Nonlinear Instability of Periodic Traveling Waves
We study the local dynamics of -perturbations to the zero solution of spatially -periodic coefficient reaction-diffusion systems. In this case the spectrum of the linearization about the zero solution is purely essential and may be described via the point spectrum of a one-parameter family of Bloch operators. When this essential spectrum is unstable, we characterize a large class of initial perturbations which lead to nonlinear instability of the trivial solution. This is accomplished by using the Bloch transform to construct an appropriate projection to capture the maximum amount of linear exponential growth associated to the initial perturbation arising from the unstable eigenvalues of the Bloch operators. This result is also extended to dissipative systems of conservation laws.
We consider a reaction-diffusion type system of real-valued differential equations of the form
posed on , an appropriate Sobolev subspace of , where is an -th order linear differential operator of the form
with real-valued -periodic coefficient functions . Other periods may be considered by rescaling . For simplicity we also assume that is a sectorial operator. We assume the nonlinear operator satisfies the following polynomial estimate,
Systems of the form (1.1) naturally arise when considering the local dynamics of constant coefficient systems of reaction-diffusion systems near a spatially-periodic equilibrium solution. That is, supposing an evolution equation has a spatially-periodic equilibrium solution , with , we investigate the behavior of solutions that begin as small perturbations of , i.e. solutions with initial conditions of the form . Typically one writes the solution as , substitutes this ansatz into , then uses the condition to find the “perturbation equations” that govern the evolution of the perturbation . In this case equation (1.1) would specifically be this perturbation equation.
For the local dynamics, we use the following definition of stability,
Let be an equilibrium solution and be a perturbation (as above) which satisfies (1.1). The equilibrium solution is said to be stable (in the norm ) if for all there exists a so that requiring ensures that for all time. Otherwise is said to be unstable.
Traditionally the focus has been showing that equilibrium solutions are stable. In contrast, we are particularly interested in the case when the spectrum of , , is “spectrally unstable”: when . In [8, 9] it is shown that spectral instability leads to instability. This is done by constructing a specific initial perturbation which results in a poorly behaved solution, thus precluding stability. In particular in  the initial perturbation is taken to be a perturbation which “activates” the most unstable part of , roughly speaking that it projects into the most unstable subspaces.
Where this paper differs is that we allow the initial perturbations to be as arbitrary as possible and attempt to characterize which could be used to preclude stability. To this end, we define stability for an initial perturbation (in contrast to an equilibrium solution ).
With this definition in mind,  shows that the initial perturbation which activates the rightmost part of is unstable. But what if an initial perturbation activates any other part of ? A naive guess would be that if activates it is unstable, and if it does not it is stable. If that were the case, then one would obtain stability of for a wide range of initial perturbations.
Our main result is a step in this direction with Theorem 7 which concludes that if an initial perturbation activates an appropriate subset of , then that initial perturbation is unstable. This result applies to many reaction-diffusion type systems with periodic equilibrium solutions, including but not limited to scalar reaction diffusion, FitzHugh-Nagumo , the Klausmeier model for vegetation stripe formulation [18, 17], and the Belousov-Zhabotinskii reaction . This methodology is robust enough that in Theorem 10 we show how it may be extended to dissipative systems of conservation laws such as Kuramoto-Sivashinsky [3, 10] and the St. Venant equation [2, 15].
To see how a spectral instability may lead to a nonlinear instability, we recall from Duhamel’s equation the solution of (1.1) can be decomposed as
which uses the solution semigroup to write the solution in terms of a linear part and a nonlinear part.
The stability of the linear part is directly influenced by point spectrum of . Suppose had an eigenfunction with eigenvalue with : then choosing as an initial perturbation, the linear part would be and we would have exponential growth. While in general the spectrum of will not contain such eigenvalues, as the coefficient functions in (1.2) were taken to be -periodic then Floquet theory gives the spectrum of as the collection of point spectrum of the one-parameter family of operators with [11, Proposition 3.1]. The respective domains and are connected through the Bloch Transform. (This theory and its preliminaries are developed in Subsection 2.1, and that specific spectral result is given in Proposition 3). In Subsection 2.2 we use the Bloch Transform to define the projection (2.10) which allow us to use this unstable point spectrum of to conclude exponential growth for the linear part of (1.4).
