Introduction

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,

$U_t = U_{xx} + F(U),$ (1)

with $U(x,t): {\bf R} \times {\bf R}^+ \to {\bf R}$ and $F(U)$ a nonlinear function of $U$. A stationary pulse solution to this PDE corresponds to a so-called `localized' solution $u(x)$ of the ordinary differential equation (ODE) associated to the stationary problem: $u_{xx} + F(u) = 0$. A `localized' solution is a solution that is close to a critical point of the ODE except on bounded $x$-intervals. Here, a critical point of the ODE corresponds to a solution $U(x,t) \equiv U_0$ of the PDE (1) with $F(U_0) = 0$; such a solution is called a (trivial) `background state'. In the terminology of dynamical systems, a pulse solution corresponds to a homoclinic solution $u_h(x)$ of the ODE; it satisfies $\lim_{x \to \pm \infty} u_h(x) = U_0$.

The stability of the pulse can be established by linearizing around the pulse, i.e. by introducing the eigenvalue $\lambda \in {\bf C}$ and the perturbation $v: {\bf R} \to {\bf C}$ through $U(x,t) = u_h(x) + e^{\lambda t} v(x)$, substituting this `Ansatz' into (1), and neglecting the nonlinear terms in $v$:

$[{\cal L}(x) - \lambda]v = v_{xx} + [F'(u_h(x)) - \lambda] v = 0.$ (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 ${\cal L}(x)$ consists of essential spectrum, in this case given by the set $\{ \lambda \in {\bf R}: \lambda \leqslant F'(U_0) \}$ and a finite number ($N$) of discrete eigenvalues $\lambda_N \leqslant ... \leqslant \lambda_1$ [11]. Thus, the essential spectrum is determined by the background state $U_0$ [8]. A discrete eigenvalue $\lambda_i$ corresponds to a localized and integrable eigenfunction/perturbation $v_i(x)$. It can be expected that the homoclinic pulse solution is an asymptotically stable solution to the PDE if $F'(U_0) < 0$ and $\lambda_i < \lambda_1 =0$ for all $i = 2, ..., N$ (note that $\lambda_i = 0$ is always an eigenvalue with eigenfunction $\frac{d}{dx} u_h(x)$) - 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 $F(U) = U^2 - U$, which will be considered in the next section, $U_0 = 0$, $u_h(x) = \frac32 {\rm sech}^2(\frac12 x)$, $N = 3$ and $\lambda_1 = \frac54$, $\lambda_2 = 0$, $\lambda_3 = -\frac34$ [10].