1. Introduction
In what follows, we distinguish between stationary waves (pulses) and wave fronts. A traveling wave
is a front if it travels at speed
and
On the other hand, for a stationary pulse (with
) and
In order to determine the stability of the solutions of a given partial differential equation, we linearize it about the wave solution. Then, with a view to recognizing the stability, it suffices to determine that the spectrum in the left-half plane (
) corresponds to stable places and that the spectrum in the right-half plane (
) corresponds to unstable places. There are many applications of the Legendre polynomials as they arise in mathematical models of the heat conduction and fluid flow problems in spherical coordinates. The novelty of the proposed method via the finding of the eigenvalues exactly is shown in
Section 4. This is the advantage of our proposed method over a previous method (see, for details [
1]).
This paper is constructed as follows. It is devoted to a systematic study of the stability of traveling waves through the use of the exponential dichotomies and the Evans function. In
Section 2, we will discuss the stability of pulses. In
Section 3, we illustrate the results by considering two examples. In
Section 4, we will introduce the direct approach for analyzing the stability of traveling wave solutions and compute the eigenvalues for the associated Legendre equation arising from the stability analysis of traveling waves after making a convenient transformation. Finally, conclusions will be presented in
Section 4.
We first introduce the stability analysis of these waves by using exponential dichotomies.
1.1. Exponential Dichotomies
First of all, we consider the following set of first-order ordinary differential equations:
where
and
If the eigenvalues of the matrix
A have nonzero real parts, then the space
or
is split into two stable or unstable eigenspaces according to whether the real parts of the eigenvalues are negative or positive, respectively. Equation (
1) has an exponential dichotomy on a subspace of
with the following evaluation:
where
Now, for the Equation (
1), we have
where
A fundamental set of solutions of the Equation (
2) is a set of
n linearly independent vectors
. The square matrix
, which is constructed with columns consisting of the vectors
is the fundamental matrix of the differential Equation (
2). We then have the following representation (see [
2]):
and
where
Equation (
4) for the determinant of the fundamental matrix may serve as an introductory definition of the Evans function.
1.2. Evans Function
The Evans function is an important tool for studying and investigating the stability of nonlinear waves (see, for example, [
3,
4]). The Evans function is an analytic function whose zeros correspond exactly, in location and multiplicity, to the eigenvalues of the linearized operator. The Evans function was first formulated by Evans for a specific class of systems and is a generalization which is suited for systems of partial differential equations of the transmission coefficient from quantum mechanics (see, for details, [
5,
6,
7]). Evans paid remarkable attention toward studying the stability of nerve impulses, which he then classified as the category of nerve impulse equations. This class has an important property that leads naturally to the formulation of blue the Evans function in a clear and straightforward manner. The
notation was used by Evans to refer to the determinant which, in fact, played the same role as the determinant of an eigenvalue matrix in problems of finite dimensions. Jones in [
8] applied the stability of the traveling pulse (nerve impulse) of the Fitzhugh-Nagumo system by following Evans’ idea. In fact, Jones gave the name
“Evans function" as well as the notation
which is now in common usage. The authors in [
9] presented the first general definition of the Evans function
, which is based upon the idea of Evans, with its placement in a new conceptual form to clearly give a general definition.
We consider the following eigenvalue problem:
and assume that (
5) is rewritten in the form given by
Firstly, we consider the following equation:
and assume that the eigenvalues of
are
and that the corresponding eigenvectors are
Whenever
and the
are distinct for
the space
is represented as follows:
We assume that
and that
while
and
Solutions belonging to
are bounded as
, while the solutions belonging to
are bounded as
We may then label these solutions as
or
, according to whether they are bounded as
or
Also, these solutions satisfy the following limit relationship:
We mention that there exist a set of values
and a common subspace of
and
such that
We now give the following definition.
Definition 1. is an eigenvalue of (5) if the following condition holds true: We mention here that the values of
which satisfy the equation
determines the essential spectrum of the Equation (
5).