This clarifies the notion of “ activating an unstable part of ” as “the Bloch Transform of contains some sufficiently unstable eigenspace of some .” A further area of study would be to see if this Bloch transform view gives any insight into specifically how the initial perturbation goes unstable.
To handle the nonlinear part of (1.4), we apply the reverse triangle inequality to obtain the following lower bound for the solution,
In Section 3 we prove an upper bound on the growth of the nonlinear part, which when contrasted with the exponential growth of the linear part gives instability.
2 Spectral Properties
2.1 Characterization of the Spectrum
The operator from (1.1) is a linear differential operator with -periodic coefficients, so standard results in Floquet theory [5, Section 2.4] tell us that there are no eigenfunctions: the spectrum is entirely essential. Furthermore, the spectrum of can be determined from the following one-parameter family of Bloch operators ,
Given the form of in (1.2), the Bloch operators take the following explicit form,
As the resolvent of each with domain is a compact operator, then the spectrum of is a countable set of isolated eigenvalues with finite multiplicity . Note that from (2.2), appears in as a polynomial and so is holomorphic as a function of , and thus [13, Theorem 1.7 from VII-§1] given a closed curve that separates the spectrum, its corresponding spectral projection is holomorphic in and [13, Theorem 1.8 from VII-§1] any finite system of eigenvalues which depend holomorphically on . See Figure 2.1 for a depiction of the spectral picture.
In particular, let be an eigenvalue of and be a curve that contains and no other eigenvalue of . Then there is some interval , with , so that the following spectral projection
is holomorphic on , where is the resolvent of .
The Bloch transform may be used to relate the domain of the Bloch operators to the domain of . To explain the former domain, first fix and consider . The Bloch transform requires the map be , identifying this as the domain of the Bloch operators . We define the Bloch transform of to be the unique function which satisfies
We have uniqueness because there is an explicit formula for ; starting from the Fourier inversion formula, if we break the integral into blocks of the form with ,
From Parseval’s theorem we see that the Bloch transform is an isometry,
We can also use the Bloch transform to write the linear evolution in terms of the linear evolution of the Bloch operators [12, Equation 1.15]. Specifically, given an we have
2.2 Bloch-Space Projections
Our first goal is to use the unstable spectrum of to show that the linear part of (1.4) has some sort of exponential growth. We assume that the spectrum of is unstable, so let . In Proposition 3 we characterized the spectrum of in terms of the eigenvalues of the one-parameter family of Bloch operators : there must be some so that is an eigenvalue of with eigenfunction . Note that when considered as an initial perturbation has exponential growth as , albeit in . We use the Bloch Transform (2.4) to extend into some function in that has exponential growth.
First we extend to more values than just . Using the spectral projection around a single eigenvalue of introduced in (2.3), for in some interval we may continuously define . We restrict the contour and interval to be sufficiently small so that
then we can use the Bloch Transform to define the following function in ,
The intuition behind this is constructing an initial perturbation that is close to the eigenfunction . In the sequel our strategy changes to instead defining a projection that recognizes when such “eigenfunctions” are present in an initial perturbation . This has the advantage of being widely applicable to all initial perturbations rather than just a constructed few.
As a technical issue we require any such “eigenfunctions” to have a sufficient level of exponential growth. To that end we first define the following quantity which will be the maximum rate of exponential growth an initial perturbation contains,
where is a spectral projection to the eigenspace of as defined in equation (2.3). Note that the condition in (2.9) is analogous to requiring “,” but accounts for the technicality that if is an eigenvalue for multiple then as defined in (2.3) is not unique.
