\documentstyle[12pt]{article}
\setlength{\parindent}{0.2in}
\oddsidemargin .5in
\evensidemargin 0in
\topmargin 0in
\headheight 0in
\headsep
\headheight
\textwidth 5.75in
\textheight 23cm
\footskip 21.0mm
\footheight 10mm
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\renewcommand{\baselinestretch}{1.4}

\begin{document}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\ep}{\mbox{$\varepsilon$}}
\newcommand{\fracd}[2]{\frac{\displaystyle{#1}}{\displaystyle{#2}}}
\newcommand{\half}{\mbox{$\frac{1}{2}$}}
\newtheorem{thm}{\em\bf Theorem}
\newtheorem{lem}{\bf Lemma}
\begin{center}
{\large \bf Existence and construction of
travelling wavefront solutions of Fisher equations with fourth order
perturbations
  } \\ 
\vspace*{1cm}
S.A. Gourley and M.V. Bartuccelli \\
\vspace{4mm}
Department of Mathematics and Statistics,
University of Surrey,
Guildford,
Surrey GU2 5XH, U.K. \\
\end{center}

\vspace{.4in}
{\bf Abstract:} We consider a generalised Fisher equation containing a
fourth order spatial derivative term. Of interest is the question
of the existence of travelling front solutions of the equation and
their qualitative form. When the fourth order term has sufficiently
small coefficient existence of such fronts is shown using invariant
manifold theory. Asymptotic analysis is employed to construct an
expression for the travelling front when its speed is large.

\vspace{.4in}


All correspondence to Dr. S.A. Gourley at above address

\vspace{.1in}
{\bf E-mail}: s.gourley@mcs.surrey.ac.uk \\
{\bf Fax}: 01483 876071

\newpage

\setcounter{equation}{0}
\section{Introduction}
It is well known that Fisher's equation
\be
\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}+u(1-u)
\label{eq1}
\ee
admits travelling front solutions which connect the two uniform steady
state solutions $u=0$ and $u=1$. Indeed, setting $u(x,t)=U(z)$ where
$z=x-ct$, these fronts satisfy the ODE
\be
U''+cU'+U(1-U)=0.
\label{eq2}
\ee
If the boundary conditions for the latter are taken as $U(-\infty)=1$
and $U(\infty)=0$ then the waves are necessarily rightward moving
($c>0$) and, furthermore, if $c\geq 2$ the waves are strictly monotonically
decreasing for all $z\in(-\infty,\infty)$. 

It is the object of the present paper to discover to what extent these
travelling fronts are preserved when one adds a fourth order spatial
derivative term to~(\ref{eq1}), so that the equation becomes
\be
\frac{\partial u}{\partial t}=-\ep^2\frac{\partial^4 u}{\partial x^4}+
\frac{\partial^2 u}{\partial x^2}+u(1-u).
\label{eq3}
\ee
Our approach, after converting~(\ref{eq3}) to travelling wave form, is
first to show that an invariant manifold exists for the
four-dimensional system of ODE's that the desired front must
satisfy. The question of the existence of the front in this manifold
is then resolved using Fredholm theory. When $\ep$ is small the front
still looks like the Fisher front. We then compute how
the front changes qualitatively when $\ep$ is not so
small. Perturbation theory is used to obtain an asymptotic expression
for the front when the wave speed $c$ approaches infinity.

Equations like~(\ref{eq3}) are not entirely new. A slight
generalisation of~(\ref{eq3}) has been studied by Cohen \&
Murray~\cite{Cohen81} as a possible model for pattern formation in a
single species population. They carry out linear stability and
bifurcation analyses about the uniform steady states. Their Laplacian term
$u_{xx}$ can be 
given a negative coefficient, when the term has a 
destabilising effect on a uniform steady state. On the other hand, a
fourth order term with a negative coefficient always has a stabilising
influence. They conclude that the interplay between these two terms can
be a mechanism for spatial pattern formation. Biologically, fourth
order derivative terms are considered to model {\em long range}
effects. Murray~\cite{Murray93} explains how equations with fourth
(and even higher) order derivatives can be approximations of equations
with {\em nonlocal} terms.

