# The evolution problem for the 1D nonlocal Fisher-KPP equation with a top hat
kernel. Part 1. The Cauchy problem on the real line

D. J. Needham  School of Mathematics, University of Birmingham, UK, B15 2TT  ,
J. Billingham  School of Mathematical Sciences, University of Nottingham, UK,
NG7 2RD  ,  N. M. Ladas  School of Mathematics, University of Birmingham, UK,
B15 2TT  and  J. C. Meyer  School of Mathematics, University of Birmingham,
UK, B15 2TT

###### Abstract.

We study the Cauchy problem on the real line for the nonlocal Fisher-KPP
equation in one spatial dimension,

$u_{t}=Du_{xx}+u(1-\phi*u),$

where $\phi*u$ is a spatial convolution with the top hat kernel,
$\phi(y)\equiv H\left(\frac{1}{4}-y^{2}\right)$.

After observing that the problem is globally well-posed, we demonstrate that
positive, spatially-periodic solutions bifurcate from the spatially-uniform
steady state solution $u=1$ as the diffusivity, $D$, decreases through
$\Delta_{1}\approx 0.00297$ (the exact value is determined in section 3). We
explicitly construct these spatially-periodic solutions as uniformly-valid
asymptotic approximations for $D\ll 1$, over one wavelength, via the method of
matched asymptotic expansions. These consist, at leading order, of regularly-
spaced, compactly-supported regions with width of $O(1)$ where $u=O(1)$,
separated by regions where $u$ is exponentially small at leading order as
$D\to 0^{+}$.

From numerical solutions, we find that for $D\geq\Delta_{1}$, permanent form
travelling waves, with minimum wavespeed, $2\sqrt{D}$, are generated, whilst
for $0<D<\Delta_{1}$, the wavefronts generated separate the regions where
$u=0$ from a region where a steady periodic solution is created via a distinct
periodic shedding mechanism acting immediately to the rear of the advancing
front, with this mechanism becoming more pronounced with decreasing $D$. The
structure of these transitional travelling wave forms is examined in some
detail.

##  1\. Introduction

Nonlocal reaction-diffusion equations arise in many different scientific areas
(see, for example, [7, 8, 9, 12, 21, 27] and [18, 28]). Many of these
applications are biomedical, including tumour modelling and models for
evolution and speciation (see [2] for an extensive review). The most studied
of these equations is the nonlocal dimensionless Fisher-KPP equation

(1) $\frac{\partial u}{\partial t}=D\frac{\partial^{2}u}{\partial
x^{2}}+u\left\\{1-\int_{-\infty}^{\infty}\phi\left(x-y\right)u\left(y,t\right)\,dy\right\\},$

where lengths have been scaled on the nonlocal length scale, and the
dimensionless parameter $D$ represents the square of the ratio of the
diffusion length scale to the nonlocal length scale. As shown in [3], this has
permanent form travelling wave solutions for all wavespeeds greater than or
equal to $2\sqrt{D}$, with this minimum wavespeed fixed by the behaviour of
the solution when $u\ll 1$. The linearisation of (1) when $u$ is small is the
same as that of the local Fisher-KPP equation,

(2) $\frac{\partial u}{\partial t}=D\frac{\partial^{2}u}{\partial
x^{2}}+u\left(1-u\right),$

and the minimum wavespeed exists for the same reason, namely that no strictly
positive travelling wave solutions exist for wavespeed less than $2\sqrt{D}$.

Although much of the literature on (1) has focussed on kernels with
$\phi(y)>0$ for all $y\in\mathbb{R}$ (see, for example, [5, 15]), there has
also been some interest in kernels with finite support. For example, in [2,
24, 17, 13], the authors show that for small enough diffusivity such kernels
lead to steady solutions with spatial patterning, with either large, narrow
spikes (treated as delta functions in [24]) or spatial patterns with width and
height of $O(1)$ as $D\to 0$, which we discuss below. The canonical example of
a kernel with compact support is the top hat kernel given below by (8). Our
aim in this series of papers is to develop a thorough understanding of the
nature of typical evolutionary dynamics of (1) with the top hat kernel, and
the mechanisms involved therein. More generally, systems such as (1) with
$D\ll 1$, where the characteristic diffusion length scale is much smaller than
the length scale associated with nonlocal interation, have long been known to
have solutions with localised spatial patterning, [14], and (1) with $D\ll 1$
is no exception. It is also important to note here that the key features
explored in this paper for the top hat kernel are robust to the addition of
general piecewise continuous perturbations to this kernel from
$L^{\infty}(\mathbb{R})\cap L^{1}({\mathbb{R}})$, and which are sufficiently
small in $L^{1}(\mathbb{R})$. This point will be addressed by the authors at a
later stage in this series of papers.

In order to begin our study, it is convenient to introduce some notation that
will be used throughout the paper. For any $T>0$ we introduce
${D}_{T}\subset\mathbb{R}^{2}$ by

(3)
$\displaystyle{D}_{T}=\\{(x,t)\in\mathbb{R}^{2}:x\in\mathbb{R},\,t\in(0,T]\\}$

with closure $\overline{D}_{T}$, and also,

(4)
$\displaystyle{D}_{\infty}=\\{(x,t)\in\mathbb{R}^{2}:x\in\mathbb{R},\,t\in\mathbb{R}^{+}\\}$

with closure $\overline{D}_{\infty}$. The Cauchy problem we consider is that
concerned with classical solutions $u:\overline{D}_{T}\to\mathbb{R}$ to the
semilinear, nonlocal, evolution problem,

(5) $\displaystyle u_{t}=Du_{xx}+u(1-\phi*u),{\quad\text{on }{D}_{T};}$ (6)
$\displaystyle u(x,0)=Ag(x)=:u_{0}(x),\quad\forall\,x\in\mathbb{R}$ (7)
$\displaystyle u(x,t)\to 0\text{ as }|x|\to\infty\text{ uniformly for
}t\in[0,T].$

Here $A>0$, $g\in C(\mathbb{R})\cap L^{\infty}(\mathbb{R})$ and non-negative,
$\norm{g}_{\infty}=1$ whilst $\operatorname{supp}(g)\subseteq[-x_{0},x_{0}]$
$(x_{0}>0)$. It is also worth noting that the majority of theory developed
hereafter will also apply when the initial data has unbounded support, but
decays sufficiently rapidly as $|x|\to\infty$ (for example, the Gaussian
initial data used in Section 2). Throughout, with $A$, $x_{0}$ and $g$
prescribed, we will regard a solution to (5)-(7) as being a solution in the
classical sense. As discussed above, we further restrict attention to the
situation when the nonlocal kernel $\phi:\mathbb{R}\to\mathbb{R}$ has the
simple top hat structure,

(8) $\displaystyle\phi(y)$ $\displaystyle=\begin{cases}1,&-\frac{1}{2}\leq
y\leq\frac{1}{2}\\\ 0,&\text{elsewhere},\end{cases}$

after which, for $(x,t)\in\overline{D}_{T}$,

(9)
$\displaystyle(\phi*u)(x,t)=\int_{x-\frac{1}{2}}^{x+\frac{1}{2}}{u(y,t)}dy.$

The main focus of the paper will be both the qualitative and quantitative
study of the Cauchy problem (5)-(7) with (8) and (9). Of particular interest
will be the large-$t$ structure of the solution. For brevity, we will refer to
this Cauchy problem as (IBVP) for the rest of the paper. With this objective
in mind, the paper is structured in the following way:

$\mathbf{In~{}Section}~{}\ref{section2}$ we briefly review the fundamental
questions of uniqueness and global existence for (IBVP), together with some
very general basic bounds on the solution. These are readily and rigorously
established by applying the results developed in [17] to (IBVP). Then we
present detailed illustrative numerical solutions to (IBVP), which enable us
to formulate a number of structural and mechanistic conjectures concerning the
evolution of the solution to (IBVP) as $t\to\infty$; in particular in relation
to the propagation of travelling wavefronts which, below a critical value of
$D$ (which is determined explicitly in section 3), leave a stationary
spatially periodic state in their wake. Particular attention is paid to the
wavelength selection mechanism for this emerging stationary periodic steady
state. It is determined, via both numerical computation and large-t
asymptotics, that this mechanism changes as the diffusivity $D$ decreases.
Specifically, it is shown that for moderately small values of $D$, close to
the linear stability margin of the equilibrium state $u=1$, the mechanism
approximates to a selection of the most unstable linear wavelength, and
operates at a distance to the rear of the wavefronts. However, the mechanism
changes significantly for very small values of $D$ into what we refer to as a
_hump formation_ mechanism, which is controlled by the wavefront and its
immediate exponentially small precession, and takes place immediately to its
rear, and involves both nonlinear and nonlocal effects. This mechanism is
analysed and illuminated in detail via large-t asymptotics.

$\mathbf{In~{}Section~{}\ref{section4}}$, motivated by section 2 and the
references therein, we examine the temporal stability of the two equilibrium
states $u=0$ and $u=1$ to the nonlocal PDE (5) with (9) in detail (via
linearization, which is underpinned by classical rigorous linearization
theorems away from the marginal stability case), and how this, and its
consequences, may relate to (IBVP). In this way, we briefly confirm that the
unreacted equilibrium state $u=0$ is temporally unstable at all $D>0$. The
linearized analysis at this equilibrium state also confirms that the solution
to (IBVP) should develop wavefronts at locations $x\sim\pm
2\sqrt{D}t+O(\log{t})$ as $t\to\infty$. This accords with the spreading speeds
established in [17] (Theorem 1.5), as well as the numerical solutions
considered in section 2. We then provide a detailed linearized analysis of the
fully reacted equilibrium state $u=1$, and observe that this equilibrium state
bifurcates from temporally stable to temporally unstable as $D$ decreases
through the critical value $\Delta_{1}$, with the value $\Delta_{1}\approx
0.00297$ determined exactly, and in excellent agreement with the numerical
solutions presented in section 2. In addition, we present a detailed analysis
of the nature of this temporal instability when $0<D<\Delta_{1}$, which
indicates and supports the observations in section 2 that the solution to
(IBVP) will develop into a stationary spatially periodic steady state at the
rear of the propagating wavefronts. We determine the wavelength of the
harmonic Fourier mode which has maximum temporal growth rate and confirm that,
_at least at values of $D$ close to the marginal stability value
$\Delta_{1}$_, the selected wavelength of the steady periodic state which
emerges in (IBVP) is very close to the wavelength of this Fourier mode with
maximum linear growth rate. The analysis in this section affords us the
opportunity to propose two fundamental conjectures concerning the large-$t$
asymptotic structure of the solution to (IBVP), which we label as (P1) and
(P2). These conjectures motivate the direction of each of the remaining
sections of the paper.

$\mathbf{In~{}Section~{}\ref{section5}}$, motivated by (P2), we consider in
detail the existence of spatially periodic steady states to the nonlocal
Fisher-KPP equation featuring in (IBVP), with particular attention devoted to
the most interesting situation when $D$ is very small. This section gives a
detailed unfolding, in the case of the top hat kernel, of the generic
existence result established in [17] (Theorem 1.1) regarding spatially
periodic steady states. With $\lambda$ representing _fundamental wavelength_ ,
we establish, using local bifurcation/weakly nonlinear theory that a unique
(up to spatial translation) nontrivial spatial periodic steady state exists at
each point $(\lambda,D)\in\bigcup_{i=1}^{\infty}{\Omega_{i}}$, where the
family $\Omega_{i}$ are bounded, open, simply connected and pairwise disjoint
subsets of the first quadrant of the $(\lambda,D)$ plane, and are constructed
explicitly. The periodic steady states are created by a family of steady state
pitchfork bifurcations, from the equilibrium state $u=1$, as the boundary of
each subdomain is crossed into its interior. Standard weakly nonlinear
approximations to these periodic steady states are recorded for points close
to the bifurcation boundary, and these bifurcation curves are path followed
numerically to generate the complete bifurcation surface above each subdomain
$\Omega_{i}$. In addition we prove that each of the periodic steady states is
strictly positive, and represents an oscillation about $u=1$. Of particular
interest is the nature of these periodic steady states as $D\to 0^{+}$. In
this limit we have developed a detailed and intricate theory, via the method
of matched asymptotic expansions, to asymptotically approximate the periodic
steady states. This enables us to explicitly construct detailed asymptotic
approximations to the periodic steady states as $D\to 0^{+}$, which are
spatially uniform over their wavelength. A key observation is that the
structure of the periodic steady states develops into a distinct form of
localised hump regions where $u=O(1)$ separated by dead regions where $u$ is
exponentially small in $D$, as $D\to 0$. The asymptotic constructions are
shown to be in excellent agreement with numerically determined approximations.

$\mathbf{In~{}Section~{}\ref{section6}}$, in relation to the nature of the
propagating fronts identified in the large-$t$ structure of (IBVP), and
referred to in (P2), we consider the possibility that the nonlocal Fisher-KPP
equation featuring in (IBVP) can support nontrivial, positive, _propagating_ ,
spatially periodic travelling waves which bifurcate from the equilibrium state
$u=1$. It is straightforward for us to establish that no such local
bifurcations take place, and as such, a possible role of spatially periodic
nontrivial travelling weaves in (IBVP) can be ruled out.

$\mathbf{In~{}Section~{}\ref{section7}}$ we consider the role in (IBVP) of
non-negative travelling waves which have steady profile and represent a
transition from the unreacted equilibrium state ahead to the fully reacted
equilibrium state to the rear (which we refer to a _transitional permanent
form travelling wave_ abbreviated to (TPTW) throughout). We establish that
such a (TPTW) exists at each $D>0$ when the propagation $v$ speed has $v\geq
2\sqrt{D}$, which is in accord with the wavefront evolution speed in (IBVP),
as recorded in sections 2 and 3, and in relation to the reported spreading
speed for the similar initial value problem reported in [17] (Theorem 1.5).
The remaining analysis establishes some further elementary properties, and
then examines in detail the behaviour to the rear of a (TPTW) with particular
emphasis on monotone decay, oscillatory/monotone decay and oscillatory decay.
This theory develops, expands and provides explicit critical transition
points, for the case of the top hat kernel, in relation to the earlier general
theory in [10].