We will construct this projection analogously to (2.8): as was extended to by prepending , we shall do so here as well. We choose our so that is arbitrarily close to . That is, let with . Then choose an eigenvalue of with so that , and restrict the contour and interval containing so that and is holomorphic on . See Figure 2.2 (a) for an illustration. We may then use the Bloch Transform (2.4) to define the following projection,
To see that is a projection, first note that by the uniqueness of (2.4) we see that , and so . Secondly, applying 2.5 and that gives that . Furthermore the spectral projection commutes with the semigroup for each [19, Theorem 3.14.10], which we can use to show that commutes with as well.
As is a projection, then
So it suffices to show this exponential growth for the projected linear part . Set and recall the choice of in (2.10), restricting the interval so that for all . As is a linear combination of eigenfunctions of with eigenvalues that satisfy , , and after applying (2.5),
Our instability argument requires that be sufficiently large to attain a certain minimum level of exponential growth. To define this level, we first need to determine an upper bound of over all choices of initial perturbations ,
Note that as was assumed to be sectorial, then is necessarily finite. As part of the upcoming Hypothesis 5 we assume that this quantity is finite. We later determine in Theorem 7 that a sufficient level of exponential growth is attained if , where is the power of the nonlinearity as in equation (1.3). Put another way, if we define
then there is a sufficient amount of exponential growth if for some , an eigenvalue of , we have that . This is what is precisely meant by saying “activates” the unstable part of the spectrum. We now make a hypothesis on , that eigenvalues do not enter and exit it too many times.
is constant for .
For a visual depiction of this latter assumption, see Figure 2.2 (b), where eigenvalues are allowed to enter and exit a contour enclosing only — and no other part of — only finitely many times. In [7, Figure 6] [16, Figure 3] a numerical calculation of the point spectrum of appears to agree with this hypothesis.
As it stands, a naive exponential growth upper bound for — that is, obtained solely by looking at the spectrum of — would be . If we bound the nonlinear part in (1.4) by this exponential function , then it would overshadow the lesser growth that (2.10) can provide for the linear part. But we can take advantage of the fact that for all with , then (for any choice of ). Thus intuitively should “ignore” that part of the spectrum, and the exponential growth upper bound should instead be .
We start by using Hypothesis 5 to find intervals a finite partition of so that the number of eigenvalues of (counted by multiplicity) in is constant for each . Let be a curve that encloses all of and no other part of (see Figure 2.2 (b)). Then we define the following spectral projection,
Note that by construction . Combining this with (2.6),
and [19, Theorem 3.14.10] we see that commutes with and . This gives the growth estimate. ∎
3 Nonlinear Instability
We now state our main instability result. With Definition 2 in mind we start by defining , for , to be the solution to the following evolution equation,
Showing that the initial perturbation is unstable is equivalent to showing that cannot be made arbitrarily small by taking arbitrarily small. In our instability theorem we find an explicit time where the solution fails to be arbitrarily small.
Consider the initial value problem (1.1), with a sectorial operator with -periodic coefficients as defined in (1.2), and a nonlinear operator satisfying the polynomial estimate (1.3). Assume that Hypothesis 5 holds and let be an initial perturbation with . Then is unstable in the sense of Definition 2 as there exist and sufficiently small so that for all , at the time when
Recall from the introduction that our general strategy was to use (1.5) to pit the exponential growth of the linear term obtained from the unstable spectrum of against the nonlinear term’s slower growth. The former was developed in Lemmas 4 and 6, so we handle the latter below.
For as in Theorem 7, then if we have
For we define the quantity
To prove the result, it is sufficient to show that is uniformly bounded for . We start by taking the norm of (1.4). From Lemma 6 we have that , and as is sectorial111The assumption that is sectorial may be relaxed so long as we have this same semigroup estimate and is finite. then . Then after recalling (1.3), we have
Then we multiply and divide the nonlinear term by , apply (3.3), evaluate the integral, and note that is monotone increasing to obtain
Recall the hypothesis , and note that it implies
so we may focus solely on the larger exponential growth.