There are, in fact, various papers that study equations
like~(\ref{eq3}) from the point of view of the steady
solutions. For example, Peletier and Troy~\cite{Peletier} show that
fourth order diffusivity can give rise to chaotic steady spatial
patterns.
There are, however, fewer papers that consider the question of the
existence of travelling fronts and their qualitative
profiles. However, Dee \& van Saarloos~\cite{dee88} consider
equation~(\ref{eq3}) with a cubic nonlinearity.
Akveld \& Hulshof~\cite{Akveld} consider a fourth order equation
similar to~(\ref{eq3}) but again with a different nonlinearity. The
latter authors, like
us, use dynamical systems theory to construct an invariant manifold for the
travelling wave equations. However, their approach to confirming the
existence of a travelling front within the manifold is quite
different from ours, involving the use of the implicit function
theorem. In the present paper, we prove existence of the front by employing
the Fredholm alternative. Further, the classical asymptotic analysis
we use later in this paper is not restricted to $\ep$ small.

Methods similar to those of the present paper may also be used to
prove existence of travelling front solutions to certain delay
equations which can be reformulated as systems which are of higher
order but without delay, and such that the ODE's for travelling
front solutions will have order three or higher~\cite{Ashwin}.

\setcounter{equation}{0}
\section{Preliminaries}
If we convert~(\ref{eq3}) to travelling wave form, by writing
$u(x,t)=U(z)$, $z=x-ct$, we obtain
\be
-\ep^2U''''+U''+cU'+U(1-U)=0
\label{eq4}
\ee
and, defining $\phi=U'$, $p=\phi'$ and $q=\ep p'$, we can
recast~(\ref{eq4}) as a system:
\be
\begin{array}{l}
U'=\phi, \\ \phi'=p, \\ p'=\frac{1}{\ep}q, \\
q'=\frac{1}{\ep}(p+c\phi+U(1-U)). 
\end{array}
\label{eq5}
\ee
The equilibrium states (steady states) of this system are
$(U,\phi,p,q)=(0,0,0,0)$ and $(1,0,0,0)$ and these states are, of
course, independent of $\ep$.

Of course, the ODE for Fisher fronts, namely~(\ref{eq4}) with $\ep=0$,
is only second order and thus can be analysed in the $(U,U')$ phase plane in
which the required travelling wave is simply a connection between a
saddle point at $(1,0)$ and (if $c\geq 2$) a stable node at the
origin. With $\ep>0$ we are working in a four dimensional phase
space. The plausibility of a heteroclinic connection between the two
equilibrium states can be seen by calculating the dimensions of the
local stable and unstable manifolds of these two equilibria. We then
rigorously confirm the existence of a front in the subsequent sections
of the paper.

The linearised (Jacobian) matrix $J$ of the system~(\ref{eq5}) is
\[
J(U,\phi,p,q)=\left(\begin{array}{cccc} 0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1/\ep \\ (1-2U)/\ep & c/\ep & 1/\ep & 0
\end{array}\right)\]
and, at the steady state $(0,0,0,0)$ the eigenvalues $\lambda$ of this
matrix satisfy
\[\ep^2\lambda^4-\lambda^2-c\lambda-1=0.\]
For $\ep$ sufficiently small it is easily seen that (i) if $c\geq 2$
this equation has three real negative roots and one real positive
root; (ii) if $0<c<2$ it has two real roots of opposite sign, and two
complex roots with negative real part. In either case the dimension of
the stable manifold of the steady state $(0,0,0,0)$ is three.
Similarly it can be seen that for $\ep$ sufficiently small the
eigenvalues at the steady state $(1,0,0,0)$ are all
real (two positive and two negative). The unstable manifold
at this steady state therefore has dimension two.

The sum of the dimensions of the stable and unstable manifolds
mentioned above is five, while the phase space has dimension
four, so in principle they can intersect along a
one-dimensional curve in ${\bf R}^4$. In the next sections we confirm
the existence of a connection between the steady states $(1,0,0,0)$
and $(0,0,0,0)$, i.e., a travelling front solution for~(\ref{eq3}). 

For the time being we shall fix $c\geq 2$ so that a monotone
travelling front exists for the unperturbed ($\ep=0$) problem. Our aim
is to show that such a front still exists for the perturbed problem
when $\ep$ is sufficiently small. Later it will be argued that, in the
perturbed problem, monotone fronts also exist for values of $c$ just
below~2.

