Figure 1: The homoclinic pulse solution of (4) for and . Here, the scaled pulses and are plotted on the long spatial scale . The singular `sharp' pulse represents the -component, it is close to a homoclinic solution of the scalar equation (1) with (Theorem 1). This figure has been obtained by direct numerical simulation of (4) using the moving-grid code described in [2].

Although pulse solutions of scalar equations
can thus never be stable,
there are many systems of PDEs,
such as the Gray-Scott model for autocatalytic
reactions [7] and the Gierer-Meinhardt
model, that exhibit pulse solutions that are
very similar to scalar pulses, see Figure 1. These systems can be written
in the following form

(3)

where
is an asymptotically small parameter. Since
both and are assumed to be bounded for
- a natural condition for solutions to PDEs on unbounded domains -
it follows that
, a constant, as
. Hence,
(3) reduces to (1), with
in
the limit
. Note that a coupled system of reaction-diffusion
equations in which the ratio of the diffusion coefficients can be
assumed to be small, as is the case for the Gray-Scott and the
Gierer-Meinhardt model,
can always be written in the form (3).

The limit
intuitively yields two conclusions
for the situation
:
*(i)* it can be expected that (3) indeed has solutions
of pulse
type of which the -component
is close to a pulse solution of a
scalar PDE (1); *(ii)* the pulse solution
can also not be stable: the spectrum of the linearization
around
should be close to that of the scalar case
(which has an
unstable eigenvalue). In general,
neither of these assertions is (completely) true. Moreover,
both the existence *(i)* and the stability *(ii)* question
require the development of novel ideas and techniques. For instance,
the scalar problem (1) with
will have
a pulse solution for open -sets, the limit procedure is expected
to select only a discrete number of the 's and the corresponding
pulses
. However, the ODE associated to the stationary problem
for the PDE (3) is a dynamical system in . A priori,
there are no general techniques by which the existence of homoclinic orbits
in such problems can be established.
Furthermore, the stability problem yields a
linear system that is neither of Sturm-Liouville type, nor self-adjoint,
which implies that one should expect complex eigenvalues .
Moreover, the Sturm-Liouville equation (2) associated to the scalar
limit indeed is a singular limit,
the `perturbations' due to the coupling to
the additional slow diffusion equation for have a leading order
effect on the eigenvalues. In other words: the coupling of
to a slow diffusion equation for might be able to
bring the unstable eigenvalue
of the scalar limit problem to the stable
half-plane.

As an example we present here two theorems that settle both the
the existence problem and the stability problem
in the special case of the classical Gierer-Mienhardt problem,

(4)

It should be remarked, however, that this case is less special
than suggested by comparing (3) to (4). In fact,
(4) can be seen as a prototypical system: under certain
(generic) conditions equation (3) can be scaled into a
*normal form* of the type (4) [3].
Essential in the derivation
of the normal form is that both the -component
as well as the -component of the pulse solution to
(3) scale with a certain negative power of
as
, i.e. the amplitude of the pulse
is in general asymptotically large [3].

**Theorem 1.** [3] *For any there exists an
such that for
all
there
exists a homoclinic pulse solution
to (4). Both components are of
, i.e.
both
and
exist (and are not 0). Moreover,
,
the homoclinic solution of the associated
scalar limit problem (1) with
and
.*

See Figure 1.
The proof of this theorem is based on the ideas of *geometric singular
perturbation theory*, see [9].

**Theorem 2.** [3] *The homoclinic pulse solution
of (4) is
unstable for
, and
spectrally stable for
.*

The proof of this result relies heavily on the *Evans function approach*
as was developed in [1]. This method was extended in [4]
to systems of the type (3),
in the context of an explicit model problem - the Gray-Scott equation.
Later, this method
has been generalized, so that it is possible to consider
the general equation (3). Note that
one can go from a situation where the above intuitive argument *(ii)*
is valid to a bifurcation that contradicts the intuition,
by varying the parameter : if is small enough
there is a unstable eigenvalue
that merges with the unstable eigenvalue
of the
scalar limit problem (2) in the limit [3].
However, at a certain critical value
merges with another real
(positive) eigenvalue
and a pair of complex conjugated eigenvalues is formed. This pair crosses the
imaginary axis as passes through
. Furthermore,
it follows from the methods developed in [4,3] that the number
of eigenvalues of the linearized stability
problem associated to (4) is , two of which can be complex, one more
than that of the limit problem (2), which only has real eigenvalues.
Hence, the relation between (4) and its scalar limit (1)
with
is more singular than expected at first
sight.