The spectrum of the Equation (
5) generally consists of the pure point spectrum, isolated eigenvalues of finite multiplicity, and the essential spectrum. The essential spectrum is contained within the parabolic curves of the continuous spectrum (see [
10]). In many cases, the essential spectrum can be shown to be contained in the left-half complex plane and hence does not contribute to linear instability. Now, we search for solutions of the Equation (
5), namely
together with the boundary conditions given by
The Evans function is defined after the solutions of (
11) as follows:
where
and
As a consequence of Abel’s formula, the Evans function
given by (
11) is independent of
z and
By fixing the orientation of the orthonormal basis of the subspaces
and
, the Evans function can be made unique and the following theorem holds true (see [
2,
6,
9,
11]).
Theorem 1. The Evans functionis analytic onand satisfies the following properties:
- (i)
whenever
- (ii)
if and only if α is an eigenvalue of (
5)
- (iii)
The order ofas a zero of the Evans functionis equal to the algebraic multiplicity ofas an eigenvalue of (
5)
In applications, one finds that
so that the Evans function
reduces to
The importance of the manner in which the Evans function
is constructed is seen by the following argument. Suppose that
for some
. It is then clear that
for some
. Hence, clearly, there is a localized solution of the Equation (
10) when
, such that
is an eigenvalue. Similarly, if
is an eigenvalue, then it is not difficult to convince oneself that
In general, the Equation (
11) is not explicitly given as a function of
It is then necessary to evaluate the Evans function numerically (see, for details, [
1,
12,
13,
14,
15,
16]). In this case, the boundary conditions for the Equation (
10) are approximately used at infinity by approximating the boundary conditions (see [
17]). An alternative approximation is to set the Equation (
10) on a bounded domain, namely, on
and
, and then impose the exact asymptotic boundary conditions for boundedness of the solution of the Equation (
10) at these finite end-points. The use of approximate boundary conditions usually has a dramatic impact on the essential (continuous) spectrum (see [
18]). One of the most useful numerical techniques is the exterior numerical computations of discrete eigenvalues of the Equation (
10), which has no effect on the essential spectrum (see [
18]). In this case, the exact boundary conditions are applied at finite values
which are taken to be sufficiently large (see [
19]). In the extended space of exterior products, the Equation (
11) becomes
where
and
The Evans function
is defined here as follows:
After the above fundamental theorem on the Evans function , we give the following definition.
Definition 2. A traveling wave is said to be linearly unstable (or spectrally unstable) if, for somewiththere exits a solution of (
5)
which satisfies the following limit relationship: 2. Stability of Stationary Traveling Waves (Pulses)
Consider the following reaction-diffusion equation:
For the steady-state solution, we have
which satisfies the following condition:
We assume that
where
is a polynomial at least of degree 2 in
Exact solutions of the Equation (
17) are given in terms of Jacobi elliptic functions if
is a polynomial of degree up 6 in
u. In general, if the terms of pulse solutions, this Equation (
17) admits a solution of the following form:
where
is assumed to be a positive integer and
n is the degree of
. Specially, if
, where
b is a constant, then (
16) and (
17) become
We mention that, if in the first equation in (
19), we confine ourselves to solutions
, then the results of this section show that the necessary condition for traveling wave generation is
and
or
and
. In the case when
and
, the second equation in (
19) admits the solution
where
But, if
and
, then (
19) has the solution given by
where
,
and
k satisfy an over determined set of algebraic equations.
When solving these equations, we find that solutions in the form (
21) exist only when
or
and are given by
and
We now introduce a perturbation around the solution
as follows:
with
. Upon substituting from (
24) into (
16), we find, up to first order in
, that
Equation (
25) is a Sturm-Liouville eigenvalue problem. Our aim now is to find
, where
We note that
is an eigenvalue of (
25), because, if we differentiate the second equation of (
19) with respect to x, it becomes
We then find that (
26) satisfies (
25) with
and
This is a result of the fact that (
20) is translationally invariant (see [
20]). We assume that
and construct the system of equations given by
where
and
An important remark is that
which is independent of
x and
. To continue our investigation, we distinguish two cases:
or
. Firstly, we assume that
,
and in (
27) and that
If
or
then the solution is stable.