Finally, in Section 7, we bring all of the subsequent results together, with
emphasis on their bearing on (IBVP).

##  2\. General Setting and Numerical Solutions for (IBVP)

To begin this section we briefly recall fundamental preliminary results
concerning (IBVP), which can be obtained directly from application of the
theory developed in [17]. We first observe, with key details established in
[17] (Theorem 1.3), that (IBVP) has a unique, global (on
$\overline{D}_{\infty}$) solution, which, for each $T>0$, depends continuously
on intial data, throughout $\overline{D}_{T}$. Moreover,
$||u(\cdot,t)||_{\infty}$ is uniformly bounded on $[0,\infty)$ by a constant
depending only upon $A$ and $D$ (although, for given $A\geq 0$ we record from
[17] (Remark 1.4) that this constant blows up as $D\to 0^{+}$). In addition,
it is trivially established, via the strong maximum principle and comparison
theorem, that, for any $t_{0}$ sufficiently large,

(10) $\displaystyle
0<u(x,t)\leq(t+t_{0})^{-\frac{1}{2}}e^{(t+t_{0})}e^{-x^{2}/4D(t+t_{0})}\quad\forall\,(x,t)\in
D_{T},$

which provides useful information in later sections. However, we may observe
immediately from (10) that should an identifiable wavefront location and
structure develop in the solution to (IBVP) as $t\to\infty$, say at location
$|x|\sim S(t)$ (beyond which $u$ is exponentially small as $t\to\infty$),
then,

(11) $\displaystyle 0<S(t)\leq 2\sqrt{D}t-\frac{1}{2}\sqrt{D}\log t+O(1),$

as $t\to\infty$. We should remark here that Bouin, Henderson and Ryzhik [6]
establish an asymptotic estimate giving wavefront location at large-$t$ for
the solution to the nonlocal Fisher-KPP equation, when the initial data is
front-like and localized to the (wlog) left half-line, and in the case of the
top hat kernel, this gives a form which satisfies the above inequality, with
the coefficient of the $\log t$ term being $-\frac{3}{2}$, as in the Bramson
correction for the same evolution problem with the classical local Fisher-KPP
equation. We, tentatively, anticipate that in the present situation, $S(t)$
will be similarly asymptotic to this form as $t\to\infty$.

In the remainder of this section we begin our principal study of the detailed
qualitative and quantitative features of the solution to (IBVP), and in
particular how these properties respond to decreasing the diffusion parameter
$D$. We will see that significant changes in structure and mechanism occur,
particularly as $D$ decreases from moderately small ($\sim 10^{-3}$) to
extremely small ($\sim 10^{-6}$). With this in mind, we develop a numerical
scheme to approximate solutions to (IBVP), and use this to investigate the
qualitative and quantitative structure of solutions to (IBVP), with particular
attention to the structure of the solution as $t\to\infty$. We note that
similar evolutionary computations have been made and presented in [22],
concerning a stability question relating to permanent form travelling wave
structures connecting two unstable states of equation (1). Here our aim is to
form a preliminary approach to elucidating the qualitative and quantitative
properties of the evolution problem (IBVP). For the first set of numerical
solutions that we present, we take $x_{0}=\frac{1}{2}$ and
$g:\mathbb{R}\to\mathbb{R}$ as

(12) $\displaystyle g(x)$
$\displaystyle=\begin{cases}(1-2x)^{2}(1+2x)^{2},&|x|\leq\frac{1}{2}\\\
0,&|x|>\frac{1}{2}.\end{cases}$

We discretise $u$ on a uniform spatial grid of $N$ points, truncated to $0\leq
x\leq L$, approximating the second derivative using central finite
differences. The convolution term is evaluated using the trapezium rule, in
other words assuming a linear variation of $u$ between grid points, dealing
carefully with cases where the edge of the support of the top hat kernel lies
between grid points. We also take into account the symmetry of the solution
about $x=0$ and assume that $u=0$ for $x>L$. In the simulations discussed
below, we take $L=10$ and $N=1000$. Timestepping is done using the midpoint
method (second order accurate), with time step chosen adaptively so that the
maximum change in $u$ at each step is below $10^{-2}$.

Figure 1 shows the solution of (IBVP) when $t=9/2\sqrt{D}$, which is just
before the wavefront reaches the edge of the truncated domain, with $A=0.01$.

Figure 1. The numerical solution of (IBVP) for various values of $D$ with
$A=0.01$ (solid black line), along with the minimum speed travelling wave
(broken blue line).

For all values of $A$ that we investigated, indeed for all localised initial
inputs of $u$ that we tried, the solution was qualitatively similar to those
shown in Figure 1. In each case, a wavefront propagates in the positive
$x$-direction. Also shown is the corresponding minimum speed travelling wave
solution (see section 6 for details of the minimum speed travelling wave
solution), calculated numerically using the same finite difference method to
set up the discretised equations and ‘fsolve’ in Matlab to solve them. The
equilibrium state $u=1$ is temporally unstable for $D<\Delta_{1}\approx
0.00297$ whilst temporally stable for $D$ larger than this critical value (see
section 3 below). For $D>\Delta_{1}$ the minimum speed travelling wave
solution emerges, which leaves the equilibrium state $u=1$ in its wake. For
$0<D<\Delta_{1}$ however, a non-propagating stationary, spatially periodic
state is left in the wake of the wavefront. For moderately small values of
$D$, as shown in Figure 1, the wavelength of this periodic state is close to
$0.7$, which is close to the most unstable wavelength in the linear stability
theory for the equilibrium state $u=1$, but as we will see, this is only
remains so at moderately small values of $D$, with the wavelength selection
mechanism and value changing significantly for much smaller values of $D$ (see
Section 3, but also Figure 5 and subsequent discussion below). Movies of the
numerical solutions illustrated in Figures 1 and 4 can be found here.

The temporal periodicity of the creation of the stationary spatially periodic
state behind the wavefront is also reflected in a weak periodic variation in
the position of the wavefront (defined to be the largest value of $x$ at which
$u=\frac{1}{2}$), but with an average speed of approximately $2\sqrt{D}$, the
minimum wavespeed, which is clearly illustrated in Figure 2 (see sections 3
and 6, regarding the specific notion of minimum wavespeed in the present
context).

Figure 2. The numerically-calculated position of the wavefront for various
values of $D$. The broken line has slope $2\sqrt{D}$, the minimum wavespeed.

For the solutions shown in Figure 2 with $D=0.003$ and $D=0.002$, the
formation of a stationary periodic state behind the wavefront does not cause
oscillations in the position of the wavefront, whereas for $D=0.001$ the
magnitude of the oscillation is large enough that there is a weak effect.
Although choosing a smaller value of $u$ at which to define the position of
the wavefront would eliminate these traces of the oscillations, as $D\to 0$,
the value of $u$ required to do so becomes exponentially small. It therefore
seems reasonable to use $u=\frac{1}{2}$ as our qualitative definition of the
position of the wavefront, and thereby retain an indication of the
fundamentally oscillatory nature of the observable solution in our record of
the progress of the travelling wave.

For smaller values of $D$ than those used in Figures 1 and 2, we find that the
creation of the humps, which ultimately form the stationary and spatially
periodic state to the rear of the wavefront, initiates, periodically in $t$,
just ahead of the wavefront and, as we shall see below, can be related to the
dynamics of that part of the solution profile that becomes exponentially-small
with distance ahead of the wavefront. It is now this periodic mechanism that
selects the final spatial wavelength of the stationary periodic state which
forms at the rear of the wavefront. In order to accurately compute this part
of the solution, we use a different numerical method, solving instead for
$W=\log u$, which satisfies the evolution equation

(13) $\frac{\partial W}{\partial t}=D\frac{\partial^{2}W}{\partial
x^{2}}+D\left(\frac{\partial W}{\partial x}\right)^{2}+1-\phi*e^{W}.$

We also use an FFT to calculate the convolution term (and periodic boundary
conditions with periodicity large enough that the effect on the travelling
wave dynamics is negligible) and use five point stencils for the derivatives
for greater accuracy. In addition, since we cannot take the logarithm of an
initial condition with compact support, we instead use a Gaussian of width
$w>0$ as the initial condition, so that

(14) $g(x)=e^{-x^{2}/w^{2}}.$

Figure 3 shows a snapshot of the evolving solution for $D=10^{-8}$, $A=0.01$
and $w=0.1$.

Figure 3. The numerical solution of (IBVP) for Gaussian initial data with
$A=0.01$ and width $w=0.04$, and $D=10^{-8}$, when $t=6000$. The upper panel
shows $\log_{10}u$. New spikes are initiated ahead of the wave at the point
where $u$ is close to $10^{-700}$, which can only be captured accurately by
solving for $\log u$ instead of $u$.

Although the creation of a spatially periodic steady state behind the
wavefront is qualitatively similar to that shown for larger values of $D$ in
Figure 1, we can now see how the behaviour ahead of the wavefront leads to the
creation of the humps, which have unit weight, width of $O(D^{1/2})$ and
height of $O(D^{-1/2})$ for $D\ll 1$ (see Section 4.3 and Figure 6). Far ahead
of the wavefront, the solution evolves as

(15)
$u(x,t)\sim\frac{wA}{\sqrt{w^{2}+4Dt}}\exp\left(t-\frac{x^{2}}{w^{2}+4Dt}\right)~{}~{}\mbox{for
$x\gg t\geq O(1)$.}$

Note that (15) complies with the upper bound given by (10) with
$t_{0}=w^{2}/4D$. In a spatial interval of unit width, centred on the leading
hump, the term $1-\phi*u$ is small. However, ahead of this interval, it
becomes close to one. This has the effect of turning on the term $e^{t}$ in
(15), which manifests itself in the solution shown in Figure 3 as a rapid
change in $\log_{10}u$, ahead of which $u$ grows until a new hump is formed.
This is most easily observed in the animation that can be found here. This
hump formation process, generated by the dynamics of the exponentially small
part of the solution ahead of the wavefront, is similar to that studied in [4]
for a nonlocal reaction-diffusion equation with a different local reaction
term. For $D\ll 1$, this hump forms when the logarithm of the far field
solution (15) becomes small, and we can approximate this location as the point
where the logarithm is zero, namely $x=x_{f}(t)$, with

(16) $x_{f}(t)=\sqrt{w^{2}+4Dt}\sqrt{t+\log(Aw)-\frac{1}{2}\log(w^{2}+4Dt)}.$

Figure 4 shows the position of the wavefront along with $x_{f}(t)$ for various
values of $D$.

Figure 4. The numerically-calculated position of the wavefront for
$D=10^{-4}$, $10^{-5}$, $10^{-6}$, $10^{-7}$, $10^{-8}$ and $10^{-9}$, with
$w=0.1$ and $A=0.01$. The broken line is the function $x_{f}(t)$, defined in
(16).

As can be seen, $x_{f}(t)$ gives an excellent prediction of the position of
the wavefront, and we observe that it conforms with the wavefront bound given
in (11). We also note that this hump formation process naturally leads to a
stationary, spatially periodic solution behind the wavefront with wavelength
close to $\frac{1}{2}$. Figure 5 shows how the wavelength varies with $D$,
consistent with this observation.

Figure 5. The wavelength of the spatially-periodic steady state left behind
the wavefront, calculated numerically as a function of $D$. The broken line
shows the most unstable wavelength given by the linearized theory. Figure 6.
The inverse of the height of the spikes behind the wavefront, calculated
numerically as a function of $D$. The broken line is 100/$\sqrt{D}$.

Also shown is the most unstable wavelength according to the linearized theory
of Section 3, which is close to the wavelength observed for moderately small
values of $D$, when the periodic states emerges behind the wavefront, _but not
for smaller values of_ $D$, where the hump formation mechanism described above
takes over and creates this state, and determines its wavelength to be
approaching $\frac{1}{2}$ as $D\to 0^{+}$. Finally, it is anticipated that
additional details relating to this hump formation mechanism may be made
available by a more detailed development of the large-$t$ asymptotic form (15)
via the method of matched asymptotic coordinate expansions (see Leach and
Needham [20]) or the approximation methods detailed in van Saarloos [25].
However, for the present paper we regard the above details as sufficient to
elucidate this particular mechanism. Many of the additional distinct features
described above, particularly when the diffusivity is very small, have not
been reported in earlier literature, and their illumination is the focus of
the following sections.

