Research Article | Open Access
-Nonlinear Stability of Nonlocalized Modulated Periodic Reaction-Diffusion Waves
Assuming spectral stability conditions of periodic reaction-diffusion waves , we consider -nonlinear stability of modulated periodic reaction-diffusion waves, that is, modulational stability, under localized small initial perturbations with nonlocalized initial modulations. -nonlinear stability of such waves has been studied in Johnson et al. (2013) for by using Hausdorff-Young inequality. In this note, by using the pointwise estimates obtained in Jung, (2012) and Jung and Zumbrun (2016), we extend -nonlinear stability () in Johnson et al. (2013) to -nonlinear stability. More precisely, we obtain -estimates of modulated perturbations of with a phase function under small initial perturbations consisting of localized initial perturbations and nonlocalized initial modulations .
Many evolutionary PDEs possess spatially periodic traveling waves and their stability has been widely studied in recent years. In this paper, by using the results of [1–3], we consider -nonlinear modulational stability of spatially periodic traveling waves in a system of reaction-diffusion equations: with , , and , where is sufficiently smooth. Suppose that is a spatially -periodic traveling wave of (1) with a wave speed . By substituting into (1), one can say that is a stationary -periodic solution of Indeed, is a -periodic profile of traveling wave ODEs:
Compared with other types of traveling wave solutions such as front or pulse, the main difficulty of study of stability of periodic traveling waves is that the linearized operator considered on the whole line has only essential spectrum; that is, the spectrum is continuous up to zero (see Section 1.1 for details). This gives no spectral gap from the origin. The spectral gap plays a very important role in the study of (linear and nonlinear) stability because it gives exponential decay of the linearized operator. This is the reason why stability of the periodic traveling waves has been an open problem for a long time.
However, in the late 1990s, nonlinear stability of bifurcating periodic traveling waves of Swift-Hohenberg equation with respect to the localized perturbation has been studied in [4, 5] by using renormalization techniques. Stable diffusive mixing of periodic reaction-diffusion waves has been obtained in  based on a nonlinear decomposition of phase and amplitude variables and renormalization techniques. Johnson, Zumbrun, and their collaborators also showed nonlinear modulational stability of periodic traveling waves of systems of reaction-diffusion equations and of conservation under both localized and nonlocalized perturbations in [1, 7–9]. By using pointwise linear estimates together with a nonlinear iteration scheme developed by Johnson-Zumbrun, pointwise nonlinear stability of such waves has been also studied in [2, 3, 10]. For other related works on modulated periodic traveling waves, we refer readers to [11–13].
To begin with, we first review the concept of stability of of (2). Roughly speaking, we say that is (boundedly) stable if any other solutions of (2) which are initially near stay near for all . More precisely, we understand the following (bounded) stability definition.
Definition 1. Let and be Banach spaces and suppose that is a stationary solution of nonlinear partial differential equation of the form where includes all differential, linear, and nonlinear terms. Then we say is (boundedly) stable if, for all , there is such that, for all with , the corresponding solutions (with ) of (4) satisfy for all . In particular, if converge to in as tends to infinity, then we say that is asymptotically stable.
Here, we call as a perturbation of . Finally, in order to study stability of , we need to estimate perturbations in under initial perturbations in .
In this paper, we study modulational stability by estimating modulated perturbations in with an appropriate modulation with when the initial perturbation is not localized. To understand intuitively, suppose that for some nonlocalized with ; that is, nearby solutions are obtained by shifting slightly out of phase at . Then the nonlocalized data might be approximated by a localized initial modulational perturbation plus a nonlocalized initial modulation . Thus, our main investigation is that we choose an appropriate nonlocalized modulation with such that the modulated perturbation remains small in for all time, showing modulational stability of , when initial modulated perturbations and are sufficiently small.