3. A Set of Examples
If, in (
19), we set
,
and
, then the Equation (
19) becomes
and the Equation (
25) becomes
The stability of the stationary pulse is determined by the spectrum of the following operator:
This spectrum consists of a point spectrum of isolated eigenvalues and an essential spectrum (see, for details, [
21,
22]). Because, for the stationary solution
as
, the location of the essential spectrum on the spectral plane follows from considering the limits of the operator
On looking for modal solutions
, it follows that the essential spectrum is given by
Upon solving for
, we obtain
which shows how the spatial wave-numbers
depend on the temporal growth rate
. The absolute spectrum given by
consists of all points
for which the corresponding spatial wave-numbers
and
have the same real part. The transition to instability occurs when a discrete eigenvalue moves from the left-half plane to the right-half plane.
A stationary pulse solution of the Equation (
28) corresponds to a so-called “localized” solution
of the Equation (
19) and it is (
20). Thus, clearly, the Equation (
20) reduces to
Consequently,
and
become
respectively.
We now show how the Evans function can be used in order to deduce the same conclusion.
It is important to note here that
and that the decay is exponentially fast. For the rest of this discussion, it will be assumed that
The eigenvalues of
are given by
and the associated eigenvectors are as follows:
One can construct solutions
of the Equations (
27) and (
30), which satisfy the following limit relationship:
It is noted that the construction implies that
The Evans function
is given by
and, by Abel’s formula, it is independent of
x, namely, the Wronskian of
or
is given by
where
c constant and
Now, the bases of the stable and unstable subspaces can be determined numerically in the following manner. We calculate the eigenvalues of
with negative real part and their corresponding eigenvectors. Then, by choosing a sufficiently large number
L, we solve the following homogeneous equation:
in
starting from the right-end point with the initial condition given by
Hence, we obtain linearly independent (approximate) solutions of the differential equation. Therefore, their values at
give a basis for
. Similarly, by solving the differential equation in
we obtain a basis of
and the determinant defining the Evans function can be computed. The Evans function for the Equation (
28) is shown in
Figure 1, where we find that the Evans function has two discrete eigenvalues at
and
Thus, the spectrum is given as mentioned above.
We now consider
and
in (
19), so that
and
Consequently,
and
become
respectively.
The Evans function
is computed numerically for the Equation (
35) and is shown in
Figure 2. We find from
Figure 2 that the Evans function
has three discrete eigenvalues at
,
and
. Since there exists a positive eigenvalue for the Evans function
of the linearized stability problem for a pulse solution of (
24), therefore, this pulse solution is unstable. It is also appropriate to compare our analytical and numerical results with the results in [
1]. We can thereby see an excellent agreement between the results developed in this article and the earlier results in [
1].
4. A Direct Approach for Analyzing the Stability of
Traveling Wave Solutions
Here, in this section, we present an approach for solving the Sturm-Liouville problem arising from a stability analysis for traveling waves. The idea behind this approach is inspired by some results for the Legendre functions and the Legendre polynomials (see also the recent investigations [
23,
24] involving applications of the substantially more general Jacobi polynomials).
The Legendre operator is given by
We consider the following eigenvalue problem:
or, equivalently,
This equation has regular singular points at
However, if
then the solution of the Equation (
37) or (
38) is finite as
In this case, the solution of (
32) is known as a Legendre polynomial
We also note that
is an eigenvalue of the problem (
38).
We now consider the associated Legendre operator
and consider the following eigenvalue problem:
We note that
are regular singular points of (
39). The solution of (
39) is finite as
if
so that
is an eigenvalue of (
39). In order to find the relationship between the stability analysis and the results for the Legendre functions, we consider the general example given by the Equation (
19). Thus, for instance, we assume that
and
, so that the solution
of the second equation in (
19) is given by
where
Now, the perturbed equation for (
19) is given by
By using the transformation
, (
41) becomes
Equation (
42) has two regular singular points at
In the original variable
x, they correspond to
and are regular singular points of (
19). Indeed, (
42) can be written in the form:
Equation (
43) has the solution given by
In the Equation (
44), the finite solutions require that
and
are positive integers. We search now for solutions of (
44) which satisfy the following limit relationship:
This condition determines the solution of the eigenvalue problem (
41). We remark that this solution does not depend on the coefficient of the nonlinear term in (
19), namely,
b.