##  3\. Equilibrium States, Stability Characteristics and linear evolution

The nonlocal PDE (5), with (9), has two equilibrium states. The _unreacted_
state with

(17) $\displaystyle u(x,t)=0\quad\forall\,{(x,t)}\in\overline{D}_{\infty},$

and the _fully reacted_ state

(18) $\displaystyle u{(x,t)}=1\quad\forall\,{(x,t)}\in\overline{D}_{\infty}.$

We have seen in Section 2 that these equilibrium states play a key role in the
large-$t$ structure of the solution to (IBVP). Of particular significance is
the temporal stability of these equilibrium states. In this section we examine
the linearized temporal stability of each of these equilibrium states in
detail, and consider the consequences thereof in terms of spatio/temporal
evolution. To this end, we formulate a linearized initial value problem. We
write,

(19) $\displaystyle
u(x,t)=u_{e}+\delta\overline{u}(x,t),\quad(x,t)\in\overline{D}_{\infty},$

with $\delta\ll 1$ and $u_{e}=1$ when considering the fully reacted state or
$u_{e}=0$ when considering the unreacted state. On substituting from (19) into
(5), and neglecting terms of $O(\delta^{2})$ as $\delta\to 0$, we obtain a
linear evolution equation for $\overline{u}$, namely

(20)
$\displaystyle\overline{u}_{t}=D\overline{u}_{xx}+L(\overline{u}),{\quad\text{on
}{D}_{\infty}},$

where

(21) $\displaystyle L(\overline{u})=\begin{dcases}\overline{u};&u_{e}=0\\\
-\int_{x-\frac{1}{2}}^{x+\frac{1}{2}}\overline{u}(y,t)dy;&u_{e}=1\end{dcases}$

after using (9). The linearized initial value problem is then composed of
(20), with associated initial and far field conditions,

(22)
$\displaystyle\overline{u}(x,0)=\overline{g}(x)\quad\forall\,x\in\mathbb{R}$
(23) $\displaystyle\overline{u}{(x,t)}\to 0\text{ as }|x|\to\infty\text{
uniformly on }\overline{D}_{T}\text{ (each }T>0).$

Here $\overline{g}\in C^{1}(\mathbb{R})\cap L^{\infty}(\mathbb{R})$ and non-
negative, $\norm{\overline{g}}_{\infty}=1$ whilst
$\operatorname{supp}(\overline{g})\subseteq[-x_{0},x_{0}]$ $(x_{0}>0)$. This
problem will be referred to as $\text{(LIVP)}_{0}$ when $u_{e}=0$, and
$\text{(LIVP)}_{1}$ when $u_{e}=1$. We remark that aspects of the following
linearized theory have been discussed in [2] in relation to equilibrium state
temporal stability. Here we develop this further in terms of detailing the
full qualitative structure associated with the linearized evolution problems,
which provides additional and significant information towards the analysis of
(IBVP). We now consider $\text{(LIVP)}_{0}$ and $\text{(LIVP)}_{1}$ in turn.

###  3.1. Analysis of (LIVP)0

We first seek elementary solutions to (20) and (21) in the form,

(24) $\displaystyle\overline{u}{(x,t)}=e^{ikx-
wt},\quad{(x,t)}\in\overline{D}_{\infty}$

with $k\in\mathbb{R}$ and $w\in\mathbb{C}$. On substitution from (24) in (20)
and (21) we obtain the dispersion relation

(25) $\displaystyle w=w_{0}(k)=Dk^{2}-1\quad\forall\,k\in\mathbb{R}$

and so, as expected, $\text{(LIVP)}_{0}$ is _nondispersive_
($w_{0}(k)\in\mathbb{R}$ $\forall$ $k\in\mathbb{R}$), and moreover, for each
$D>0$,

(26) $\displaystyle
w_{0}(k)<0\quad\forall\,k\in\left(\frac{-1}{\sqrt{D}},\frac{1}{\sqrt{D}}\right)$

with

(27) $\displaystyle w_{0}(0)=-1=\inf_{k\in\mathbb{R}}w_{0}(k).$

We immediately conclude that the equilibrium state $u_{e}=0$ is temporally
unstable, according to the linearized theory, at each $D>0$ (this conclusion
can readily extended to apply to the fully nonlinear and nonlocal PDE (5) with
(9), via an application of the parabolic comparison theorem to the operator
$N(w):=w_{t}-Dw_{xx}-w$ on ${D}_{T}$; for brevity we omit the details). The
Fourier integral theorem allows us to write down the solution to
$\text{(LIVP)}_{0}$ and this can then be estimated, via steepest descents, to
obtain,

(28)
$\displaystyle\overline{u}{(x,t)}\sim\frac{\sqrt{\pi}}{\sqrt{D}t^{\frac{1}{2}}}\widehat{g}(0)e^{t}e^{-x^{2}/4Dt}$

as $t\to\infty$ uniformly for $x\in\mathbb{R}$, with

$\displaystyle\widehat{g}(0)=\frac{1}{2\pi}\int_{-x_{0}}^{x_{0}}\overline{g}(s)ds\quad(>0).$

We observe from (28) that, as $t\to\infty$, there are two symmetric
‘wavefronts’ where

(29) $\displaystyle|x|\sim 2\sqrt{D}t\text{ as }t\to\infty,$

and behind the wavefronts $\overline{u}$ is growing exponentially in $t$,
whilst ahead of the wavefronts $\overline{u}$ is decaying exponentially in
$t$. This structure is consistent with (11), and the numerical solutions to
(IBVP) in section 2. It is also consistent with the rigorous result in [6] for
the corresponding evolution problem when the initial data is front-like and
localized to the left half-line.

###  3.2. Analysis of (LIVP)1

We seek elementary solutions to (20) and (21) in the form of (24), with again
$k\in\mathbb{R}$ and $w\in\mathbb{C}$. This now leads directly to the
dispersion relation

(30) $\displaystyle
w=w_{1}(k)=Dk^{2}+\frac{2}{k}\sin\frac{1}{2}k\quad\forall\,k\in\mathbb{R}$

and we observe that $w_{1}(k)$ is an even function of $k$. Moreover
$w_{1}(k)\in\mathbb{R}$ for all $k\in\mathbb{R}$ and so $\text{(LIVP)}_{1}$ is
_nondispersive_. In further analysing (30), it is convenient to introduce the
function $\Delta:\mathbb{R}^{+}\to\mathbb{R}$ such that

(31)
$\displaystyle\Delta(X)=-\frac{2}{X^{3}}\sin\frac{1}{2}X\quad\forall\,X\in\mathbb{R}^{+}.$

The zeros of $\Delta(X)$ are at

(32) $\displaystyle X=2n\pi,\quad n\in\mathbb{N},$

whilst the turning points are at

(33) $\displaystyle X=\delta_{n},\quad n\in\mathbb{N},$

with $2n\pi<\delta_{n}<2(n+1)\pi$, and

(34) $\displaystyle\delta_{n}\sim(2n+1)\pi\text{ as }n\to\infty.$

Furthermore, $\delta_{n}$ is a local maximum when $n$ is odd and a local
minimum when $n$ is even. At each local maximum point we write,

(35) $\displaystyle\Delta_{r}=\Delta(\delta_{2r-1}),\quad r=1,2,\dots.$

We observe that,

(36) $\displaystyle 0<\Delta_{r+1}<\Delta_{r},\quad r=1,2,\dots$

and

(37) $\displaystyle\Delta_{r}\sim\frac{2}{\pi^{3}(4r-1)^{3}}\text{ as
}r\to\infty,$

whilst a straightforward numerical calculation gives ${\Delta}_{1}\approx
0.00297$.

We can now readily interpret the dispersion relation (30). We first observe
that for

(38) $\displaystyle D>\Delta_{1}$

then $w_{1}(k)>0$ for all $k\in\mathbb{R}$ and so the equilibrium state
$u_{e}=1$ is temporally asymptotically stable. However, for

(39) $\displaystyle 0<D<\Delta_{1}$

then

(40) $\displaystyle\inf_{k\geq
0}w_{1}(k)=w_{1}(k_{m})=k_{m}^{2}(D-\Delta(k_{m}))<0,$

and so the equilibrium state $u_{e}=1$ is now temporally unstable. We note
that $k=k_{m}$ is uniquely determined as the smallest positive root of the
transcendental equation,

(41) $\displaystyle w_{1}^{\prime}(k)\equiv
2k(D-\Delta(k))-k^{2}\Delta^{\prime}(k)\equiv(2Dk^{3}-2\sin\frac{1}{2}k+k\cos\frac{1}{2}k)k^{-2}=0$

and it is straightforward to establish from this that,

(42) $\displaystyle k_{m}\to\begin{cases}\delta_{1}\text{ as
}&D\to\Delta_{1}^{-}\\\ k_{0}\text{ as }&D\to 0^{+}\end{cases}$

where $k=k_{0}$ is the smallest positive root of the equation

(43) $\displaystyle\tan\frac{1}{2}k=\frac{1}{2}k$

so that $2\pi<k_{0}<3\pi$, and

(44) $\displaystyle w_{1}(k_{m})\to\frac{2}{k_{0}}\sin\frac{1}{2}k_{0}\text{
as }D\to 0^{+}.$

We note that in both cases,

$\displaystyle w_{1}(k)\sim Dk^{2}\text{ as }|k|\to\infty$

and so $\text{(LIVP)}_{1}$ is well-posed. Figure 5 shows the wavelength of the
most unstable mode as a function of $D$.

The solution to $\text{(LIVP)}_{1}$ is readily obtained as

(45)
$\displaystyle\overline{u}{(x,t)}=\int_{-\infty}^{\infty}\widehat{g}(k)e^{-w_{1}(k)t}e^{ikx}dk$

for ${(x,t)}\in\overline{D}_{\infty}$, with $\widehat{g}$ being the Fourier
transform of $\overline{g}$. We obtain from (45), via Laplace’s method, that,

(46)
$\displaystyle\overline{u}{(x,t)}\sim\frac{\sqrt{\pi}2^{3/2}|\widehat{g}(k_{m})|}{(w_{1}^{\prime\prime}(k_{m}))^{\frac{1}{2}}t^{\frac{1}{2}}}e^{-w_{1}(k_{m})t}\cos{(k_{m}x+\arg(\widehat{g}(k_{m}))}$

for $|x|=O(1)$ as $t\to\infty$. It should be noted that further regions in the
large-$t$ structure of $\overline{u}$ are required when $|x|=O(t)$ as
$t\to\infty$, which gives the transition into the spatially exponentially
decaying far field for $|x|\gg O(t)$. We observe, from (46), that when
$0<D<\Delta_{1}$, the solution to $\text{(LIVP)}_{1}$ evolves into an
exponentially growing harmonic periodic state with spatial wave number
$k_{m}$, which depends upon $D$, and temporal exponential growth rate
$|w_{1}(k_{m})|$. This periodic state is stationary, and evolves when
$|x|=O(1)$ as $t\to\infty$, behind transition into the far field when
$|x|=O(t)$ as $t\to\infty$.

In relation to (IBVP), the above analyses indicate that when $t$ is large, two
symmetric permanent form wavefronts propagate to left and right, with
asymptotic propagation speeds of $\pm 2\sqrt{D}$. However, _in the region to
the rear of the two wavefronts at_ $|x|\sim 2\sqrt{D}t$, the large-$t$
structure to $\text{(LIVP)}_{1}$ indicates that, as $t\to\infty$ in (IBVP),
and specifically when $|x|=O(1)$ as $t\to\infty$, _there are two possible
steady spatial structures which emerge as a consequence of wavefront passage_
, depending upon $D$. With $u:\overline{D}_{\infty}\to\mathbb{R}$ being the
solution to (IBVP), these possibilities are:

  * (P1)

when $D>\Delta_{1}$, then $u{(x,t)}\to 1$ as $t\to\infty$, uniformly with
$|x|=O(1)$

  * (P2)

when $0<D<\Delta_{1}$, then $u{(x,t)}\to P(x)$ as $t\to\infty$, uniformly with
$|x|=O(1)$. Here $P:\mathbb{R}\to\mathbb{R}$ is a steady periodic solution to
equation (5) with (9), which is positive, has $1\in\imaginary(P)$ and has
wavelength which
(i) is close to $2\pi/k_{m}(D)$ _when $D$ is close to $\Delta_{1}$_, and is
selected well behind the emerging wavefronts by a _maximum linear growth rate
mechanism_ on the now weakly linearly unstable equilibrium state $u=1$,
and
(ii) moves away from this value as $D$ decreases, decreasing and drifting
towards the minimum wavelength of $1/2$ (see Figure 5 and section 4) as $D\to
0^{+}$, with _this wavelength now selected via the hump formation mechanism_ ,
which operates to the immediate rear of the wavefronts (see section 2).

We recall that both (P1) and (P2) are supported by the numerical solutions to
(IBVP) presented in section 2 in conjunction with the detailed linearized
theory of this section.

As a consequence of the theory in this section, and in particular to further
enable the development of theory to support and extend the conjecture (P2),
the next natural key step is to investigate the existence and structural
nature of the positive periodic steady solutions to the nonlocal equation (5)
with (9), which are conjectured to emerge in the large-$t$ development of the
solution to (IBVP). The starting point for this study begins by investigating
the emergence of periodic steady solutions via steady state bifurcations from
the equilibrium solution $u_{e}=1$. Thereafter, the detailed structure of
these fully nonlinear and nonlocal periodic solutions is developed in detail
for significant asymptotic limits. In particular, a fully developed theory is
obtained in the small $D$ limit, when the nonlocal length scale is much larger
than the diffusion length scale, leading to spatially periodic states
consisting of separated, periodically distributed, localised humps,
characterised by nonlocal effects which are regulated by weak diffusion.