\setcounter{equation}{0}
\section{Existence of an invariant manifold}
Written slightly differently, system~(\ref{eq5}) is
\be
\begin{array}{l}
dU/dz=\phi, \\ d\phi/dz=p, \\ \ep dp/dz=q, \\
\ep dq/dz=p+c\phi+U(1-U). 
\end{array}
\label{eq6}
\ee
Writing $\zeta=z/\ep$,~(\ref{eq6}) becomes
\be
\begin{array}{l}
dU/d\zeta=\ep \phi, \\ d\phi/d\zeta=\ep p, \\ dp/d\zeta=q, \\
dq/d\zeta=p+c\phi+U(1-U),
\end{array}
\label{eq7}
\ee
which we call the dual ``fast system'' associated with~(\ref{eq6}).

If in~(\ref{eq6}) $\ep$ is set to zero, then $U$ and $\phi$ are
governed by
\[\frac{d^2U}{dz^2}+c\frac{dU}{dz}+U(1-U)=0,\hspace{.4in}\phi=\frac{dU}{dz}\]
(as for the Fisher front) while $p$ and $q$ must lie on the set
\be
M_0:=\{(U,\phi,p,q)\in{\bf R}^4\,:\hspace{.1in}
q=0\hspace{.1in}\mbox{and}\hspace{.1in}
p+c\phi+U(1-U)=0\}
\label{eq8}
\ee
which is a two-dimensional submanifold of ${\bf R}^4$. The system only
makes sense on this manifold, and the manifold is invariant for the
system.

What we claim is that, for $\ep$ sufficiently small, there exists a
two-dimensional submanifold 
$M_{\ep}$ of ${\bf R}^4$ which is within distance $\ep$ of $M_0$ and
which is invariant for the flow~(\ref{eq6}). By Fenichel's invariant
manifold theory (Fenichel~\cite{Fenichel}) such a perturbed invariant
manifold $M_{\ep}$ will exist if $M_0$ is {\em normally
  hyperbolic}. A manifold $M_0$ is said to be normally
hyperbolic if the linearisation of the fast system, restricted to
$M_0$, has exactly ${\rm dim}\,M_0$ eigenvalues on the imaginary axis,
with the remainder of the system hyperbolic.

The linearisation of the fast system~(\ref{eq7}), restricted to $M_0$
(i.e. $\ep=0$) has matrix
\[
\left(\begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\ 1-2U & c & 1 & 0\end{array}\right)
\]
and the eigenvalues of this are $0,0,1,-1$. Thus, $M_0$ is normally 
hyperbolic and, by Fenichel's theory, for sufficiently small $\ep$
there exists a perturbed manifold $M_{\ep}$ with the properties
listed above.

We should like to determine the dynamics on $M_{\ep}$. We can write
\[
M_{\ep}=\{(U,\phi,p,q)\in{\bf R}^4\,:\hspace{.1in}
q=g(U,\phi,\ep)\hspace{.1in}\mbox{and}\hspace{.1in}
p=-c\phi-U(1-U)+h(U,\phi,\ep)\}
\]
where the functions $g$ and $h$ (to be found) satisfy 
\[g(U,\phi,0)=h(U,\phi,0)=0.\]
Substitution into~(\ref{eq6}) yields that $g(U,\phi,\ep)$ and
$h(U,\phi,\ep)$ satisfy the partial differential equations
\[
\ep\left((2U-1)\phi+cU(1-U)+c^2\phi-ch+\phi\frac{\partial h}{\partial
    U}+(-U(1-U)-c\phi+h)\frac{\partial h}{\partial\phi}\right)=g,
\]
\[
\ep\left(\phi\frac{\partial g}{\partial U}+(-c\phi-U(1-U)+h)\frac{\partial
    g}{\partial\phi}\right)=h.
\]
Next, we expand $g$ and $h$ in Taylor series in the variable $\ep$:
\[g(U,\phi,\ep)=g(U,\phi,0)+\ep g_{\ep}(U,\phi,0)+\half\ep^2
g_{\ep\ep}(U,\phi,0)+\cdots,\]
and similarly for $h$ and substitute into the above partial
differential equations. Powers of $\ep^0$ give
\[g(U,\phi,0)=h(U,\phi,0)=0\]
as expected. Powers of $\ep$ give
\[
\begin{array}{l}g_{\ep}(U,\phi,0)=(2U-1)\phi+c^2\phi+cU(1-U), \\
h_{\ep}(U,\phi,0)=0,\end{array}
\]
and powers of $\ep^2$ give
\[
\begin{array}{l}\half g_{\ep\ep}(U,\phi,0)=0, \\
\half
h_{\ep\ep}(U,\phi,0)=\phi(2\phi+c(1-2U))+(-U(1-U)-c\phi)(2U-1+c^2).\end{array}
\]
Thus
\[
h(U,\phi,\ep)=\ep^2 h_1(U,\phi,\ep)\]
where
\[
h_1(U,\phi,\ep)=\phi(2\phi+c(1-2U))+(-U(1-U)-c\phi)(2U-1+c^2)+O(\ep)
\]
and the system~(\ref{eq6}) becomes
\be
\begin{array}{l}
dU/dz=\phi, \\
d\phi/dz=-U(1-U)-c\phi+\ep^2(\phi(2\phi+c(1-2U))+(-U(1-U)-c\phi)(2U-1+c^2)),
\end{array}
\label{eq9}
\ee
which determines the dynamics on the slow manifold $M_{\ep}$. We shall
study this system in the next section.