If
is a positive integer, then we find from the properties of associated Legendre polynomials that
or
The finite solutions of (
42) as
(or
) are given by
where
and
are integers and
is a constant. The required values of
are determined by the following equation:
Among the values of
which satisfy (
45), we select the values which satisfy the eigenvalue condition. To make these results clear, we consider the following special cases:
- (I)
If
,
and
, the first equation in (
19) becomes the Newell-Whitehead equation. Then, from (
45), we obtain
. The corresponding eigensolutions are given, respectively, by
and
We remark that the above results for
are exactly the zeros of the Evans function
which we have found numerically (see
Figure 1). But
is not an eigenvalue as the corresponding solution
does not satisfy the eigenvalue condition. Thus, clearly, the only eigenvalues are
and
, so that the solution is unstable. To examine the effects of varying the parameter
a, we take
,
and
. We find that the values of
such that finite solutions as
exist are
or
, besides
, with eigensolutions given by
and
We find again that is not an eigenvalue. Thus, the effect of increasing is that the positive eigenvalue shifts to the right on the -axis.
- (II)
If
,
and
, then (
19) becomes the Fisher equation.
From (
45), we find that
,
, 0,
. The corresponding eigensolutions are
and
Here,
is again not an eigenvalue, while
, 0,
are eigenvalues and they are exactly the zeros of the Evans function
(see
Figure 2). We remark that the solution of the Sturm-Liouville eigenvalue problem for (
19) is too simple, because it is directly related to the associated Legendre polynomials.
We summarize these results in the following table. We fix the value and b is any arbitrary positive real number.
From
Table 1, we remark that the number of real eigenvalues for the Equation (
41) is
. We now present an approach which may enable us to treat general problems. It is based mainly on polynomial solutions of differential equations (or truncation of the series solution).
We should remark that, in order to obtain the solution of (
43) in the form of the associated Legendre polynomials, the solution expansion is taken near the regular point
. In what follows, the solution expansions are taken near a regular singular point at either plus or minus infinity (
). In the Equation (
40), we assume that
, so that it becomes
In connection with the Equation (
46), we remark that
are regular singular points and
corresponds to
, which are regular singular points. We search for solutions of (
46) in the form of
The indicial equation gives rise to
Here
is arbitrary, while
. In order to obtain finite solutions, we take the upper sign in the last equation for
d. The general recursion formula is given by
where
Equation (
47) admits a polynomial solution if there exists an integer
and
such that the coefficient of
vanishes. This holds true for
Certainly, when analyzing the Equation (
47) by taking
we find the same results as before. So, instead, we can find results from (
47) and (
48).
In view of (
47), we have a lower bound for
and from (
47) we obtain the least upper bound for
(or the dominant value of
), namely,
when
. This is true because, for
, the values of
d decrease, and then
decreases. In fact, when
, we obtain
and the following result holds true:
The last result suggests the introduction of the following general localization concepts of eigenvalues of the Sturm-Liouville problem.
We now define what is meant by a dominant eigenvalue and a dominant solution. A dominant eigenvalue
is the least upper bound of the eigenvalues
. A dominant solution is a solution which corresponds to a dominant eigenvalue
. In the solution expansion near a regular singular point, we conjecture that a dominant solution is obtained at the lowest-order truncation of the series, namely, at
. Consequently, the dominant eigenvalue is determined by the solution of the recursion formulas for
. We return to (
47) in order to check that we get the same results found previously. To this end, we reconsider the same examples.
For
,
and
, we find that (
48) gives
For
and
we have
and
, respectively. For
,
and
, (
48) gives rise to
Thus, we obtain , for , respectively. The corresponding eigensolution can be obtained as well.
We remark that we have obtained the same results as above. On the other hand, polynomial solutions found through an expansion near a regular singular point gives rise to values of
. We mention that, when analyzing the stability of pulse solutions of the first equation in (
19) when
and
, we find that the eigenvalues are non-positive. These solutions are then stable.