##  4\. Positive Periodic Steady States

We begin by considering the steady state form of the nonlocal equation (5)
with (9), and in particular we seek to identify steady state bifurcations from
the equilibrium state $u_{e}=1$, which give rise to periodic steady states.
For the general form of the nonlocal Fisher-KPP equation, under quite general
conditions on the kernel, generic considerations concerning local bifurcations
to small amplitude bounded steady states on the real line, as the associated
equilibrium state loses its temporal stability, have been developed in [11]
and a detailed consideration of the spectral theory on the real line of the
linear operators thereby encountered has been made by Volpert and Vougalter
[29]. By restricting attention to bifurcations to _periodic steady states_ ,
we effectively restore compactness to the operators, and can then develop a
standard weakly nonlinear bifurcation theory. Indeed, under the periodic
restriction, these operator theoretic results conform with the basic weakly
nonlinear theory outlined below for the current specific situation, which we
then develop in detail onto the global bifurcation branch. Any steady state to
the nonlocal equation (5) with (9), in the present context, is a function
$F:\mathbb{R}\to\mathbb{R}$, with $F\in C^{2}(\mathbb{R})\cap
L^{\infty}(\mathbb{R})$, and such that,

(47) $\displaystyle
DF_{xx}+F\left(1-\int_{x-\frac{1}{2}}^{x+\frac{1}{2}}F(y)dy\right)=0,\quad
x\in\mathbb{R}.$

We restrict attention to considering the existence of positive periodic steady
states, which oscillate about the equilibrium state $u_{e}=1$. Specifically,
we will fix $D>0$, and consider the bifurcation to periodic steady states from
$u_{e}=1$, with fundamental wavelength $\lambda$, as the bifurcation
parameter. As a preliminary we first observe, via bootstrapping in (47), that
when $F\in C^{2}(\mathbb{R})\cap L^{\infty}(\mathbb{R})$ is a steady state,
then in fact, $F\in C^{\infty}(\mathbb{R})$. Moreover (see for example [16])
this can be improved to

(48) $\displaystyle F\in C^{\omega}(\mathbb{R})\cap L^{\infty}(\mathbb{R}).$

Now, let $F=F_{p}(x,\lambda,D)$ be a positive, periodic, steady state at
diffusivity $D$ and with fundamental wavelength $\lambda>0$. We define

(49)
$\displaystyle\alpha\equiv\max_{x\in[0,\lambda)}F_{p}(x,\lambda,D)-\min_{x\in[0,\lambda)}F_{p}(x,\lambda,D).$

the peak-to-trough magnitude of this periodic steady state. In general, we
anticipate that $\alpha=\alpha(\lambda,D)$. For fixed $D>0$, we now suppose
that a bifurcation to periodic steady states occurs from $u_{e}=1$ as
$\lambda$ passes through $\lambda=\lambda_{b}$ $(>0)$. To determine the
possible values of $\lambda_{b}$, we consider those values
$\lambda=\lambda_{b}$ when the linearized form of (47) has solution

(50) $\displaystyle F(x)=1+\alpha\cos(\frac{2\pi}{\lambda_{b}}x+\phi),\quad
x\in\mathbb{R},$

as $\alpha\to 0^{+}$, with $\phi$ being a constant phase. On substitution from
(50) into (47), and allowing $\alpha\to 0^{+}$, a non-trivial solution
requires that $\lambda_{b}$ satisfies the transcendental equation, with fixed
$D>0$,

(51)
$D=-\frac{\lambda_{b}^{3}}{4\pi^{3}}\sin\frac{\pi}{\lambda_{b}}=\Delta\left(\frac{2\pi}{\lambda_{b}}\right)$

with $\Delta(\cdot)$ as introduced in (31). We can now interpret the root
structure of (51), using (32)-(38). In particular, for each $r=1,2,\dots$,
then, with

(52) $\displaystyle\Delta_{r+1}<D<\Delta_{r},$

equation (51) has exactly $2r$ positive roots, which we label as
$\lambda_{i}^{\pm}(D)$ for $i=1,2,\dots,r$, with

(53)
$\displaystyle\frac{1}{2i}<\lambda_{i}^{-}(D)<\frac{2\pi}{\delta_{2i-1}}<\lambda_{i}^{+}(D)<\frac{1}{(2i-1)}$

for each $i=1,\dots,r$. We note that, for fixed $k\in\mathbb{N}$, then
$\lambda^{\pm}_{k}(D)$ exist and are continuous for $D\in[0,\Delta_{k})$.
Moreover, the following properties are readily established:

(54) $\displaystyle\lambda_{k}^{\pm}(D)\to\frac{2\pi}{\delta_{2k-1}}\text{ as
}D\to\Delta_{k}^{-},$ (55) $\displaystyle\lambda_{k}^{+(-)}(D)\text{ is
monotone decreasing (increasing) with }D\in[0,\Delta_{k}),$ (56)
$\displaystyle\lambda_{k}^{+}(D)\to\frac{1}{(2k-1)}^{-}\text{ as }D\to 0^{+},$
(57) $\displaystyle\lambda_{k}^{-}(D)\to\frac{1}{2k}^{+}\text{ as }D\to
0^{+}.$

Now, fix $r\in\mathbb{N}$, and fix $\Delta_{r+1}<D<\Delta_{r}$. At this fixed
$D$, there are thus $2r$ local bifurcation points to periodic steady states,
namely at critical wavelengths

(58) $\displaystyle\lambda=\lambda_{k}^{\pm}(D),\quad k=1,2,\dots,r.$

Further lengthy, but straightforward calculations (standard weakly nonlinear
asymptotics as the bifurcation point is approached) establish (which can be
made rigorous in a standard way within the classical local bifurcaton theory,
as developed in, for example, Kuznetsov [19]) that each bifurcation is a
steady state pitchfork bifurcation from $u_{e}=1$, being subcritical at
$\lambda=\lambda_{k}^{+}(D)$ and supercritical at $\lambda=\lambda_{k}^{-}(D)$
$(k=1,2,\dots,r)$. Additional investigation (numerical continuation of the
weakly nonlinear bifurcation branches) reveals that, for each $k$, the two
bifurcated branches at $\lambda_{k}^{\pm}(D)$ are connected in the
$(\lambda,\alpha)$ bifurcation diagram. We can represent this curve as
$\alpha=\alpha(\lambda;D)$ for
$\lambda_{k}^{-}(D)\leq\lambda\leq\lambda_{k}^{+}(D)$ (for each
$k=1,2,\dots,r$). Close to the bifurcation points
$\lambda=\lambda_{k}^{\pm}(D)$ a very standard weakly nonlinear local
bifurcation theory is readily developed (details are omitted for brevity as
they follow a now standard routine) to establish that, as should be expected
for a nondegenerate bifurcation,

(59)
$\displaystyle\alpha(\lambda,D)=C_{k}^{\pm}(D)\left|\lambda-\lambda_{k}^{\pm}(D)\right|^{\frac{1}{2}}+O(\left|\lambda-\lambda_{k}^{\pm}(D)\right|)$

as $\lambda\to(\lambda_{k}^{+}(D))^{-}$ or as
$\lambda\to(\lambda_{k}^{-}(D))^{+}$ respectively. Here $C_{k}^{\pm}(D)$ is a
positive constant. The bifurcated periodic steady states have the form

(60) $\displaystyle
u=F_{p}(x,\lambda,D)=1+\alpha(\lambda,D)\cos(\frac{2\pi}{\lambda}x+\phi)+O(\alpha^{2}(\lambda,D))$

as $\lambda\to\lambda_{k}^{\pm}(D)$ respectively, with $\alpha(\lambda,D)$ as
in (59), and $\phi\in[0,2\pi)$ being an arbitrary phase (representing
translational invariance of (47) in $x$). In addition, numerical investigation
confirms that, with $D$ fixed, then $\alpha(\lambda,D)$ has a single
stationary point for $\lambda\in[\lambda_{k}^{-}(D),\lambda_{k}^{+}(D)]$,
which is a maximum point.

The numerically-calculated $(\lambda,\alpha)$ bifurcation diagram is given in
Figure 7 for various values of $D$.

Figure 7. The $(\lambda,\alpha)$ bifurcation diagram with $D=3\times 10^{-6}$,
$10^{-5}$, $3\times 10^{-5}$, $10^{-4}$, $3\times 10^{-4}$, $10^{-3}$,
$2\times 10^{-3}$. The amplitude, $\alpha$, increases as $D$ decreases.

It is now straightforward to construct the bifurcation locus in the
$(\lambda,D)$ plane. This is given directly by the intersection of the curve

$\displaystyle D=-\frac{\lambda^{3}}{4\pi^{3}}\sin\frac{\pi}{\lambda}$

with the positive quadrant of the $(\lambda,D)$ plane. This intersection is in
the form of a countably infinite sequence of ‘tongue like’ curves based on the
$\lambda-$axis, and with monotone decreasing ‘height’. The $i^{th}$ ‘tongue’
has base points at $\lambda=1/(2i-1)$ and $\lambda=1/2i$, with ‘height’
$D=\Delta_{i}$. We label the set of interior points of the $i^{th}$ ‘tongue’
as $\Omega_{i}$, and its boundary as
$\partial\Omega_{i}=\overline{\Omega_{i}}\setminus\Omega_{i}$. We write

(61)
$\displaystyle\Omega=\bigcup_{i=1}^{\infty}\Omega_{i},\quad\partial\Omega=\bigcup_{i=1}^{\infty}\partial\Omega_{i}.$

At each point $(\lambda,D)\in\Omega$ there is a _unique_ (up to translation in
$x$) and _nontrivial_ periodic steady state, with fundamental wavelength
$\lambda$ and amplitude $\alpha(\lambda,D)$. Since the equation (47) is
invariant under the transformation $x\mapsto-x$, this allows for the existence
of a translation in $x$, so that at each $(\lambda,D)\in\Omega$ we may select
a representative periodic steady state $u=F_{p}(x,\lambda,D)$ which is an
_even function_ of $x\in\mathbb{R}$, so that

(62) $\displaystyle F_{p}(-x,\lambda,D)=F_{p}(x,\lambda,D)\quad\forall
x\in\mathbb{R}$ (63) $\displaystyle F_{p}^{\prime}(0,\lambda,D)=0$

and

(64) $\displaystyle
F_{p}^{\prime}\left(\pm\frac{1}{2}\lambda,\lambda,D\right)=0.$

A plot of $\Omega$ in the $(\lambda,D)$ plane is shown in Figure 8.

Figure 8. The region $\Omega$ lies below these curves, or tongues.

The numerically-calculated maximum value of $u=F_{p}(x,\lambda,D)$, which we
denote by $u_{max}$, is shown in Figure 9 for the first four tongues of
$\Omega$. This is a more suitable measure of amplitude than $\alpha$ for the
purposes of this semi-logarithmic plot, since $\alpha$ is zero at the edges of
the region where the nonlinear solutions exist.

Figure 9. Semilogarithmic plots of $u_{max}$ in the first four tongues of
$\Omega$.

We now examine the structure of the periodic steady states with
$(\lambda,D)\in\overline{\Omega}$. Firstly, on the bifurcation locus, with
$(\lambda,D)\in\partial\Omega\cap(\mathbb{R}^{+})^{2}$ we have

(65) $\displaystyle F_{p}(x,\lambda,D)=1\quad\forall x\in\mathbb{R},$

whilst it follows from (59) and (60) that

(66) $\displaystyle
F_{p}(x,\lambda,D)=1+O(|\lambda-\lambda_{0}|^{\frac{1}{2}}+|D-D_{0}|^{\frac{1}{2}})\text{
as }(\lambda,D)\to(\lambda_{0},D_{0})$

uniformly for $x\in\mathbb{R}$, with
$(\lambda_{0},D_{0})\in\partial\Omega\cap(\mathbb{R}^{+})^{2}$ and
$(\lambda,D)\in\Omega$. In terms of regularity, it follows from (48) and (47)
(see for example, [16]) that

(67) $\displaystyle F_{p}\in
C^{\omega,0,0}(\mathbb{R}\times(\overline{\Omega}\cap(\mathbb{R}^{+})^{2})).$

We can now establish the following bounds:

  * (B1)

For any $(\lambda,D)\in\Omega$, then,

$\inf_{x\in\mathbb{R}}F_{p}(x,\lambda,D)>0$

###### Proof.

Set $(\lambda^{*},D^{*})\in\Omega$, then $(\lambda^{*},D^{*})\in\Omega_{i}$
for some $i\in\mathbb{N}$. The point $(\lambda^{*},D^{*})$ can then be
connected, in $\Omega_{i}$, to a point
$(\lambda_{0},D_{0})\in\partial\Omega_{i}\cap(\mathbb{R}^{+})^{2}$ via a
straight line segment $L$. Now, suppose that

(68) $\displaystyle\inf_{x\in\mathbb{R}}F_{p}(x,\lambda^{*},D^{*})\leq 0.$

We recall from (65) that

(69) $\displaystyle\inf_{x\in\mathbb{R}}F_{p}(x,\lambda_{0},D_{0})=1.$

Thus, from (67), it follows that there exists $(\lambda_{1},D_{1})\in
L\cap\Omega_{i}$, such that

(70) $\displaystyle\inf_{x\in\mathbb{R}}F_{p}(x,\lambda_{1},D_{1})=0.$

As a consequence (since $F_{p}$ is periodic in $x\in\mathbb{R}$, together with
(67)) there exists $x_{1}\in\mathbb{R}$ such that

(71) $\displaystyle F_{p}(x_{1},\lambda_{1},D_{1})=0.$

It then follows, via (70), (71), (67) and (47), that

(72) $\displaystyle
F_{p}^{(n)}(x_{1},\lambda_{1},D_{1})=0\quad\forall\,n\in\mathbb{N}.$

It is then a consequence of (72) with (67) that,

$\displaystyle F_{p}(x,\lambda_{1},D_{1})=0\quad\forall\,x\in\mathbb{R}.$

However, $(\lambda_{1},D_{1})\in\Omega_{i}$, and we have established earlier
in this section that as such $F_{p}(x,\lambda_{1},D_{1})$ must be a _non-
trivial_ periodic function of $x\in\mathbb{R}$, with fundamental wavelength
$\lambda_{1}$, and so cannot be the trivial function. Thus we arrive at a
contradiction. The result follows. ∎

We next have

  * (B2)

For any $(\lambda,D)\in\Omega$, then,

$\sup_{x\in\mathbb{R}}F_{p}(x,\lambda,D)>1$

###### Proof.

Set $(\lambda,D)\in\Omega$, then $(\lambda,D)\in\Omega_{i}$ for some
$i\in\mathbb{N}$. Suppose that

(73) $\displaystyle\sup_{x\in\mathbb{R}}F_{p}(x,\lambda,D)\leq 1.$

Then since $F_{p}(x,\lambda,D)$ is a non-trivial periodic function in
$x\in\mathbb{R}$, with fundamental wavelength $\lambda$, it follows that there
exists $x^{*}\in\mathbb{R}$ such that

(74) $\displaystyle
0<F_{p}(x^{*},\lambda,D)=\inf_{x\in\mathbb{R}}F_{p}(x,\lambda,D)<1,$

with the left hand inequality following from (B1). From (73) and (74) it
follows that,

(75)
$\displaystyle\int_{x^{*}-\frac{1}{2}}^{x^{*}+\frac{1}{2}}F_{p}(y,\lambda,D)dy<1.$

Also, from (47), we have

(76) $\displaystyle
F_{p}^{\prime\prime}(x^{*},\lambda,D)=-F_{p}(x^{*},\lambda,D)\left(1-\int_{x^{*}-\frac{1}{2}}^{x^{*}+\frac{1}{2}}F_{p}(y,\lambda,D)dy\right)<0,$

via (74) and (75). However, via (74) and (67), $x=x^{*}$ is a local minimum
point for ${F_{p}(x,\lambda,D)}$, and so

$F_{p}^{\prime\prime}(x^{*},\lambda,D)\geq 0$

which contradicts (76). Thus,

$\sup_{x\in\mathbb{R}}F_{p}(x,\lambda,D)>1,$

as required. ∎

Correspondingly, we now have

  * (B3)

For any $(\lambda,D)\in\Omega$, then,

$\inf_{x\in\mathbb{R}}F_{p}(x,\lambda,D)<1$

###### Proof.

This follows, with the obvious adjustments, that of (B2). ∎

To complete this section, we next consider in detail the structure of the
periodic steady states in the $(\lambda,D)$ plane as $D\to 0^{+}$. We begin by
examining this limit for the periodic steady states in the principal tongue
$\Omega_{1}$, and then give the corresponding results for $\Omega_{i}$,
$i=2,3,4,\dots$ .

