An evolutionary equation is one of the form
(11.3.1)
where the function f may be quite a complicated function of its arguments and could even depend on the spatial derivatives of u. Equations of the form (11.3.1) have already arisen many times in the context of biological modelling (see Chapter 4) and will be taken up again in Chapter 12.
The expression “simple evolutionary equations” used for this section is somewhat illusory. It does not mean that the problems we shall consider are particularly simple or easy, but refers to equations in which the function f depends only on the dependent variable u and is of a simple form such as a polynomial. Thus the equations we shall consider will be of the type
(11.3.2) If f(u) is linear, say
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where A and B are constants, then in many instances the equation (11.3.2) can be solved by the separation of variables method. However if, as in many of the applications considered in this book, f(u) is non-linear, say
(11.3.3)
then the problem is much more intractable. Indeed, it is not usually possible to obtain “general” analytic solutions and one must solve such problems numerically (see Mitchell and Griffiths, 1980). Despite this, however, many evolutionary equations have “special” or “particular” solutions, which are of fundamental importance to our understanding of biological phenomena modelled by evolutionary equations. In this section we shall explore some of these particular solutions and demonstrate their use in the section to follow.
Suppose that f(u) is a polynomial in u and that f(u) has real roots at u=α, β, γ,… Then it is obvious that in this case u=a, u=β, u=γ, etc., are all constant solutions of (11.3.2). Such solutions are of importance in the treatment of the pure initial value problem
(11.3.4)
Here, for example, it often happens that the solution u(x, t) evolves as t→∞ into one of the asymptotic states u=α, β, γ,…Which state is reached depends crucially on the form of the initial data and that of f. Indeed some of these constant asymptotic states are stable to small perturbations of while others are not.
As an example of this, consider the problem (11.3.4) with f(u) given by (11.3.3). Then it can be shown (see Section 11.4) that if
u(x, t)<a for all . From this it may be proved that u(x, t)→0 (exponentially) as t→∞. By a similar argument it can also be shown that if
supu(x, t)>a for all t>0 and that u(x, t)→1 as t→∞. In other words, the problem (11.3.4) has u=0 or u=1 as asymptotic states depending on the magnitude of the initial data. The number a is called a “threshold”
parameter.
Another fundamentally important set of solutions to equation (11.3.4) is the class of “travelling wave” solutions. Here we look for solutions of the form
where c is a constant, positive or negative, called the “wave speed.” If we make the transformation ξ=x+ct in (11.3.4) then we see that V(ξ) satisfies
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the non-linear ordinary differential equation
(11.3.5) where the primes represent differentiation with respect to ξ.
We have encounted travelling waves several times already (Chapters 4, 7 and 8) and have analysed them using the phase plane methods of Chapter 5. They are also of fundamental importance in the study of asymptotic states of the problem (11.3.4). Indeed we may ask the question: When does the solution u(x, t) “evolve” into the travelling wave V(x+ct)? Mathematically we can state the problem as: determine conditions on the initial data so that
In practice, there may be several such travelling waves, some stable and some unstable, and the determination of the appropriate stable travelling wave is often a challenging task.
Travelling waves can be classified as follows: (a) wave trains—V(ξ) periodic;
(b) wave fronts—V(−∞) and V(∞) exist and are unequal;
(c) pulses—V(−∞) and V(∞) exist and are equal and V(ξ) is not constant.
Apart from the use of phase plane methods the problem of finding travelling wave solutions is usually solved by numerical methods. However, in some cases, if we are fortunate, travelling waves can be determined analytically.
Example 11.3a Here
(11.3.6) where
In this example V(−∞)=0, V(∞)=1 and so in this case V(ξ) is a wave front travelling from right to left with
speed .
Example 11.3b
(11.3.7)
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If we make the substitution ξ=x+ct then we obtain equation (11.3.6) with f defined by (11.3.8). Again we look for a non-decreasing (i.e., monotone increasing) solution satisfying the conditions 0≤V(ξ)≤1, V(−∞)=0, V(∞)=1.
