Linghai Zhang
 

We study the asymptotic stability of traveling wave solutions of
nonlinear systems of integral differential equations. It has been
established that nonlinear stability of traveling waves is equivalent
to linear stability. Moreover if
max\{Re$\lambda:\lambda\in\sigma(L),\lambda\neq0\}\leq-c_0$,
for some positive constant $c_0$, and $\lambda=0$ is an algebraically
simple eigenvalue of $L$, then the linear stability follows,
where $L$ is the operator obtained by linearizing a nonlinear system
about its traveling wave and $\sigma(L)$ is the spectrum of $L$.
The main aim of this paper is to construct Evans function for
determining eigenvalues of operators regarding traveling wave stability.

When considering multipulse solutions, certain components of the
traveling
waves cross their thresholds for many times. These crossings cause great
difficulty in the construction of Evans function. In particular we have
to solve an overdetermined system to define the Evans function.
By using the method of variation of parameters and by investigating
boundedness on $(-\infty,\infty)$ of eigenfunction candidates,
we find a way to define the Evans function. The zeros of the Evans
function coincide with the eigenvalues of the operator $L$.

By estimating the zeros of the Evans function, we establish
the asymptotic stability of the traveling wave (multipulse solutions)
of an example from synaptically coupled neuronal networks,
describing spatially structured activity.