###  4.1. Asymptotic structure as $D\to 0^{+}$ with $(\lambda,D)\in\Omega_{1}$

We consider the asymptotic structure of $F_{p}(x,\lambda,D)$ with
$(\lambda,D)\in\Omega_{1}$ as $D\to 0^{+}$. This is achieved via a detailed
application of the method of matched asymptotic expansions. Due to the
evenness of $F_{p}$ in $x$, we need only consider this structure for
$x\in\left[0,\frac{1}{2}\lambda\right]$, with
$\lambda\in\left(\frac{1}{2},1\right)$ fixed as $D\to 0^{+}$. We recall that,
$F_{p}(x,\lambda,D)>0$ for all $x\in\left[0,\frac{1}{2}\lambda\right]$, whilst
evenness requires

(77) $\displaystyle
F_{p}^{\prime}(0,\lambda,D)=F_{p}^{\prime}\left(\frac{1}{2}\lambda,\lambda,D\right)=0.$

An examination of (47), together with a consideration of numerical solutions,
dictates that, for $\lambda\in\left(\frac{1}{2},1\right)$ fixed, the
asymptotic structure of $F_{p}(x,\lambda,D)$, as $D\to 0^{+}$, develops into a
three region structure as follows:

_Region I_ (support region)

$\displaystyle x\in[0,S-O(D^{\gamma})),\,\,F_{p}=O(1)^{+}\text{ as }D\to
0^{+}$

with $0<S<\frac{1}{2}\lambda$ and $\gamma>0$ to be determined. In general, $S$
will depend upon both $\lambda$ and $D$, with $S=O(1)^{+}$ as $D\to 0^{+}$,
and $\lambda\in\left(\frac{1}{2},1\right)$. With this in mind, we expand
$S(\lambda,D)$ as,

(78) $\displaystyle
S(\lambda,D)=\bar{S}(\lambda)+D^{\gamma}S_{1}(\lambda)+o(D^{\gamma}),\text{ as
}D\to 0^{+}$

with $\lambda\in\left(\frac{1}{2},1\right)$, which allows for weak
displacements in the location of region II below.

_Region II_ (transition region)

$\displaystyle
x\in\left(S-O(D^{\gamma}),S+O(D^{\gamma})\right),\,\,F_{p}=O(D^{\delta})^{+}\text{
as }D\to 0^{+}$

with $\delta>0$ to be determined.

_Region III_ (exponential region)

$\displaystyle
x\in\left(S+O(D^{\gamma}),\frac{1}{2}\lambda\right],\,\,F_{p}=O(E(D))^{+}\text{
as }D\to 0^{+}$

with $E(D)$ indicating terms exponentially small in $D$ as $D\to 0^{+}$.

The above structure requires, for the change in structure across region II,
that the leading order interval length of region III (which is the leading
order separation of consecutive support regions) must have the same length as
the half span of the nonlocal term, which is $\frac{1}{2}$, and so,

$\displaystyle\bar{S}+\frac{1}{2}=\lambda-\bar{S}$

which gives,

(79)
$\displaystyle\bar{S}=\frac{1}{2}\left(\lambda-\frac{1}{2}\right)=\bar{S}(\lambda)$

and we note from this that

$\displaystyle 0<\bar{S}(\lambda)<\frac{1}{4}$

for $\lambda\in\left(\frac{1}{2},1\right)$, with,

(80)
$\displaystyle\bar{S}(\lambda)-\frac{1}{2}<-\bar{S}(\lambda)\quad\forall\,\lambda\in\left(\frac{1}{2},1\right).$

With (78)-(80), then equation (47) requires that

(81)
$\displaystyle\int_{x-\frac{1}{2}}^{x+\frac{1}{2}}F_{p}(y,\lambda,D)dy=\int_{-\frac{1}{2}}^{\frac{1}{2}}F_{p}(y,\lambda,D)dy+O(E(D))\text{
as }D\to 0^{+},$

for $x\in\left[0,\bar{S}(\lambda)-O(D^{\gamma})\right)$, whilst

(82)
$\displaystyle\int_{-\frac{1}{2}}^{\frac{1}{2}}F_{p}(y,\lambda,D)dy=1+\alpha_{1}D+\beta_{1}D^{r}+o(D^{r})$

as $D\to 0^{+}$, with the constants $r>1$, $\alpha_{1}$ and $\beta_{1}$ to be
determined.

We begin in region I and expand in the form

(83) $\displaystyle
F_{p}(x,\lambda,D)=F_{0}(x,\lambda)+D^{m}F_{1}(x,\lambda)+o(D^{m})\text{ as
}D\to 0^{+},$

with $x\in[0,S(\lambda)-O(D^{\gamma}))$ as $m>0$ to be determined. We
substitute into (47), to obtain, at leading order as $D\to 0^{+}$,

(84) $\displaystyle F_{0}^{\prime\prime}-\alpha_{1}F_{0}=0,\quad
0<x<\bar{S}(\lambda)$ (85) $\displaystyle
F_{0}^{\prime}(0,\lambda)=F_{0}(\bar{S}(\lambda),\lambda)=0$ (86)
$\displaystyle F_{0}(x,\lambda)>0,\quad 0\leq x<\bar{S}(\lambda)$

with the second condition in (85) required by asymptotic matching (via Van
Dyke’s principle, [26]) between region I and region II. Finally, (82) requires

(87) $\displaystyle\int_{0}^{\bar{S}(\lambda)}F_{0}(y,\lambda)dy=\frac{1}{2},$

using the evenness of $F_{p}(x,\lambda,D)$, and its asymptotic form in regions
II and III. Now, the problem (84)-(86) is a classical Sturm-Liouville
eigenvalue problem, with eigenvalue $-\alpha_{1}$, whilst condition (86)
requires that $-\alpha_{1}$ is the _lowest_ eigenvalue. This may be treated
directly, to obtain,

(88)
$\displaystyle\alpha_{1}=-\frac{\pi^{2}}{\left(\lambda-\frac{1}{2}\right)^{2}}$

with

(89) $\displaystyle F_{0}(x,\lambda)=B\cos\frac{\pi
x}{\left(\lambda-\frac{1}{2}\right)},\quad 0\leq x\leq\bar{S}(\lambda)$

and $B>0$ an arbitrary constant. It remains to apply condition (87), which
gives,

(90) $\displaystyle B=\frac{\pi}{(2\lambda-1)},$

and so,

(91) $\displaystyle F_{0}(x,\lambda)=\frac{\pi}{(2\lambda-1)}\cos\frac{\pi
x}{\left(\lambda-\frac{1}{2}\right)},\quad 0\leq x\leq\bar{S}(\lambda)$

and $\bar{S}(\lambda)=\frac{1}{2}\left(\lambda-\frac{1}{2}\right)$. Via (81),
(82) and (88), we also have

(92)
$\displaystyle\int_{x-\frac{1}{2}}^{x+\frac{1}{2}}F_{p}(y,\lambda,D)dy=2\int_{0}^{\frac{1}{2}}F_{p}(y,\lambda,D)dy=1-\frac{\pi^{2}}{\left(\lambda-\frac{1}{2}\right)^{2}}D+\beta_{1}D^{r}+o(D^{r})$

as $D\to 0^{+}$, with $x\in[0,S(\lambda)-O(D^{\gamma}))$. This completes the
leading order structure in region I. We now move on to consider region II, the
transition region. In region II we introduce the scaled coordinate $X=O(1)$ as
$D\to 0^{+}$, given by,

(93) $\displaystyle x=S(\lambda,D)+D^{\gamma}X$

It then follows from the expansion in region I, on moving into region II, via
(91), (83) with (87), that $F_{p}=O(D^{\gamma})$ as $D\to 0^{+}$ in region II.
Thus, we expand in the form

(94) $\displaystyle
F_{p}(X,\lambda,D)=D^{\gamma}\bar{F}_{0}(X,\lambda)+o(D^{\gamma})\text{ as
}D\to 0^{+}$

with $X=O(1)$. We remark here that the term at $O(D^{\gamma})$ in
$S(\lambda,D)$ could be removed by a constant shift in the coordinate $X$.
However, it is convenient for later matching purposes to leave this shift in
$S$; this also leads to a parameter free equation at leading order, with all
parameters shifted into the matching condition. We next consider the form of
the nonlocal term in (47) in region II, which can be written as,

$\displaystyle\int_{x-\frac{1}{2}}^{x+\frac{1}{2}}F_{p}(y,\lambda,D)dy=\int_{S(\lambda,D)-\frac{1}{2}+D^{\gamma}X}^{S(\lambda,D)+\frac{1}{2}+D^{\gamma}X}F_{p}(y,\lambda,D)dy$
(95)
$\displaystyle=\int_{S(\lambda,D)-\frac{1}{2}+D^{\gamma}X}^{S(\lambda,D)-\frac{1}{2}}F_{p}(y,\lambda,D)dy+\int_{S(\lambda,D)-\frac{1}{2}}^{S(\lambda,D)+\frac{1}{2}-O(1)}F_{p}(y,\lambda,D)dy+\int_{S(\lambda,D)+\frac{1}{2}-O(1)}^{S(\lambda,D)+\frac{1}{2}+D^{\gamma}X}F_{p}(y,\lambda,D)dy.$

Now, using the periodicity and evenness of $F_{p}$, and its structure in each
of regions I-III, we have firstly that,

(96)
$\displaystyle\int_{S(\lambda,D)-\frac{1}{2}+D^{\gamma}X}^{S(\lambda,D)-\frac{1}{2}}F_{p}(y,\lambda,D)dy=O(E(D)D^{\gamma})$

as $D\to 0^{+}$ with $X=O(1)$, whilst,

(97)
$\displaystyle\int_{S(\lambda,D)-\frac{1}{2}}^{S(\lambda,D)+\frac{1}{2}-O(1)}F_{p}(y,\lambda,D)dy=1-\frac{\pi^{2}}{\left(\lambda-\frac{1}{2}\right)^{2}}D+\beta_{1}D^{r}+o(D^{r})$

as $D\to 0^{+}$. The final term in (95) may now be written as

$\displaystyle\int_{S(\lambda,D)+\frac{1}{2}-O(1)}^{S(\lambda,D)+\frac{1}{2}+D^{\gamma}X}F_{p}(y,\lambda,D)dy=D^{\gamma}\int_{Y=-O(D^{-\gamma})}^{Y=X}F_{p}\left(S(\lambda,D)+\frac{1}{2}+D^{\gamma}Y,\lambda,D\right)dY$
$\displaystyle=D^{\gamma}\int_{Y=-O(D^{-\gamma})}^{Y=X}F_{p}(\lambda-S(\lambda,D)+D^{\gamma}Y,\lambda,D)dY$
(98)
$\displaystyle=D^{\gamma}\int_{Y=-O(D^{-\gamma})}^{Y=X}F_{p}(-S(\lambda,D)+D^{\gamma}Y,\lambda,D)dY$
$\displaystyle=D^{\gamma}\int_{Y=-O(D^{-\gamma})}^{Y=X}F_{p}(S(\lambda,D)-D^{\gamma}Y,\lambda,D)dY\sim
D^{2\gamma}\int_{Y=-\infty}^{Y=X}\bar{F}_{0}(-Y,\lambda)dY$
$\displaystyle=D^{2\gamma}\int_{z=-X}^{z=\infty}\bar{F}_{0}(z,\lambda)dz$

as $D\to 0^{+}$ with $X=O(1)$. Using (96)-(98) in (95), we finally arrive at

$\int_{S(\lambda,D)-\frac{1}{2}+D^{\gamma}X}^{S(\lambda,D)+\frac{1}{2}+D^{\gamma}X}F_{p}(y,\lambda,D)dy$
(99)
$=1-\frac{\pi^{2}}{\left(\lambda-\frac{1}{2}\right)^{2}}D+\beta_{1}D^{r}+D^{2\gamma}\int_{z=-X}^{z=\infty}\bar{F}_{0}(z,\lambda)dz+o(D^{2\gamma},D^{r},E(D))$

as $D\to 0^{+}$ with $X=O(1)$. We now substitute from (93), (94) and (99) into
equation (47). To obtain a non-trivial balance at leading order requires us to
choose

(100) $\displaystyle\gamma=\frac{1}{4}.$

At leading order, equation (47) then becomes,

(101)
$\displaystyle\Psi_{\bar{X}\bar{X}}-\Psi\int_{w=-\bar{X}}^{w=\infty}\Psi(w)dw=0,\quad\bar{X}\in\mathbb{R},$

which is both nonlinear and nonlocal. In (101) we have introduced the simple
scalings

(102)
$\displaystyle\bar{F}_{0}=(2\lambda-1)^{-\frac{3}{2}}\Psi,~{}~{}~{}X=(2\lambda-1)^{\frac{1}{2}}\bar{X}.$

Equation (101) is now completed by matching conditions between region II and
region I (as $\bar{X}\to-\infty$) and region III (as $\bar{X}\to\infty$). The
matching process is straightforward (following Van Dyke’s asymptotic matching
principle, [26]), and leads to the two boundary conditions,

(103) $\displaystyle\Psi(\bar{X})\to 0\text{ as }\bar{X}\to\infty,$ (104)
$\displaystyle\Psi(\bar{X})\sim-2\pi^{2}(\bar{X}+l)\text{ as
}\bar{X}\to-\infty.$

Here

(105) $\displaystyle S_{1}(\lambda)=(2\lambda-1)^{\frac{1}{2}}l.$

It is also required that,

(106) $\displaystyle\Psi(\bar{X})>0\quad\forall\,\bar{X}\in\mathbb{R}.$

We observe that the problem (101), (103), (104) and (106) is a nonlinear,
nonlocal eigenvalue problem, with real eigenvalue $l$. It is also free of
parameters. A numerical investigation of this problem establishes that a
solution exists if and only if $l=l^{*}$, and the solution is unique. Here

(107) $\displaystyle l^{*}\approx-\frac{3.493}{2\pi^{2}},$

and a numerically-determined graph of $\Psi(\bar{X})$ against $\bar{X}$,
calculated using a simple finite difference method and the trapezium rule, is
shown in Figure 10.