For , -estimates of such nonlocalized modulated perturbations have been already established in  and  for systems of reaction-diffusion equations and of conservation laws, respectively, by using the generalized Hausdorff-Young inequality for and , where is a Bloch transform of defined below in (12). This is the reason why their stability analysis has been restricted to . In this paper, we extend their -stability results () to -stability by using pointwise estimate of linear behaviors under localized data and nonlocalized modulational data established in [2, 3].
In order to study stability of , the spectral information of the linearization of (2) around is required; so we first linearize the PDE (2) about . In order to see the importance of linearization about , we consider again the general PDE of form (4). By setting perturbations of as , we have Here, is referred to as the linearization of (4) about ; so, from this linearization, we obtain the linear perturbation equation.
We now linearize (2) around and consider the eigenvalue problem of the form with -periodic coefficients. We consider this linear operator on with densely defined domain . In this section, we recall Bloch analysis [1, 3, 7] which is a key idea of spectral analysis of linear operators with periodic coefficients. If we apply Floquet theory  to the first-order ODE system obtained by (7), lies in -spectrum of if and only if has the form for some Floquet exponent and -periodic function in . It means that, recalling the linear operator acts on the whole line , there is no -eigenfunction of ; so -spectrum of introduced in (7) is entirely essential; that is, there is no isolated eigenvalue (point spectrum). That is why the spectrum is continuous up to zero without spectral gap.
Inserting into (7) yields one-parameter family of Bloch operators, for each : operating on with densely defined domain . That is, for any -periodic function , Here, noting first that is compactly embedded into , has only point spectrum in with eigenfunctions ; in fact, -spectrum of is given by the continuous union of these isolated eigenvalues of for all .
Continuing with this setup, we recall the inverse Bloch-Fourier representation. Applying the inverse Fourier transform, we have, for any , that so, for any , where is referred to as the Bloch transform and denotes the Fourier transform. Furthermore, noting that is 1-periodic in , (10) and (12) give us the formula of periodic coefficient solution operator for : As a starting point, we used this formula to estimate linear behaviors of in terms of the localized and nonlocalized data in  and , respectively. Indeed, pointwise estimates on Green function of , by plugging into the Dirac delta function , have been obtained in  to estimate for a localized data . Moreover, in , we established the pointwise linear behavior of on nonlocalized data .
1.2. Spectral Stability
From (D1), we see that the origin is the only neutral spectrum of . By differentiating the traveling wave ODE (3), we obtain ; so is an eigenvalue of because . The second assumption (D2) implies that, for sufficiently small , an eigenvalue of bifurcating from is analytic in . If we use Taylor series expansion with respect to , by (D3) and the complex symmetry , can be written as where and . Moreover, by the perturbation theory, the corresponding right and left eigenfunctions of , denoted by and , respectively, are also analytic in for sufficiently small . In particular, (D2) implies that one can take because . Moreover, condition (D3) was verified by direct numerical Evans function analysis in .
2. Main Result
For the nonlocalized initial modulation , we set as a piecewise defined function where and . Here, in order to make sense of Bloch transform framework for the nonlocalized data, we may assume . Indeed, for any asymptotic constants and , we have with .
Theorem 2. Let be a stationary 1-periodic solution of (2) satisfying the spectral stability conditions (D1)–(D3). For a sufficiently small number , we assume initial data and satisfy Then, for all initial data satisfying (16) and (17), the corresponding solution to (2) satisfies that, for all , for an appropriate modulation with . Here the constant depends on in Lemma 4 and we determine in Section 4.
2.1. Discussion and Open Problems
As shown in Theorem 2, is modulationally stable in under nonlocalized initial perturbations. That is, even if we perturb the underlying solution by shifting it slightly out of phase at , is still stable in with some modulation determined in Section 4. However, it is just boundedly stable, while the main theorem in  implies that the underlying periodic solution is nonlinearly “asymptotically” stable in ; that is, converges to in as goes to infinity. It makes sense if we plug into -estimates in :
We emphasize again that we use pointwise estimates of the solution operator in terms of the localized data and the nonlocalized data in order to obtain -stability, because we cannot apply the Hausdorff-Young inequality (5) as we mentioned in the Introduction. This is the reason why we need both initial conditions (16) and (17), while (16) is enough to prove -stability () in .