\setcounter{equation}{0}
\section{The flow on the manifold $M_{\ep}$}
When $\ep=0$, system~(\ref{eq9}) reduces to the coupled first order
ODEs for the classical Fisher front. We let $(U_0,\phi_0)$ be the
solution of~(\ref{eq9}) when $\ep=0$. In the $(U,\phi)$ phase 
plane, this front is a connection between a saddle point at $(1,0)$ and
(if $c\geq 2$) a stable node at $(0,0)$. Note that when $\ep>0$, the
system~(\ref{eq9}) still has these same critical points, together with
another at $(\half(1-c^2-1/\ep^2),0)$ that is not part of the region
of phase space of interest for us. We shall now use
Fredholm theory to show that when $\ep>0$ is sufficiently small a heteroclinic
connection exists between the critical points $(1,0)$ and $(0,0)$
of~(\ref{eq9}). The connection corresponds to a travelling front
solution of~(\ref{eq3}).

To seek such a connection set
\[U=U_0+\ep^2\tilde{U},\hspace{.3in}\phi=\phi_0+\ep^2\tilde{\phi},\]
and substitute into~(\ref{eq9}). To lowest order the differential
equation system governing $\tilde{U},\tilde{\phi}$ is
\be
\frac{d}{dz}\left(\begin{array}{c}\tilde{U} \\ \tilde{\phi}\end{array}
\right)-\left(\begin{array}{cc}0 & 1 \\ 2U_0(z)-1 & -c\end{array}
\right)\left(\begin{array}{c}\tilde{U} \\ \tilde{\phi}\end{array}
\right)=\left(\begin{array}{c} 0 \\ h_1(U_0,\phi_0,0)\end{array}
\right)
\label{eq10}
\ee
and we want to prove this system has a solution satisfying
\[\tilde{U},\tilde{\phi}\rightarrow
0\hspace{.2in}\mbox{as}\hspace{.2in}z\rightarrow \pm\infty.\]
Working in the space $L^2$ of square integrable functions, with the
inner product
\[\int_{-\infty}^{\infty}({\bf x}(z),{\bf y}(z))\,dz,\]
$(\cdot,\cdot)$ being the Euclidean inner product on ${\bf R}^2$,
Fredholm theory says that~(\ref{eq10}) will have a solution if and
only if
\[\int_{-\infty}^{\infty}\left({\bf x}(z),\left(\begin{array}{c} 0 \\
      h_1(U_0(z),\phi_0(z),0)\end{array}\right)\right)\,dz =0\]
for all functions ${\bf x}(z)$ in the kernel of the adjoint of the
operator defined by the left hand side of~(\ref{eq10}).

