant, hence leading to a

Laplacian (we shall not need this assumption).

Among reaction-diffusion models, bistable chemical systems have been extensively studied in connection with pattern formation and self-organization [1, 10, 11]. For spatially inhomogeneous bistable systems limited by diffusion, the evolution can be generically divided into two steps. During the first one, the relatively fast reaction processes determine the formation of spatial domains where the density is essentially constant and equals that of one of the two stable homogeneous states. In the second stage, these spatial domains evolve driven by diffusion: they grow or shrink, and eventually give rise to a completely homogeneous state. During this stage, the boundaries between domains (“domain walls”) behave as shape-preserving shock fronts with a well-defined velocity. Similar phenomena can be observed in monostable systems.

2. – The mathematical method

We are interested in finding stationary solutions to the one-dimensional version of eq. (1), which for ¯f

¯t 4 0 reduces to

]D[f(x) ] f(x)8 (81 g1F [f(x) ] 40 , (2)

where the primes indicate d

dx. Our strategy will be based on the following observation:

it is true that (in principle) the functions D(f) and F(f) are indeed functionals of x for an arbitrary f(x); but if we knew the solution f(x) to eq. (2), then we would also know

D(f) and F(f) as functions of x

(

)

[D(x) f(x)8]81g1F(x) 40 . (3)

Our aim in this work will be to cast the 1DRD equation (2) in the form of a Schrödinger equation

(

)

(

)

f(x) 4 [D(x) ]21 O2_{c(x) ,} (4)

which suppresses the terms in c 8(x). By defining

f (x) 421O2[ ln D(x) ]8 and _{a(x) 4g}1

F [f(x) ] f(x) D[f(x) ] ,

we arrive at the following equation:

c 9(x)1G(x)c(x) 40 ,

(6)

where the function G(x) is given by

G(x) 4a(x)2f2

(x) 1f 8(x) . (7)

Equation (6) can be regarded as the stationary 1D Schrödinger equation for a particle in a potential USch(x), with energy eigenvalue En, by identifying G(x) 4En2 USch(x) and c(x) 4cst

n(x) (we are assuming a discrete spectrum, for simplicity).

If f(x) represents the (absolute) concentration of some (chemical, say) species—or, in general, some density—it must be required to be positive in the domain of interest. That restricts the search to cst

0(x), since it is the only bound eigenstate which has no nodes for x  (x₂, x₁). Moreover, by normalization, cst

0(x) must be zero at x 4x1₂. This requires that a(x) F0.

In summary, given f (x) and G(x), there exists a function a(x) given by eq. (7), and the concentration will satisfy eq. (6).

3. – Some examples

In the following, two examples will be considered:

3.1. A polynomic G(x). – As a first example we consider the following function:

G(x) 4 _{2 g}b 2 g2x2

g

1_{2 g}b2

h

1 3 gx2, associated to a Schrödinger potential USch(x) 4

g2x2

g

2 g2

h

2 3 gx2.

The ground-state energy and wave function are given by [12]:

E04 b 2 g , (8) c0(x) Pexp

k

l

g

h

g

h

g

h

g

h

g

h

g

h

In the framework of Quantum Mechanics, a bound state implies the relationship a2 4 F g 2_. (12) By eq. (5), D(x) Pexp

k

k

1dx

ll

k

l

k

l

The concentration patterns f(x) for this example (assuming the coefficients g and b in USch(x) to be positive), are shown in fig. 1 a). Figure 1 b) depicts the spatial dependence of the “reaction-diffusion ratio” a(x) for this potential. Figures 2 a) and 2 b) illustrate the same magnitudes f(x) and a(x) for b 421 (in this case, g2

4 1 O2). In both cases, we have taken a 422, b41, c41 and d41 in G(x). It can be appreciated that f(x) is essentially localized at the minima of the “reaction-diffusion ratio” a(x). 3.2. An exponential G(x). – Let us now consider the case of the Morse potential

USch(x) 4S2(e22 x2 2 e2x) . (14)

The physical domain x  (x₂, x₁) is such that x_{24 0 and x14 Q. The ground-state} energy and wave function are [13]:

En4 2(S 2 n 2 1 O2 )2,

(15)

cst0(x) PLn2 S 22n21( 2 Se2x) exp [2(S2n21O2) x2Se2x] ,

(16)

where n 40, L2 S 21

0 ( 2 Se2x) are the Laguerre polynomials within S F1O2, and we choose the energetic origin at E04 0. Furthermore, we have

