Stationary and traveling pulses appear generically in the dynamics
generated by nonlinear partial differential equations (PDEs) of evolution
type. In the context of applications
they for instance represent pulses through nerves, signals through transmission
wires or optical fibers, solitary water waves, etc.
A typical example of a stationary pulse `pattern' in the Gierer-Meinhardt system,
a model that was originally introduced in the context of biological pattern
formation (`morphogenesis') [6], is shown in Figure 1.
The *existence* and
*stability* of solutions of pulse type to a given system is
a fundamental question in the study of nonlinear PDEs.

One-dimensional scalar parabolic equations
of reaction-diffusion type
are among the most simple nonlinear PDEs,

(1)

with
and a nonlinear function of .
A stationary pulse solution to this PDE corresponds to a so-called
*`localized'* solution of the ordinary differential
equation (ODE) associated to the
stationary problem:
. A `localized' solution
is a solution that is close to a critical point of the ODE
except on bounded -intervals. Here, a critical point of the ODE
corresponds to a solution
of the PDE (1) with
;
such a solution is called a (trivial) `background state'.
In the terminology of dynamical systems, a pulse solution corresponds to
a *homoclinic* solution of the ODE; it satisfies
.

The stability of the pulse can be established by linearizing around
the pulse, i.e. by introducing the eigenvalue
and
the perturbation
through
, substituting
this `Ansatz' into (1), and neglecting the nonlinear terms in :

(2)

This is a singular Sturm-Liouville eigenvalue problem, or a (linear)
Schrödinger equation.
Note that eigenvalue problems of this type also occur naturally in
quantum-mechanics [10]. The spectrum of
the differential operator
consists of essential spectrum,
in this case given by the set
and a finite number () of discrete eigenvalues
[11]. Thus,
the essential spectrum is determined by the background state [8].
A discrete eigenvalue corresponds to a localized and integrable
eigenfunction/perturbation . It can be expected
that the homoclinic pulse solution is an asymptotically stable
solution to the PDE if
and
for all
(note that
is always an eigenvalue with
eigenfunction
) - see also [8].

However, the largest eigenvalue of the linearized stability problem for a pulse solution of (1) is always positive. This implies that the pulse solution is unstable [8]. As a consequence, the pulse cannot be observed in applications, or in numerical simulations of (1). The proof of this assertion for instance follows immediately from singular Sturm-Liouville theory [11], or can be obtained by various other classical approaches, such as by the use of maximum principles. Moreover, the characteristics of the problem can sometimes be computed explicitly (with the aid of classical special function theory). For instance, for the case , which will be considered in the next section, , , and , , [10].