With the above inner product the adjoint equation is
\be
\frac{d{\bf x}}{dz}=\left(\begin{array}{cc}0 & 1-2U_0(z) \\ -1 &
    c\end{array}\right){\bf x}.
\label{eq11}
\ee
Although the matrix in~(\ref{eq11}) is non-constant, $U_0$ is the
Fisher front about which we know a great deal. Letting
$z\rightarrow\infty$ we have $U_0\rightarrow 0$ and the matrix is then
a constant matrix with eigenvalues $\lambda$ satisfying
\[\lambda^2-c\lambda +1=0.\]
Since $c\geq 2$ the eigenvalues are both real and positive. Thus as
$z\rightarrow\infty$ any solution of~(\ref{eq11}), other than the zero
solution, must grow exponentially. The only solution in $L^2$ is
therefore the zero solution ${\bf x}(z)={\bf 0}$ and consequently the
Fredholm orthogonality condition trivially holds. We have proved
existence of the desired connection in the manifold $M_{\ep}$. We
summarise our result in the following theorem.

\begin{thm}
If $c\geq 2$ there exists $\ep_0$ such that, for every
$\ep\in(0,\ep_0]$, equation~(\ref{eq3}) admits a monotone travelling
front solution $u(x,t)=U(z)$ satisfying $U(-\infty)=1$ and
$U(\infty)=0$, where $z=x-ct$.
\end{thm}

Although our analysis has required $c\geq 2$ the following formal
analysis, using the differential equations~(\ref{eq9}), suggests that
for a fixed (small) value of $\ep>0$ monotone travelling fronts also exist
for~$c$ just below~2. Recall that when $\ep=0$ the minimum wave
propagation speed, for waves that are everywhere positive, is~2. We
wish to see how this minimum speed changes when $\ep>0$ is small.
We linearise far ahead of the front where $U\rightarrow 0$, i.e.,
about the critical 
point $(0,0)$ of~(\ref{eq9}). The linearised matrix about this
critical point has eigenvalues $\lambda$ satisfying
\[\lambda^2+(c-\ep^2 c(2-c^2))\lambda+1-\ep^2(1-c^2)=0.\]
To have solutions decaying to the origin with $U$ remaining positive
the origin must be a node rather than a focus. Thus, we need
\[(c-\ep^2 c(2-c^2))^2\geq 4(1-\ep^2(1-c^2))\]
and equality will give the minimum speed $c_{\rm min}$. Since $\ep$ is
small we attempt the ansatz
\[c_{\rm min}=2-\sum_{n=1}^{\infty}c_n \ep^{2n}\]
which yields $c_{\rm min}=2-\ep^2+\frac{15}{4}\ep^4+\cdots$. Thus the minimum
propagation speed is slightly lower than for the classical Fisher front.

\setcounter{equation}{0}
\section{The wave profile when $\ep$ is large} 
We have shown in previous sections that if $\ep$ is small,
equation~(\ref{eq3}) admits a rightward moving travelling front
solution that connects the uniform states $u=0$ and $u=1$. Our aim in
this section is to obtain some further information about this front
when $\ep$ is no longer small (assuming, of course, that the front
still persists). We rewrite~(\ref{eq3}) as
\be
\frac{\partial u}{\partial t}=-\alpha\frac{\partial^4 u}{\partial x^4}+
\frac{\partial^2 u}{\partial x^2}+u(1-u)
\label{eq12}
\ee
where $\alpha>0$, or, in travelling wave form,
\be
-\alpha U''''+U''+cU'+U(1-U)=0
\label{eq13}
\ee
with $U(-\infty)=1$ and $U(\infty)=0$.

We first show that for $\alpha$ sufficiently large, $U(z)$ no longer
decays {\em monotonically} to~1 as $z\rightarrow -\infty$. Linearising
about $U=1$ and seeking solutions of the linearised equation proportional to
$\exp(\lambda z)$ gives
\be
f(\lambda):=\alpha \lambda^4-\lambda^2-c\lambda+1=0.
\label{eq14}
\ee
We can prove that for $\alpha$ sufficiently large~(\ref{eq14}) has at
least two complex roots, by showing that $f'(\lambda)$ has only one
real root. Indeed, $f'(\lambda)=4\alpha\lambda^3-2\lambda-c$ and,
bearing in mind the signs of the coefficients of the latter, a
simple graphical argument shows that $f'(\lambda)=0$ will have only
one real root 
if $f'(\lambda_0)<0$ where $\lambda_0$ is the smaller of the two roots
of $f''(\lambda)=0$. Calculating $\lambda_0$ we find that
$\lambda_0=-1/\sqrt{6\alpha}$ and
\[f'(\lambda_0)=\frac{4}{3\sqrt{6\alpha}}-c\]
which indeed is negative for $\alpha$ sufficiently large.