As long as we have to solve which has the solution
(11.3.8) where A and B are constants to be determined and
provided c2≠4. In order to satisfy the condition V(−∞)=0, it is clear that we must have c>0. If c<2 then (11.3.9) is oscillatory and so we would not have a monotone solution. Consequently we must have c≥2. Suppose c>2. Then we have a solution monotone increasing with ξ. If without loss of generality V=1/2 when
ξ=0 then we require
(11.3.9) For we must solve the equation
which has the general solution
(11.3.10) where D, E are constants to be determined and
The requirement V(∞)=1 demands that D=0. In addition, the condition gives i.e.,
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Furthermore, if we require V to be continuous at ξ=0 then a differentiation of (11.3.9) and (11.3.11) provides the extra condition
(11.3.11) Equations (11.3.9), (11.3.10) can now be solved to give
Thus with A, B and E determined, (11.3.8), (11.3.10) give the complete solution to the problem for any c>2. That is,
Now suppose c=2; then the general solution (11.3.8) must be replaced by one of the form (11.3.12)
and (11.3.10) becomes
(11.3.13) By imposing the boundary and continuity conditions we have
and i.e.,
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Then, in this case
In conclusion then, (11.3.4) with f given by (11.3.7) has a one parameter family of wavefront solutions for any wave speed c≥2.
Another class of solutions to (11.3.4), which is often useful, are those independent of x. In this case we have to solve the ordinary differential equation
(11.3.14) Suppose we have the initial condition
then (11.3.14) can be solved implicitly as
(11.3.15) provided u0 and u are such that the integral exists.
Example 11.3c
Suppose f(u)=u(1−u). Then (11.3.15) has the solution (u0≠0)
We therefore obtain the explicit solution
This solution shows that if 0<u0<1 then u(t)<1 and u→1 as t→∞. Similarly if u0>1 then u(t)>1 and again u→1 as t→∞. Note that if u0=0 then u(t)≡0 and if u0=1 then u(t)≡1.
Finally consider those solutions of (11.3.4) that are independent of t. In this case we have to solve the ordinary differential equation
(11.3.16)
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In many situations it is possible to solve this equation in the following manner: if then (11.3.16) can be expressed as
i.e.,
for some arbitrary constant A. Thus if we set then
(11.3.17)
and this first-order differential equation can be solved, at least in principle, in the same way as equation (11.3.14).
Example 11.3d
Consider the positive solutions to the boundary value problem
(11.3.19)
If u is a solution to this problem with u>0 for −l/2<x<l/2 and since u=0 at the end points it follows that u must take on its maximum value m at some intermediate point x=a where −l/2<a<l/2. In other words, 0<u(x)≤u(a)=m for −l/2<x<l/2 and u′(a)=0, u″(a)≤0. From this observation, equation (11.3.18) shows that and so 0<m≤1. In fact 0<m<1 for if m=1 then u(x)=1 is the unique solution to (11.3.18) with u(a)=1, u′(a)=0. However, such a solution does not satisfy the boundary conditions (11.3.19). With 0<m<1 we see that u″(x)<0 for all x in the interval −l/2<x<l/2 and so u(x)<m except at x=a.
Let us now attempt to construct the solution to the boundary value problem via the method outlined above. Here we can write
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and so (11.3.17) can be written as
When x=a, and so A=2F(m). Thus
(11.3.20)
Notice also that F is a strictly increasing function of u for 0<u<1. Thus if x≠a we have from (11.3.20) In the first case
(11.3.21) and in the second
(11.3.22) The implicit solution defined by (11.3.21), (11.3.22) is not yet in a desirable form since it involves the two unknown constants a and m. On using the boundary conditions we have
and these identities are only compatible if a=0. That is, u(x) takes its maximum value at the midpoint of the interval −l/2≤x≤l/2. With a=0 we see that l and m are related through
(11.3.23) In summary then, the solution to the boundary value problem is given implicitly by the formula
(11.3.24) where m, as a function of l, is given by (11.3.23).
Further important information can be obtained about the problem if we analyse the relationship between m and l defined by (11.3.23). In particular it can be shown that
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(a) l is an increasing function of m for 0≤m<1; (b) l→∞ as m→1 from below;
These facts imply that there is no positive value of m satisfying (11.3.23) if l<π while for l≥π, m increases from 0 to 1 as l increases from π to ∞. Thus, for l≤π, (11.3.18) and (11.3.19) have only the trivial solution u=0, but for l>π the solution (11.3.24) appears. This type of behaviour is fundamental to the subject of
bifurcation theory and l=π is called the bifurcation point.