2S2(e22 x 2 2 e2x) 42f 2_(x) 4 1 1 2 d f (x) dx 1 a(x) . (17)

If we choose f (x) 4Ae2x_{, we obtain}

a(x) 42e22 x

g

h

g

h

The diffusion coefficient will be

D(x) PeAe2x , (19)

and the concentration pattern will be given by

f(x) Pexp [2AO2e2x_{] L}2 S 21

0 ( 2 Se2x) exp [2(S21O2) x2Se2x] . (20)

Finally, figs. 3a) and 3b) show, respectively, f(x) and a(x) for S 41O2 in eq. (14). In this case, it can be appreciated that f(x) is essentially localized at the maximum of the “reaction-diffusion ratio” a(x).

-4 -2 0 2 4 x 1.0 0.8 0.6 0.4 0.2 0.0 10 000 8000 6000 4000 2000 0 ()x ()x a f a) b)

Fig. 1. – a) Concentration pattern for a sextic symmetric potential USch(x), with positive coefficients. b) Reaction-diffusion ratio a(x) for a sextic symmetric potential USch(x), with positive coefficients. -4 -2 0 2 4 x 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00 12 000 10 000 8000 6000 4000 2000 0 ()x ()x a f a) b)

Fig. 2. – a) Concentration pattern for a sextic symmetric potential USch(x), with g positive and

b 421. b) Reaction-diffusion ratio a(x) for a sextic symmetric potential USch(x), with g positive and b 421.

0 10 20 30 40 x 50 8 6 4 2 0 ()x ()x a f a) b) 0.4 0.3 0.2 0.1 0.0

Fig. 3. – a) Concentration pattern for a Morse potential USch(x), with S41O2. b) Reaction-diffusion ratio a(x) for a Morse potential USch(x), with S 41O2.

4. – Conclusions

These two analytically solvable examples illustrate the fact that—for general one-dimensional reaction-diffusion (1DRD) systems—we can benefit from the knowledge of solutions to the 1D Schrödinger equation to extract meaningful static solutions. The method here proposed is a semi-inverse one, and the information that must be provided is—on the one hand—the spatial dependence of the diffusion coefficient, and—on the other—the spatial dependence of the reaction term. Then the ansatz in eq. (4) allows to map the original 1DRD equation into a (stationary) Schrödinger one, thus obtaining in a straightforward way—through the solution of an (obviously linear) eigenvalue problem—relevant information about a nonlinear one. In summary, the modus operandi would be the following: 1) Use the knowledge of ground-state solutions to the 1D Schrödinger equation to propose a function G(x); 2) Use the knowledge of D(x) (i.e. as a function of x) to propose a function f (x); 3) Construct the function a(x), given by eq. (7); 4) Use, e.g., the modified Riccati method [12] to obtain solutions f(x) to the “Schrödinger potentials” USch(x) associated with G(x); 5) If interested, find F(x).

Our aim in this paper has been to present the method and to illustrate its application with simple examples. In a forthcoming paper we shall deal with some other (physically more interesting) examples [13].

R E F E R E N C E S

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Bureau, New York) 1985.

[4] HAKEN H., Synergetics. An Introduction (Springer, Berlin) 1989. [5] MURRAY J. D., Mathematical Biology (Springer, Berlin) 1989.

[6] NELSON R. R. and WINTERS. G., An Evolutionary Theory of Economic Change (Harvard University Press, Cambridge) 1982; SILVERBERG B., in Thechnical Change and Economic

Theory, edited by G. DOSIet al. (Pinter, New York) 1988, p. 531.

[7] NICOLIS G. and PRIGOGINEI., Self-organization in Nonequilibrium Systems (Wiley) 1976. [8] BELOUSOVB. P., in Collection of Abstracts on Radiation Medicine (Medgiz, Moscow) 1958,

p. 145; ZHABOTINSKII A. M., Biophysics, 9 (1964) 329.

[9] FIELD R. J. and BURGER M. (Editors), Oscillation and Travelling Waves in Chemical

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[10] SCHLO¨GL F., Z. Phys., 248 (1971) 446; 253 (1972) 147.

[11] FIFEP. C., J. Chem. Phys., 64 (1976) 854; Mathematical Aspects of Reacting and Diffusing

Systems, Lect. Notes Biomath., 28 (Springer, Berlin) 1983.

[12] SALEML. and MONTEMAYORR., Phys. Rev. A, 43 (1991) 1169; MONTEMAYORR. and SALEML.,

Phys. Rev. A, 44 (1991) 7037.