In fact when $\alpha$ is sufficiently large $f(\lambda)=0$ has no real
roots at all. Indeed, for large $\alpha$ we have already established
$f'(\lambda)=0$ has 
only one real root, which will give a global minimum for $f$. The root
of $f'=0$ is easily seen to be $O(\alpha^{-1/3})$ for large $\alpha$
and, at this root, $f\sim 1>0$ so that $f=0$ has no real roots.

Next we will construct an asymptotic expression for the travelling
front $U(z)$ satisfying~(\ref{eq13}) in the limit when the wave speed
$c\rightarrow\infty$. There is no requirement that $\alpha$ be
small. This analysis will reveal the main qualitative differences
between travelling front solutions of~(\ref{eq12}) and those of
Fisher's equation. The analysis is based on ideas of
Canosa~\cite{Canosa} who constructed an asymptotic solution to
Fisher's equation when $c\rightarrow\infty$. In fact Canosa's solution
is known to be highly accurate for any $c\geq 2$
(Murray~\cite{Murray93}).

For $c$ large we define $\ep=1/c^2$ and seek a solution
of~(\ref{eq13}) of the form
\[U(z)=g(\zeta)\hspace{.3in}\mbox{where}\hspace{.3in}\zeta=z/c=\sqrt{\ep}z.\]
Equation~(\ref{eq13}) becomes
\be
-\alpha\ep^2g''''+\ep g''+g'+g(1-g)=0
\label{eq15}
\ee
with boundary conditions $g(-\infty)=1$ and $g(\infty)=0$. Since this
problem is invariant to translations in $\zeta$ we may also take
\[g(0)=\half.\]
This avoids arbitrary constants appearing in the solution. We seek a
solution to~(\ref{eq15}) in the form
\[g(\zeta)=g_0(\zeta)+\ep g_1(\zeta)+\ep^2 g_2(\zeta)+\cdots\]
where $g_0(0)=\half$ and $g_n(0)=0$ for $n\geq 1$. Powers of $\ep^0$
give
\[g_0'+g_0(1-g_0)=0,\hspace{.5in}g_0(0)=\half\]
which has solution $g_0(\zeta)=(\exp(\zeta)+1)^{-1}$. To next order,
\[g_1'+(1-2g_0)g_1=-g_0''\]
which has solution
\[g_1(\zeta)=\frac{e^{\zeta}}{(1+e^{\zeta})^2}\ln\left(\frac{4e^{\zeta}}
{(e^{\zeta}+1)^2}\right).\]
Powers of $\ep^2$ give
\[g_2'+(1-2g_0)g_2=\alpha g_0''''-g_1''+g_1^2\]
the solution of which is
\[\begin{array}{l}
g_2(\zeta)=\half e^{\zeta}(e^{\zeta}+1)^{-4}\left(-6-4\zeta+4\zeta
  e^{2\zeta}\ln 2+4\alpha e^{2\zeta}\ln(e^\zeta+1)-2\alpha\zeta
  e^{2\zeta}\right.\\
\left. 8\alpha e^\zeta\ln(e^\zeta+1)+4\ln(e^\zeta+1)+12\alpha
  e^\zeta-4\zeta e^{2\zeta}\ln(e^\zeta+1)-4\alpha\zeta e^\zeta-4(\ln
  2)^2\right. \\
\left. -4\ln 2+6 e^{2\zeta}-6\alpha-6\alpha e^{2\zeta}+4 e^{2\zeta}\ln
  2+4 e^{2\zeta}(\ln 2)^2-8 e^{2\zeta} \ln 2\ln(e^\zeta+1)-2\alpha
  \zeta\right. \\
\left. +4\alpha\ln(e^\zeta+1)+8\ln 2\ln(e^\zeta+1)-4\zeta\ln
  2-4\alpha\ln 2-4(\ln(e^\zeta+1))^2-\zeta^2-4\zeta e^\zeta\right. \\
\left. +\zeta^2 e^{2\zeta}+4 e^{2\zeta}(\ln(e^\zeta+1))^2-4
  e^{2\zeta}\ln(e^\zeta+1)+4\zeta\ln(e^\zeta+1)-8\alpha e^\zeta\ln
  2-4\alpha e^{2\zeta}\ln 2\right).\end{array}
\]
In terms of the original variables, our expression for the travelling
front is therefore
\be
U(z)=g_0(z/c)+\frac{1}{c^2}g_1(z/c)+\frac{1}{c^4}g_2(z/c)+\cdots
\label{eq19}
\ee
with $g_0$, $g_1$ and $g_2$ given as above. Figure~1 shows the graph
of this asymptotic solution.