Figure 10. The numerically-calculated solution of (101) subject to (102). The
broken line is $\Psi=-2\pi^{2}(\bar{X}+l^{*})$.

It is also straightforward to establish that

(108)
$\displaystyle\Psi(\bar{X})=-2\pi^{2}(\bar{X}+l^{*})+\Psi_{-\infty}\bar{X}^{-2}e^{-\frac{1}{2}\pi
X^{2}}(1+o(1))$

as $\bar{X}\to-\infty$, whilst,

(109) $\displaystyle\Psi(\bar{X})=\Psi_{\infty}e^{-\frac{1}{2}\pi
X^{2}}(1+o(1))$

as $\bar{X}\to+\infty$. Here $\Psi_{\infty}$ ($>0$) and $\Psi_{-\infty}$ are
parameter free, globally determined constants. In principle, these can be
determined from the numerical solution shown in Figure 10, but in practice,
since these corrections are exponentially small, we are not able to resolve
them with sufficient accuracy.

We now return to region I. To allow matching with region II we now require

(110) $\displaystyle m=\frac{1}{4},$

whilst the most structured balance in equation (47) dictates that

(111) $\displaystyle r=m+1=\frac{5}{4}.$

The problem for $F_{1}$ is then,

(112) $\displaystyle
F_{1}^{\prime\prime}+\frac{\pi^{2}}{\left(\lambda-\frac{1}{2}\right)^{2}}F_{1}$
$\displaystyle=\beta_{1}F_{0}(x,\lambda)$
$\displaystyle=\frac{\beta_{1}\pi}{(2\lambda-1)}\cos\frac{\pi
x}{\left(\lambda-\frac{1}{2}\right)},\quad
0<x<\frac{1}{2}\left(\lambda-\frac{1}{2}\right),$ (113) $\displaystyle
F_{1}^{\prime}(0,\lambda)=F_{1}\left(\frac{1}{2}\left(\lambda-\frac{1}{2}\right),0\right)=0,$
(114)
$\displaystyle\int_{0}^{\frac{1}{2}\left(\lambda-\frac{1}{2}\right)}F_{1}(y,\lambda)dy=0.$

The second of conditions (113) arises from matching with region II (with both
expansions taken up to $O(D^{\frac{1}{4}})$ in regions I and II). A solution
to (112) and (113) exists if and only if

(115) $\displaystyle\beta_{1}=0,$

after which

(116) $\displaystyle F_{1}(x,\lambda)=C\cos\frac{\pi
x}{\left(\lambda-\frac{1}{2}\right)}$

with $C$ an arbitrary real constant. However, substitution from (116) into
(114), then requires $C=0$, and so,

(117) $\displaystyle
F_{1}(x,\lambda)=0\quad\forall\,x\in\left[0,\frac{1}{2}\left(\lambda-\frac{1}{2}\right)\right].$

Thus, in region I we have

(118) $\displaystyle F_{p}(x,\lambda,D)=\frac{\pi}{(2\lambda-1)}\cos\frac{\pi
x}{\left(\lambda-\frac{1}{2}\right)}+o(D^{\frac{1}{4}})$

as $D\to 0^{+}$ with
$x\in\left[0,\frac{1}{2}\left(\lambda-\frac{1}{2}\right)\right)$.

We finally move into the exponential region, region III. In this region
$F_{p}$ is exponentially small in $D$ as $D\to 0^{+}$, and develops as a WKB
expansion. For brevity we omit details, but obtain

(119) $\displaystyle
F_{p}(x,\lambda,D)=A^{*}(\lambda)e^{-\Phi_{0}(\lambda)D^{-\frac{1}{2}}}\cosh(\frac{\Phi(x)}{D^{\frac{1}{2}}})(1+o(1))$

as $D\to 0^{+}$, with
$x\in\left(\frac{1}{2}(\lambda-\frac{1}{2})+O(D^{\frac{1}{4}}),\frac{1}{2}\lambda\right]$.
Here,

(120)
$\displaystyle\Phi(x)=\frac{1}{\sqrt{2}}\int_{x}^{\frac{1}{2}\lambda}\left(1-\sin(\frac{\pi
w}{\left(\lambda-\frac{1}{2}\right)})\right)^{\frac{1}{2}}dw\quad\forall\,x\in\left[\frac{1}{2}\left(\lambda-\frac{1}{2}\right),\frac{1}{2}\lambda\right)$

and

(121)
$\displaystyle\Phi_{0}(\lambda)=\Phi\left(\frac{1}{2}\left(\lambda-\frac{1}{2}\right)\right).$

The constant $A^{*}(\lambda)$ is independent of $D$ and is related to the
constant $\Psi_{\infty}$ and $\lambda$, via matching expansion (119) as
$x\to\frac{1}{2}\left(\lambda-\frac{1}{2}\right)^{-}$ with expansion (94) as
$X\to\infty$, which gives,

(122) $\displaystyle
A^{*}(\lambda)=\frac{2\Psi_{\infty}}{(2\lambda-1)^{\frac{3}{2}}}$

with details omitted for brevity. This completes the asymptotic structure as
$D\to 0^{+}$, with fixed $\lambda\in\left(\frac{1}{2},1\right)$.

We make the following key observations. First, up to terms exponentially small
in D, as $D\to 0^{+}$,

(123)
$\displaystyle\operatorname{supp}(F_{p}(x,\lambda,D))=[-S(\lambda,D),S(\lambda,D)]$

with

(124) $\displaystyle
S(\lambda,D)=\frac{1}{2}\left(\lambda-\frac{1}{2}\right)-\frac{3.493}{\sqrt{2}\pi^{2}}\left(\lambda-\frac{1}{2}\right)^{\frac{1}{2}}D^{\frac{1}{4}}+o(D^{\frac{1}{4}})\,\text{
as }D\to 0^{+},$

whilst,

(125)
$\displaystyle\alpha(\lambda,D)=\frac{\pi}{(2\lambda-1)}+o(D^{\frac{1}{4}})\,\text{
as }D\to 0^{+}$

and occurs at $x=0$. In addition,

(126)
$\displaystyle\int_{-S(\lambda,D)}^{+S(\lambda,D)}F_{p}(y,\lambda,D)dy=1-\frac{\pi^{2}}{\left(\lambda-\frac{1}{2}\right)^{2}}D+o(D^{\frac{5}{4}})$

as $D\to 0^{+}$. For comparison, we present in Figure 11, a numerically
determined representation of $F_{p}$, with $\lambda=0.75$ and various values
of $D$.

Figure 11. The solution $F_{p}$, calculated numerically for
$\lambda=\frac{3}{4}$ and $D=10^{-3}$, $10^{-4}$, $10^{-5}$, $10^{-6}$,
$10^{-7}$ and $10^{-8}$. The broken line shows the leading order outer
solution given by (118).

This is in excellent agreement with the asymptotic form developed above.

Finally, we observe from (123)-(126) in particular, that the asymptotic
structure developed above, for $F_{p}(x,\lambda,D)$ as $D\to 0^{+}$, with
fixed $\lambda\in\left(\frac{1}{2},1\right)$, becomes nonuniform as
$\lambda\to\frac{1}{2}$, and, specifically, when
$\lambda=\frac{1}{2}+O(D^{\frac{1}{2}})$ as $D\to 0^{+}$. For the present, we
continue by considering the asymptotic structure of $F_{p}(x,\lambda,D)$ as
$D\to 0^{+}$ in each of the remaining tongues $\Omega_{i}$, $i=2,3,\dots$ .

###  4.2. Asymptotic structure as $D\to 0^{+}$ with
$(\lambda,D)\in\Omega_{i}$, $i=2,3,\dots$

We now consider the asymptotic form of $F_{p}(x,\lambda,D)$ with
$(\lambda,D)\in\Omega_{i}$ ($i\geq 2$) as $D\to 0^{+}$. We, again, need only
consider the structure for $x\in\left[0,\frac{1}{2}\lambda\right]$, with now
$\lambda\in\left(\frac{1}{2i},\frac{1}{2i-1}\right)$ fixed, as $D\to 0^{+}$.
The structure is again as in the case of $\Omega_{1}$, and for brevity, we
restrict attention to the key support region, labelled as region I. The
analysis follows exactly that of the previous subsection, and we therefore
present only the salient features. In this case we obtain,

(127)
$\displaystyle\bar{S}(\lambda)=\frac{1}{2}i\left(\lambda-\frac{1}{2i}\right),$
(128) $\displaystyle
F_{0}(x,\lambda)=\frac{\pi}{(2i-1)(2i\lambda-1)}\cos\frac{\pi
x}{i\left(\lambda-\frac{1}{2i}\right)}\quad\forall\,x\in[0,\bar{S}(\lambda)],$
(129)
$\displaystyle(2i-1)\int_{-S(\lambda,D)}^{S(\lambda,D)}F_{p}(y,\lambda,D)dy=1-\frac{\pi^{2}}{i^{2}\left(\lambda-\frac{1}{2i}\right)^{2}}D+o(D)$

as $D\to 0^{+}$, whilst,

(130)
$\displaystyle\alpha(\lambda,D)=\frac{\pi}{(2i-1)(2i\lambda-1)}+o(1)\text{ as
}D\to 0^{+}$

with $\lambda\in\left(\frac{1}{2i},\frac{1}{2i-1}\right)$ fixed. We observe
that, as before, this structure becomes nonuniform when
$\lambda=\frac{1}{2i}+o(1)$ as $D\to 0^{+}$.

###  4.3. Asymptotic structure as $D\to 0^{+}$ with
$\lambda=\frac{1}{2}+O(D^{\frac{1}{2}})$

We return to $(\lambda,D)\in\Omega_{1}$, and now develop the structure to
$F_{p}(x,\lambda,D)$ with $\lambda=\frac{1}{2}+O(D^{\frac{1}{2}})$ and
$x\in\left[0,\frac{1}{2}\lambda\right]$ as $D\to 0^{+}$. We write

(131) $\displaystyle\lambda=\frac{1}{2}+D^{\frac{1}{2}}\bar{\lambda}$

with $\bar{\lambda}=O(1)^{+}$ as $D\to 0^{+}$. An examination of (123)-(125),
with (131), then requires,

(132) $\displaystyle x=O(D^{\frac{1}{2}})^{+}\text{ and
}F_{p}=O(D^{-\frac{1}{2}})^{+}$

as $D\to 0^{+}$ in the support region, with

(133) $\displaystyle F_{p}=O(E(D)),$

when

(134) $\displaystyle
x\in\left(O(D^{\frac{1}{2}}),\frac{1}{4}+\frac{1}{2}\bar{\lambda}D^{\frac{1}{2}}\right]$

in the exponential region. Following (132), in the support region we introduce

(135) $\displaystyle\hat{X}=\frac{x}{D^{\frac{1}{2}}}=O(1)^{+}$

as $D\to 0^{+}$, and expand in the form

(136) $\displaystyle
F_{p}(\hat{X},\bar{\lambda},D)=v(\hat{X},\bar{\lambda})D^{-\frac{1}{2}}+o(D^{-\frac{1}{2}})$

as $D\to 0^{+}$ with $\hat{X}=O(1)^{+}$. After a careful consideration of the
nonlocal term, substitution from (131), (135) and (136) into (47) gives, at
leading order, the nonlocal, nonlinear equation,

(137) $\displaystyle
v_{\hat{X}\hat{X}}+v\left(1-2\int_{-\infty}^{\infty}v(s,\bar{\lambda})ds+\int_{\hat{X}-\bar{\lambda}}^{\hat{X}+\bar{\lambda}}v(s,\bar{\lambda})ds\right)=0;\quad\hat{X}>0$

which must be solved subject to the conditions,

(138) $\displaystyle v_{\hat{X}}(0,\bar{\lambda})=0,$ (139) $\displaystyle
v(\hat{X},\bar{\lambda})\to 0\text{ as }\hat{X}\to\infty,$ (140)
$\displaystyle v(\hat{X},\bar{\lambda})>0\quad\forall\,x\geq 0,$ (141)
$\displaystyle
v(-\hat{X},\bar{\lambda})=v(\hat{X},\bar{\lambda})\quad\forall\hat{X}\geq 0.$

Here, the boundary condition (139) accommodates asymptotic matching to the
exponential region.