Note that this upper bound (3.4) also applies for all for . Multiplying both sides by and taking the supremum over all yields
Replacing the right hand side exponential term’s with , recalling (3.2), and dividing both sides by ,
Setting leads to the equivalent polynomial inequality valid for ,
This polynomial has a critical point at
And at this critical point the polynomial takes on the value
Then so long as is smaller than some expression that only involves , then the polynomial will be negative at some -value to the right of . In particular, it will have a root to the right of .
When , , so choosing sufficiently small will satisfy the polynomial inequality initially at . Then the existence of a root means that is uniformly bounded for , and hence the uniform bound for for . ∎
(Of Theorem 7)
Lemma 4, when , gives us that
From Lemma 8, we can bound the nonlinear part of by
In particular, when , we have
Then if is chosen sufficiently small so that , then by the reverse triangle inequality we have
Note that the leftmost term is a positive constant independent of : this becomes our , which completes the proof. ∎
4 Extension to Dissipative Systems of Conservation Laws
Recall that our main Theorem 7 was proven in the context of reaction-diffusion type systems of the form (1.1): specifically for systems with no derivatives in the nonlinearity. Some examples of such systems would be scalar reaction diffusion, FitzHugh-Nagumo , the Klausmeier model for vegetation stripe formulation [18, 17], and the Belousov-Zhabotinskii reaction . However, our general methodology is sufficiently robust enough that it can apply more widely to dissipative systems of conservation laws. As a specific example, consider the following Korteweg-de-Vries/Kuramoto-Sivashinsky (KdV-KS) equation
with , which arises in the context of inclined thin film flow . It was shown in [1, 10] that this equation admits periodic traveling wave solutions whose linearization satisfies Hypothesis 5. If we consider solutions of the form , we find that satisfies the following perturbation equation [3, Lemma 3.3]222A slight discrepancy arises in that  considers a modulation so that and that here we neglect such a modulation.
Our goal is to characterize which initial perturbations of our traveling wave will result in an unstable solution . Note that the nonlinearity does not in any standard Sobolev space satisfy a polynomial estimate of the form (1.3). Nevertheless we can use the following damping estimate as in [3, Proposition 3.4] to obtain a workable analogue. For completeness we reproduce the proof of this damping estimate.
Let be a solution to (4.2). Then satisfies the following nonlinear damping estimate
for some constant .
Let denote the inner product. Using integration by parts,
We can obtain from (4.2). Using Cauchy-Schwartz, Young’s inequality, and the fact that allows us to bound the nonlinear term,
The remainder of (4.4) can be handled with integration by parts and recognizing perfect derivatives, resulting in the bound
where the are non-negative constants depending on , for , and .
Before proceeding we derive a useful Sobolev interpolation inequality. Using integration by parts, for integer we have
Using Cauchy-Schwartz and Young’s inequality with an arbitrary positive constant yields the inequality
Taking a linear combination of the cases, for appropriate choices of the and a sufficiently large constant , we can obtain the following Sobolev interpolation inequality
The key ingredient to the damping estimate was that the leading term of (4.2) was negative: such an estimate is not necessarily limited to the specific PDE (4.1). In particular a damping estimate also holds for the St. Venant equation [15, Proposition 4.2], which is a hyperbolic-parabolic system of balance laws. In light of this, we introduce a general requirement on the nonlinearity, that for constants , , , , , and ,
We can then prove an alternate instability result.
Consider the initial value problem (1.1), with a sectorial operator with -periodic coefficients as defined in (1.2), and a nonlinear operator instead satisfying the estimate (4.7). Assume that Hypothesis 5 holds and let be an initial perturbation which satisfies both and , then is unstable in the sense of Definition 2 as there exist and sufficiently small so that for all , at the time when
We multiply and divide by appropriate powers of , bound by , and evaluate the integrals to obtain the analogue of (3.4),
Note that the hypotheses and imply that
so we may focus solely on the larger exponential growth.