\begin{center}
\fbox{\bf Figure 1 here}
\end{center}

 Note that the effect of the
fourth order term in~(\ref{eq3}) is only seen at second order; the
$g_0$ and $g_1$ terms in~(\ref{eq19}) do not involve $\alpha$ and are
precisely the same as are obtained when applying this solution
procedure to Fisher's equation. This suggests that travelling fronts
are very robust to the addition of fourth order terms not only in
terms of their persistence but also in that there is little change in
their qualitative form. Note however that Fig.~1 is for a wave speed of~1.6
yet the front is still positive everywhere. In Fisher's equation the
minimum speed for a positive front is~2.

\setcounter{equation}{0}
\section{Discussion}
We have shown that if $\ep$ in~(\ref{eq3}) is sufficiently small
then~(\ref{eq3}) admits travelling front solutions connecting the 
uniform states $u=0$ and $u=1$. We have also constructed an asymptotic
expression for the front when $c\rightarrow\infty$ and this does not
require the fourth order term to be small. The fact that
$\alpha$ (in~(\ref{eq12})) only appears in the second order term 
of the asymptotic solution, which has a coefficient of $c^{-4}$, with
the first two terms being the same as in the corresponding asymptotic
solution of Fisher's equation, suggests that the introduction of the
fourth order term does not have a particularly significant effect on
the profile unless $\alpha$ is large when oscillations will develop on
its leeward side. Otherwise the main effect of this term seems to be
to reduce the front's minimum speed.

We have not addressed the question of the stability of the fronts
which is, of course, a nontrivial matter even for Fisher's equation.
An interesting and worthwhile further investigation
would be to determine what classes of initial data evolve to
travelling fronts that are everywhere positive. Positivity of
solutions in fourth order equations is a genuinely difficult problem
since little is known about the sign of the fourth order term in the
evolution and thus maximum-principle and related theorems do not
apply. At present we are working on positivity preservation theorems
for~(\ref{eq3}), but on a finite spatial domain.

\newpage

\begin{thebibliography}{99}

\bibitem{Akveld} Akveld, M.E. and Hulshof, J. (1998) Travelling wave
  solutions of a fourth-order semilinear diffusion equation. {\em
  Applied Mathematics Letters} {\bf 11}, No.~3, 115-120.

\bibitem{Ashwin} Ashwin, P.B., Bartuccelli, M.V., Bridges, T.J. and Gourley,
  S.A. Travelling fronts for the KPP equation with spatio-temporal
  delay (submitted).

\bibitem{Canosa} Canosa, J. (1973) On a nonlinear diffusion equation
  describing population growth. {\em IBM. J. Res. \& Dev.} {\bf 17},
  307-313.

\bibitem{Cohen81} Cohen, D.S. and Murray, J.D. (1981) A generalised
  diffusion model for growth and dispersal in a population. {\em
  Journal of Mathematical Biology} {\bf 12}, 237-249.

\bibitem{dee88} Dee, G.T. and van Saarloos, W. (1988) Bistable systems
  with propagating fronts leading to pattern formation. {\em
  Phys. Rev. Lett.} {\bf 60}, No.~25, 2641-2644.

\bibitem{Fenichel} Fenichel, N. (1979) Geometric singular perturbation theory
  for ordinary differential equations. {\em Journal of Differential
  Equations} {\bf 31}, 53-98.

\bibitem{Murray93} Murray, J.D. (1993) {\em Mathematical Biology}
  (Springer-Verlag, Berlin).

\bibitem{Peletier} Peletier, L.A. and Troy, W.C. (1996) Chaotic spatial
  patterns described by the Extended Fisher-Kolmogorov equation.
  {\em Journal of Differential Equations} {\bf 129}, 458-508.

\end{thebibliography}
\newpage
\section*{Figure Caption}
{\bf Fig. 1:} Plot of the travelling front solution of~(\ref{eq3}) given by
expression~(\ref{eq19}). Parameter values were $c=1.6$ and
$\alpha=2.5$.

\end{document}