The problem (137)-(141) has been considered numerically, which has
demonstrated that a unique solution exists for each $\bar{\lambda}>0$. We
observe that this solution has,

(142) $\displaystyle v(\hat{X},\bar{\lambda})\sim
v_{\infty}(\bar{\lambda})e^{-\sigma_{\infty}(\bar{\lambda})\hat{X}}\text{ as
}\hat{X}\to\infty$

with

(143)
$\displaystyle\sigma_{\infty}(\bar{\lambda})=4\int_{0}^{\infty}v(s,\bar{\lambda})ds-1>0$

and $v_{\infty}(\bar{\lambda})>0$ a globally dependent constant. Figure 12
shows the numerical solution of (137) to (141) for various values of
$\bar{\lambda}$. The solution becomes larger and narrower as $\bar{\lambda}$
increases, but then becomes smaller and fatter as $\bar{\lambda}$ increases
past about five (see also Figure 14). Also shown is a comparison with the
asymptotic solutions for $\bar{\lambda}$ both large and small, with which
there is excellent agreement.

Figure 12. The numerical solution (bold curves) of (137) to (141) for
$\bar{\lambda}=0.5$, $1$, $10$, $50$ and $100$. The broken line is the
asymptotic solution for $\bar{\lambda}\ll 1$ for $\bar{\lambda}=0.5$ and $1$,
given by (156), whilst the dash-dotted line is the asymptotic solution for
$\bar{\lambda}\gg 1$ for $\bar{\lambda}=50$ and $100$, which comes from
rescaling (118).

Figure 13 shows the numerically calculated periodic solution of the full
problem for $D=10^{-3}$ and $\lambda=\frac{1}{2}+10\sqrt{D}$, along with the
corresponding asymptotic solution for $\bar{\lambda}=10$ and the solution
(118).

Figure 13. The numerically calculated periodic solution of the full problem
for $D=10^{-3}$ and $\lambda=\frac{1}{2}+10\sqrt{D}$, along with the
corresponding asymptotic solution for $\bar{\lambda}=10$ (broken line) and the
solution (118), (dash-dotted line).

The agreement is excellent, and we can also see that the $\lambda=O(1)$
asymptotic solution, (118), is not a good approximation to the solution if
$\lambda-\frac{1}{2}$ is of $O(\sqrt{D})$, as expected. The maximum of $v$ is
at $\hat{X}=0$ and a graph of $v(0,\bar{\lambda})$ against $\bar{\lambda}$ is
shown in Figure 14, which shows that there is a single maximum at
$\bar{\lambda}\approx 5.8$, with, as required,

(144)
$\displaystyle\begin{split}v(0,\bar{\lambda})&\sim\frac{\pi}{2\bar{\lambda}}\text{
as }\bar{\lambda}\to\infty,\\\ v(0,\bar{\lambda})&\to 0^{+}\text{ as
}\bar{\lambda}\to 0^{+}\end{split}$

Figure 14. The maximum value, $v(0,\bar{\lambda})$ of the numerically-
calculated solution for (137) to (141). The broken lines show the predicted
asymptotic behaviour as $\bar{\lambda}\to 0$ and $\bar{\lambda}\to\infty$.

We note that,

(145)
$\displaystyle\alpha(\bar{\lambda},D)=D^{-\frac{1}{2}}v(0,\bar{\lambda})+o(D^{-\frac{1}{2}})$

as $D\to 0^{+}$. The amplitude given by $\alpha(\bar{\lambda},D)$ is
indistinguishable from the amplitude given by the numerical solution of the
full problem, shown in Figure 7, for $D$ less than about $10^{-4}$.

To complete this subsection, we examine the problem (137)-(141) in the limit
$\bar{\lambda}\to 0^{+}$. A balancing of terms in (137) leads us to write,

(146) $\displaystyle\tilde{X}=\bar{\lambda}\hat{X}=O(1)^{+}\text{ as
}\bar{\lambda}\to 0^{+},$

and expand in the form

(147) $\displaystyle
v(\tilde{X},\bar{\lambda})=\bar{\lambda}\tilde{v}(\tilde{X})+o(\bar{\lambda})\text{
as }\bar{\lambda}\to 0^{+}.$

Using (146) and (147), the nonlocal term in (137) becomes

(148)
$\displaystyle\int_{\hat{X}-\bar{\lambda}}^{\hat{X}+\bar{\lambda}}v(s,\bar{\lambda})ds=\int_{\hat{X}-\bar{\lambda}}^{\hat{X}+\bar{\lambda}}(\bar{\lambda}\tilde{v}(\bar{\lambda}s)+o(\bar{\lambda}))ds=\int_{\bar{\lambda}\hat{X}-\bar{\lambda}^{2}}^{\bar{\lambda}\hat{X}+\bar{\lambda}^{2}}(\tilde{v}(w)+o(1))dw=2\bar{\lambda}^{2}\tilde{v}(\tilde{X})+o(\bar{\lambda}^{2})$

as $\bar{\lambda}\to 0^{+}$. The most structured balance at leading order then
requires,

(149)
$\displaystyle\int_{-\infty}^{\infty}v(\tilde{X},\bar{\lambda})d\tilde{X}=\frac{1}{2}\bar{\lambda}+\tilde{I}\bar{\lambda}^{3}+o(\bar{\lambda}^{3})$

as $\bar{\lambda}\to 0^{+}$ with $\tilde{I}$ a constant to be determined. On
substitution from (146)-(149) into (137)-(141), we arrive at the leading order
problem,

(150)
$\displaystyle\tilde{v}_{\tilde{X}\tilde{X}}-2\tilde{v}(\tilde{I}-\tilde{v})=0,\,\tilde{X}>0,$
(151) $\displaystyle\tilde{v}_{\tilde{X}}(0)=0,$ (152)
$\displaystyle\tilde{v}(\tilde{X})\to 0\text{ as }\tilde{X}\to\infty,$ (153)
$\displaystyle\tilde{v}(\tilde{X})>0\quad\tilde{X}\geq 0,$ (154)
$\displaystyle\int_{0}^{\infty}\tilde{v}(w)dw=\frac{1}{4}.$

This problem is a nonlinear eigenvalue problem, with eigenvalue
$\tilde{I}\in\mathbb{R}$, which can be solved directly. There is a single
eigenvalue given by

(155) $\displaystyle\tilde{I}=\frac{1}{72}$

with the associated eigenfunction uniquely determined as

(156)
$\displaystyle\tilde{v}(\tilde{X})=\frac{1}{48}\sech^{2}\left(\frac{1}{12}\tilde{X}\right)\quad\forall~{}\tilde{X}\geq
0.$

Thus, (147) and (156) give

(157) $\displaystyle v(0,\bar{\lambda})\sim\frac{1}{48}\bar{\lambda}\,\text{
as }\bar{\lambda}\to 0^{+}$

which is shown in Figure 14, and (154) and (155) lead to

(158)
$\displaystyle\int_{-\infty}^{\infty}v(\hat{X},\bar{\lambda})d\hat{X}=\frac{1}{2}+\frac{1}{72}\bar{\lambda}^{2}+o(\bar{\lambda}^{2})$

as $\bar{\lambda}\to 0^{+}$. Equations (156)-(158) are in excellent agreement
with the numerical solution to problem (137)-(141), when $\bar{\lambda}$ is
small

A final remark in this section relates to the temporal stability of the
periodic steady states identified with $(\lambda,D)\in\Omega$. In this
respect, we have performed a limited stability analysis of
$u=F_{p}(x,\lambda,D)$ with $(\lambda,D)\in\Omega$ when $D$ is small by
considering the exponentially small part of the solution. Without presenting
details, this analysis reveals that each such periodic steady state is
locally, temporally, asymptotically stable to perturbations in the region
between the $O(1)$ parts of the solution, where we would expect any
instability to occur (extended intervals where $u$ remains uniformly very
small, when long enough, can destabalise as they inherit the linear
instability characteristics of the zero equilibrium). In relation to (IBVP),
we observe that the results of this section support the possibility (P2), for
the case when $0<D<\Delta_{1}$.
We readily verify that the wavelength selected by (IBVP) has, _when $D$ is
close to $\Delta_{1}$_,

(159)
$\displaystyle\lambda\approx\lambda_{m}(D)=\frac{2\pi}{k_{m}(D)}\in\Omega_{1}$

and so we have, in (P2),

(160) $\displaystyle P(x)=F_{p}(x,\lambda_{m}(D),D),~{}x\in\mathbb{R},$

However this wavelength decreases as $D$ decreases, _approaching the limiting
value of $\frac{1}{2}$ as $D\to 0^{+}$_, and so now in (P2),

(161) $\displaystyle P(x)=F_{p}(x,\lambda(D),D),~{}x\in\mathbb{R},$

_where now $\lambda(D)\to\frac{1}{2}^{+}$ as $D\to 0^{+}$_, which therefore
has the structure given in detail in section 4.3. In particular the periodic
solution selected therefore is of the nature of a spike, with base width of
$O(D^{\frac{1}{2}})$ and height of $O(D^{-\frac{1}{2}})$. This is in full
accord with the numerical solution of section 2. A final point to note is that
the detailed analysis of the existence and structure of the family of steady
periodic solution in the first tongue $\Omega_{1}$ on the $(\lambda,D)$ plane
given in this section not only provides substance to the conjecture (P2) in
the present context, but also plays a crucial role in studying the
corresponding Cauchy problem on a closed finite spatial interval (with either
Dirichlet or Neumann endpoint conditions), which we present in the second of
this series of papers, [23].

We now move on to consider bifurcations to periodic travelling waves from the
equilibrium solution $u_{e}=1$. This enables us to investigate in more depth
the structure of the solution to (IBVP) when $t$ is large, now in the region
of the propagating wavefronts identified in section 4. In particular, the next
section first rules out the possibility of _nondecaying, permanent form
periodic waves travelling with, and to the rear of, the principal wavefronts._

##  5\. Positive Periodic Travelling Waves

We seek to identify the bifurcation to _spatially periodic travelling wave_
solutions from the equilibrium state $u_{e}=1$. At fixed $D>0$, a periodic
travelling wave solution, with propagation speed $v=v_{b}\neq 0$ will
bifurcate from $u_{e}=1$, at wavelength $\lambda_{b}>0$ if and only if,
$(\lambda_{b},v_{b})$ satisfies the complex equation,

(162) $\displaystyle 4\pi^{2}D-2i\pi
v_{b}\lambda_{b}+\frac{\lambda_{b}^{3}}{\pi}\sin\frac{\pi}{\lambda_{b}}=0$

which is readily obtained from equation (5) with (9), when written in the
travelling wave coordinate $z=x-vt$, and linearized about $u_{e}=1$. It is
immediate that (162) has _no solutions_ which have $v_{b}\neq 0$ and
$\lambda_{b}>0$. We can therefore conclude that there are no bifurcations to
periodic travelling waves from the equilibrium state $u_{e}=1$.
We are now able to use this result in moving on to consider transitional
permanent form travelling waves, which accommodate the transition from the
equilibrium state $u=0$ ahead of the wavefront to the equilibrium state $u=1$
or to a spatially periodic permanent form travelling wave, to the rear of the
wavefront.

##  6\. Transitional Travelling Waves

A _transitional travelling wave_(or wavefront), in the present context, is
defined as: _a travelling wave of permanent form, propagating with constant
propagation speed $v>0$. The permanent wave form is nonnegative, and ahead of
the wavefront the wave form approaches the equilibrium state $u_{e}=0$, whilst
to the rear of the wavefront, the wave form either approaches the equilibrium
state $u_{e}=1$ or approaches a non-trivial, positive periodic travelling wave
(with the same propagation speed $v>0$)._ We anticipate that such structures
will play a key role in the large-$t$ evolution of the solution to (IBVP),
when $|x|=O(t)$ as $t\to\infty$. This motivates the present consideration of
transitional travelling waves. Following Section 5, we can immediately
eliminate the existence of transitional travelling waves of the second type
described above, and we therefore concentrate on the existence of transitional
travelling waves of the first type, which we will henceforth refer to as
(TPTW) solutions to equation (5) with (9). For a different treatment of
permanent form transitional travelling waves to (1), that lead up to the
concept of generalised travelling waves, the reader is referred to [1] and [2]
together with the references therein. In addition, in this context, numerical
computations of transitional travelling waves have been considered in [22],
relating particularly to such structures connecting two unstable steady
states.

We introduce the travelling coordinate $z=x-vt$ with $v>0$ being the constant
propagation speed. We represent a (TPTW) solution with propagation speed $v>0$
as $u=u_{T}(z,v)$, with $u_{T}(\cdot,v)\in C^{2}(\mathbb{R})$ satisfying,

(163) $\displaystyle
Du_{T}^{\prime\prime}+vu_{T}^{\prime}+u_{T}\left(1-\int_{z-\frac{1}{2}}^{z+\frac{1}{2}}u_{T}(s,v)ds\right)=0,\quad
z\in\mathbb{R}$ (164) $\displaystyle u_{T}(z,v)\geq
0\quad\forall~{}z\in\mathbb{R},$ (167) $\displaystyle
u_{T}(z,v)\rightarrow\left\\{\begin{array}[]{lll}1&\text{ as
}&z\rightarrow-\infty\\\ 0&\text{ as }&z\rightarrow+\infty.\end{array}\right.$

The problem (163)-(167) will be henceforth refered as (BVP). We begin with
some qualitative results.

  * (T1)

Let $u_{T}(\cdot,v):\mathbb{R}\to\mathbb{R}$ be a (TPTW), then
$u_{T}(\cdot,v)\in C^{\omega}(\mathbb{R})$.

###### Proof.

This follows by induction, following repeated differentiation through (163),
with the condition that $u_{T}(\cdot,v)\in C^{2}(\mathbb{R})$. ∎

  * (T2)

Let $u_{T}(\cdot,v):\mathbb{R}\to\mathbb{R}$ be a (TPTW), then $u_{T}(z,v)>0$
for all $z\in\mathbb{R}$.