Pointwise estimates give us more elaborate behaviors of the perturbation like an exact solution of linear perturbation equation ; so pointwise estimate is a common method to get -stability. However, we need to make more effort to obtain pointwise bounds. As shown in , we need condition (17) in order to obtain pointwise estimate of for the case with a sufficiently large constant . This issue did not arise in  because only localized modulations with were considered in . However, the primary difficulty is to obtain pointwise estimates on the Green function of the linear operator for the case by using Bloch decomposition. If we solve the difficulty, we might delete condition (17) in Theorem 2, even in the main theorem of . This problem was discussed in more detail in . Pointwise nonlinear stability under nonlocalized perturbations in systems of conservation is an open problem.
The dynamics of modulated periodic traveling waves have been studied in  by the WKB approximations. We consider that the wave number of the periodic traveling waves is modulated by the function . As described in Theorem 2 and [1, 3], is nonlinearly stable under small initial perturbations (16) and (17), with a heat kernel rate of decay in the wave number . One can clearly see this in the integral representation of in Section 4.
3. Review of Previous Results
This section provides the previous results in [2, 3], particularly how to estimate the solution operator of in terms of the localized and the nonlocalized data in pointwise sense. For some unknown modulation , we first define modulated perturbations of the underlying periodic solution as for any solution of (2) near . Recalling the linear operator in (7), we review the nonlinear perturbation equation about established in [1, 7].
Lemma 3 (nonlinear perturbation equation). The modulated perturbation satisfies where with
The goal of this paper is to estimate (23) in for an appropriate nonlocalized modulation . The most important and difficult part is estimating pointwise bounds of with respect to the localized data and the nonlocalized data . For the localized data , we use pointwise bounds on Green’s function of in  where uniformly on , for some sufficiently large constants and . Here is the periodic left eigenfunction of at discussed in Section 1.2 and is a smooth cutoff function such that for and for .
For the nonlocalized data , we decompose the solution operator as follows:with Here, denotes the Fourier transform. This kind of decomposition was not needed in  because we considered only localized modulations with . Moreover, the decomposition of here is rather different from the decomposition in  because we need pointwise estimates of obtained in  in order to prove -stability. To compare, we state the decomposition of the solution operator in  with Here, we have the worst term of when . By using normalization and normalization , one can actually see that those two decomposition types are not different in the sense that and .
We now review the pointwise estimates of on nonlocalized data . Our main purpose is to estimate in terms of , , or which are small data. We first estimate which has slower decay than .
Recalling the Fourier transform of the th derivative, , we have
Remark 5. We notice that the condition is needed only for the case of .
4. Proof of the Main Theorem
In this section, we determine the modulation satisfying the nonlocalized initial data . We first recall the representation of obtained from Lemma 3: We now define to cancel the bad estimate parts from ; that is, Then we can easily check because of a cutoff function in and the formula of . If we substitute into (23), we rewrite as We notice that the source term consists of , , , and ; so we consider derivatives of for and :
Remark 6. The main idea of determining is to cancel and in and , respectively, because they are too big to handle, especially the source terms. By Lemma 4 and (30)~(32), we intuitively know that . Compared with -bounds () in , we need more elaborate decomposition of the solution operator in order to estimate . This is the reason why the decomposition of here and in  is rather different from the decomposition in .
We now prove the main theorem with modulations defined in (36).
Proof of Theorem 2. Assume that the initial perturbation and the initial modulation satisfy (16)~(17). We begin by estimating : By Lemma 4, for fixed , Since , , and for any and any , we obtain By recalling pointwise bounds on in (25), we have for all , so that we estimate as In order to estimate , we first review -nonlinear stability estimate for in : for all , so Then integration by parts implies that By separating the last integral into and , we can easily calculate .
Similarly, we estimate derivatives of in . From the definition of in (36),