###### Proof.

Suppose there is $z_{0}\in\mathbb{R}$ such that $u_{T}(z_{0},v)=0$, then via
(164) $u_{T}^{\prime}(z_{0},v)=0$. Thus, via induction on (163),
$u_{T}^{(n)}(z_{0},v)=0$ for each $n=2,3,\dots$ . Since $u_{T}(\cdot,v)\in
C^{\omega}(\mathbb{R})$, via (T1), it then follows that $u_{T}(z,v)=0$ for all
$z\in\mathbb{R}$, contradicting (167). The result follows. ∎

  * (T3)

The existence of a (TPTW) requires $v\geq 2\sqrt{D}$.

###### Proof.

Let $u_{T}(\cdot,v):\mathbb{R}\to\mathbb{R}$ be a (TPTW) with propagation
speed $v>0$. Via (163)-(167) and the linearization theorem (see for example
[16]), we must have,

(168) $\displaystyle Du_{T}^{\prime\prime}+vu_{T}^{\prime}+u_{T}=0,\quad z\gg
1,$ (169) $\displaystyle u_{T}(z,v)\to 0\,\text{ as }z\to\infty$ (170)
$\displaystyle u_{T}(z,v)>0~{}\text{for}~{}z\gg 1$

with the strict inequality in (170) following from (T2). As a consequence of
(168) and (169) there exists constants $A_{\infty}$ and $B_{\infty}$, not both
zero, such that,

(171) $\displaystyle u_{T}(z,v)\sim
A_{\infty}e^{\lambda_{+}(v)z}+B_{\infty}e^{\lambda_{-}(v)z},$

with

(172)
$\displaystyle\lambda_{\pm}(v)=-\frac{1}{2D}\left(v\pm\left(v^{2}-4D\right)^{1/2}\right),$

and the obvious modification when $v=2\sqrt{D}.$ It then follows from (171)
and (172) with (170), that the propagation speed $v\geq 2\sqrt{D}$, as
required. ∎

In the remainder of this section we suppose that
$u_{T}:\mathbb{R}\to\mathbb{R}$ is a (TPTW) which has the minimum possible
propagation speed

(173) $\displaystyle v=2\sqrt{D}.$

which we refer to as $u_{T}(z)$. Our intention is now to examine in detail the
form of $u_{T}(z)$ as $z\to\ -\infty$, using the linearization theorem (see,
for example, [16]). We remark here that Fang and Zhao [10] establish that
there exists a critical value of $D$ such that $u_{T}(z)$ is monotone if and
only if $D$ exceeds this critical value. We extend this here by obtaining
details of this critical value, and establishing the existence of a second,
and lower, critical value of $D$, for which the behaviour of $u_{T}(z)$ as
$z\to-\infty$, although not globally monotone, is ultimately monotone
exponential decay for $D$ above this second critical value, but becomes
harmonically oscillating exponential decay for $D$ below this critical value.
First we write,

(174) $\displaystyle u_{T}(z)=1+\overline{u}(z),$

after which $\overline{u}(z)$ must satisfy

(175) $\displaystyle
D\overline{u}^{\prime\prime}+2\sqrt{D}\overline{u}^{\prime}-\int_{z-\frac{1}{2}}^{z+\frac{1}{2}}\overline{u}(s)ds=0,\quad(-z)\gg
1,$ (176) $\displaystyle\overline{u}(z)\to 0\,\text{ as }z\to-\infty.$

The form of $\overline{u}(z)$ as $z\to-\infty$ is determined by examining
elementary solutions in the form

(177) $\displaystyle\overline{u}(z)=e^{\sigma z}$

with $\sigma\in\mathbb{C}$ to be determined, and $\mathrm{Re}(\sigma)>0$, to
satisfy condition (176) (both the real and imaginary parts of (177) provide
real-valued solutions to the real linear equation (175)). On substitution from
(177) into (175) we find that $\sigma\in\mathbb{C}$ allows (177) to solve
(175) and (176) if and only if

(178) $\displaystyle
D\sigma^{2}+2\sqrt{D}\sigma-\frac{2}{\sigma}\sinh{\frac{1}{2}\sigma}=0,\text{
with }\mathrm{Re}(\sigma)>0.$

For each fixed $D>0$, the transcendental equation (178) has a countably
infinite number of roots in the complex plane, which we label as

(179) $\displaystyle\sigma_{n}(D)\in\mathbb{C}\text{ for each
}n\in\mathbb{Z}\setminus\\{0\\}$

and, in addition

(180) $\displaystyle\sigma_{n}(\cdot)\in PC^{1}(\overline{\mathbb{R}}^{+})\cap
C(\overline{\mathbb{R}}^{+}).$

with,

(181)
$\displaystyle\begin{split}&\mathrm{Re}(\sigma_{\pm(2r-1)}(D))>0,\,r\in\mathbb{N},\\\
&\mathrm{Re}(\sigma_{\pm 2r}(D))<0,\,r\in\mathbb{N}.\end{split}$

We next observe that,

(182) $\displaystyle\sigma_{n}(D)=2n\pi
i-(-1)^{n}8n^{2}\pi^{2}\sqrt{D}+O(D)\,\text{ as }D\to 0^{+}$

with $n\in\mathbb{Z}\setminus\\{0\\}$. Also,

(183) $\displaystyle\begin{split}&\sigma_{\pm(2r-1)}(D)=2\log\frac{1}{2}D\pm
4(r-1)\pi i+o(1),\\\ &\sigma_{\pm 2r}(D)=-2\log\frac{1}{2}D\pm 4(r-1)\pi
i+o(1),\end{split}$

as $D\to\infty$, with $r\in\mathbb{N}\setminus\\{1\\}$. The full locus of
$\sigma_{n}(D)$ is sketched in Figure 15 for various values of $n$.

Figure 15. The locus of the eigenvalues $\sigma_{n}(D)$ in the complex plane.
Note that the imaginary part has been scaled by a factor of $2\pi$. These
start from $\sigma_{n}=2\pi ni$ when $D=0$ (shown as circles) and move through
the complex plane as $D$ increases. For $n=\pm 1$, $\pm 2$, $\sigma_{n}$
reaches the real axis at the points marked with a square, with $\sigma_{1}$,
$\sigma_{2}\to 0$ as $D\to\infty$, and $\sigma_{-1}\to\infty$,
$\sigma_{-2}\to-\infty$ as $D\to\infty$. Both $\sigma_{-1}$ and $\sigma_{-2}$
are shown as broken lines.

We observe from (178) that, in fact,

(184) $\displaystyle\sigma_{-n}(D)=\bar{\sigma}_{n}(D)$

for $n\in\mathbb{Z}\setminus\\{\pm 1,\pm 2\\}$.

We now consider the cases $n=\pm 1,\pm 2$. We concentrate on $\sigma_{\pm
1}(D)$. For $0<D<D_{+}$, we have that $\sigma_{-1}(D)=\bar{\sigma}_{1}(D)$ and
$\mathrm{Im}(\sigma_{1}(D))>0$. At $D=D_{+}$,
$\sigma_{-1}(D_{+})=\sigma_{1}(D_{+})=\sigma_{+}\in\mathbb{R}^{+}$. For
$D>D_{+}$, $0<\sigma_{1}(D)<\sigma_{+}<\sigma_{-1}(D)$, with

(185)
$\displaystyle\begin{split}&\sigma_{1}(D)\sim\frac{(\sqrt{2}-1)}{\sqrt{D}},\\\
&\sigma_{-1}(D)\sim 2\log\frac{1}{2}D,\end{split}$

as $D\to\infty$, as can be seen in Figure 15. In the above,
$(\sigma_{+},D_{+})\in(\mathbb{R}^{+})^{2}$ is the solution to the
transcendental equations

(186)
$\displaystyle\begin{split}&D\sigma^{3}+2\sqrt{D}\sigma^{2}-2\sinh\frac{1}{2}\sigma=0,\\\
&3D\sigma^{2}+4\sqrt{D}\sigma-\cosh\frac{1}{2}\sigma=0,\end{split}$

which gives, via numerical approximation,

(187) $\displaystyle(\sigma_{+},D_{+})\approx(4.437,2.824\times 10^{-2}).$

We are now able to interpret these results in relation to a (TPTW) with
minimum propagation speed $v=2\sqrt{D}$. In particular, _translation
invariance in $z$_ can, _and henceforth will be assumed to be_ , fixed so that

(188) $\displaystyle u_{T}(z)\sim ze^{-\frac{1}{\sqrt{D}}z}\,\text{ as
}z\to\infty$

via (171) and (172), whilst, for $0<D<D_{+}$, there are global constants
$A_{\infty}\neq 0$ and $\phi_{\infty}\in[0,2\pi)$ such that,

(189) $\displaystyle u_{T}(z)\sim
1-A_{\infty}e^{a(D)z}\cos(b(D)z+\phi_{\infty})\,\text{ as }z\to-\infty$

with

(190) $\displaystyle a(D)=\mathrm{Re}(\sigma_{1}(D)),\quad
b(D)=\mathrm{Im}(\sigma_{1}(D))$

which are both positive. In particular,

(191) $\displaystyle a(D)\sim\begin{cases}8\pi^{2}\sqrt{D}&\text{ as
}D\rightarrow 0^{+},\\\ \sigma_{+}-O\left(\left(D_{+}-D\right)\right)&\text{
as }D\rightarrow D_{+}^{-}\end{cases}$ (192) $\displaystyle
b(D)\sim\begin{cases}2\pi+O(D)&\text{ as }D\rightarrow 0^{+},\\\
O((D_{+}-D)^{\frac{1}{2}})&\text{ as }D\rightarrow D_{+}^{-}.\end{cases}$

However, for $D>D_{+}$, there is a global constant $A_{\infty}^{\prime}\neq 0$
such that

(193) $\displaystyle u_{T}(z)\sim
1-A_{\infty}^{\prime}e^{\sigma_{1}(D)z}\,\text{ as }z\to-\infty$

recalling that $0<\sigma_{1}(D)<\sigma_{+}$ and decreasing for
$D\in(D_{\infty},\infty)$, with

(194)
$\displaystyle\sigma_{1}(D)\sim\begin{cases}\sigma_{+}-O((D-D_{+})^{\frac{1}{2}})&\text{
as }D\to D^{+}_{+}\\\ \frac{(\sqrt{2}-1)}{\sqrt{D}}&\text{ as
}D\to\infty.\end{cases}$

Principally we see that a (TPTW) with minimum speed is monotone as
$z\to-\infty$ when $D\in(D_{+},\infty)$, but has decaying oscillations as
$z\to-\infty$ when $D\in(0,D_{+})$. Numerical solutions of (IBVP) with
$v=2\sqrt{D}$ and $D>0$ are shown in Figure 1, and are consistent with the
existence of a (unique by fixing translation invariance with
$u_{T}(0)=\frac{1}{2}$) (TPTW) at minimum propagation speed $v=2\sqrt{D}$, and
that this is the travelling wave generated in the initial value problem (5) to
(7) for $D>\Delta_{1}$. For $D<\Delta_{1}$, as discussed in Section 3, for
moderate values of $D$, the minimum speed (TPTW) provides a good approximation
to the solution of (IBVP) at the wavefront, which moves at speed $2\sqrt{D}$,
ahead of a stationary periodic state.

##  7\. Conclusions

In this paper we have studied various aspects of the Cauchy problem for the
nonlocal Fisher-KPP equation with a top hat kernel. We showed that the problem
is globally well-posed and investigated the positive periodic steady state
solutions that bifurcate from the uniform steady state $u=1$ at dimensionless
diffusivity $D=\Delta_{1}\approx 0.00297$. These have wavelength $\lambda$ and
exist in a sequence of ’tongues’ in $(\lambda,D)$ parameter space. As $D\to
0^{+}$, each of these periodic solutions has finite amplitude and wavelength,
with $\lambda<1$, and we constructed the asymptotic solution in detail for
solutions in the tongue with largest wavelengths.

We also investigated permanent form travelling wave solutions, which are known
to exist for all wavespeeds $v$ with $v\geq 2\sqrt{D}$, [3]. Numerical
solutions suggest that, from localised initial conditions with finite support,
a pair of diverging travelling wavefronts with minimum wavespeed, $2\sqrt{D}$,
is generated, and that for $D\geq\Delta_{1}$, the solution asymptotes to the
stable uniform state, $u=1$, between the wavefronts. For $0<D<\Delta_{1}$ and
moderately small ($\sim 10^{-3}$), a periodic static steady state, with
wavelength around $0.7$, is generated between the wavefronts. The wavelength
selection mechanism is not clear, but it is notable that $0.7$ is
approximately the linearly most unstable wavelength of the uniform steady
state $u=1$, and that this instability is not dispersive. However, as $D$
becomes very small ($\sim 10^{-6}$), this wavelength drifts towards $0.5$, the
minimum available wavelength in the first tongue, with spike formation
initiating in the region ahead of the wavefront where $u$ is exponentially
small, and a rational argument for this mechanism has been presented in
section 2.

In the next paper in this series, [23] we will report on the global
bifurcation structure for the problem set on a finite domain with various
boundary conditions. Oscillatory solutions are also generated, with a
structure similar to those discussed in the present paper, but with nontrivial
dependence on the form of the boundary conditions.

An interesting extension is to the Cauchy problem in higher dimensions, for
example with the two dimensional kernel

$\phi(r)=\left\\{\begin{array}[]{cc}\frac{4}{\pi}&\mbox{for
$r<\frac{1}{2}$,}\\\ 0&\mbox{otherwise,}\end{array}\right.$

in polar coordinates $(r,\theta)$. The numerical solution of initial value
problems in higher spatial dimensions is significantly more challenging than
in one dimension. We would expect that spatially-localised solutions would be
generated for small enough $D$, and that they would be amenable to asymptotic
analysis.

## Acknowledgements

N.M. Ladas was supported by the EPSRC Doctoral Training Partnership,
Reference: EP/R513167/1.

The authors would like to thank the referees for very helpful comments, which
have led to a significant improvement in the paper.

## Competing Interests

The authors declare none.

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