Spectral stability and spatial dynamics in partial differential equations
SSpectral stability and spatial dynamics in partial differentialequations
Margaret Beck ∗ October 18, 2019
This article is focused on two related topics within thestudy of partial differential equations (PDEs) thatillustrate a beautiful connection between dynamics,topology, and analysis: stability and spatial dynam-ics . The first is a property of solutions that de-scribes the extent to which they can be expected topersist, and hence be observed, over long time scales.The second is a perspective that has been used tostudy various properties, such as stability, of non-linear waves and coherent structures, the term oftenused to describe the solutions of interest in the classof PDEs that will be considered here.To fix ideas, let’s focus on systems of reaction-diffusion equations, u t = ∆ u + f ( u ) , (1)where u : Ω × [0 , ∞ ) → R n , Ω ⊂ R d , f : R n → R n ,∆ = ∇ · ∇ = ∂ x + · · · + ∂ x d , and there are accompa-nying initial conditions and possibly also boundaryconditions for ∂ Ω, which for the moment I will leaveunspecified. I will assume f and ∂ Ω are smooth.Reaction-diffusion equations are a class ofparabolic PDEs for which it is interesting to studythe dynamics specifically because well-posedness isknown: under reasonably mild assumptions, unique ∗ Margaret Beck is a professor of mathematics at BostonUniversity. Her email address is [email protected]. The connection between these two concepts was also de-scribed in the talk entitled “Stability for PDEs, the Maslov In-dex, and Spatial Dynamics,” which the author gave at MSRIin 2018. That talk can be accessed via solutions exist and depend smoothly on the initialdata and the function f . This means that one canfocus on the resulting behavior of solutions as timeevolves, and in many cases obtain quite detailed in-formation. They are also relevant because they ap-pear in a wide variety of applications, for examplein chemistry, biology, and ecology, which means thatnot only are there specific models in which to testthe theory, but there are also important open ques-tions originating in other sciences that can point tointeresting new mathematical directions.It is worth noting that many of these propertiesthat have just been described are also present in othertypes of PDEs, such as the nonlinear Schr¨odingerequation and the Korteweg-de Vries equation, bothdispersive evolution equations, and so much of whatwill be discussed below can be applied not just toreaction-diffusion equations but also more broadly.See [CDB11] for a variety of examples related to thecontext of this article. Stability
In order to describe the dynamics of the PDE oneoften begins by identifying specific solutions, such asstationary or time-periodic patterns, and then seek-ing to understand the extent to which such solutionswill be observed in the long-time dynamics. Withinthis context, one might ask about two types of sta-bility. The first is related to robustness of the so-lution to perturbation in the system parameters, orin other words to perturbations within the PDE it-self. This type of stability is referred to as structural1 a r X i v : . [ m a t h . D S ] O c t tability, and it typically falls within the sub-field ofbifurcation theory. The second type of stability, andthe one that is a focus of this article, is stability intime, or dynamic stability: can one expect to observethis solution in the dynamics of a fixed PDE as timeevolves? This has to do with robustness of the so-lution to perturbations in the initial condition, or toperturbations in the current state of the system. Inthis sense, stable solutions attract (or at least do notrepel) nearby data. Unstable states repel (at leastsome) nearby data, which will be driven away to somestructure that is dynamically attracting. Structuraland dynamic stability are of course connected; onecould for example ask how dynamic stability is af-fected by changes in system parameters. But for theremainder of this article, stability will always refer tostability in time.Let’s suppose that we are given a stationary solu-tion of (1), ϕ ( x ), so that0 = ∆ ϕ + f ( ϕ ) , (2)and we want to investigate its stability. We can writethe solution to (1) as u ( x, t ) = ϕ ( x ) + v ( x, t ) andderive an evolution equation for the perturbation v : v t = ∆ v + df ( ϕ ) v (cid:124) (cid:123)(cid:122) (cid:125) =: L v + [ f ( ϕ + v ) − f ( ϕ ) − df ( ϕ ) v ] (cid:124) (cid:123)(cid:122) (cid:125) =: N ( v ) . If v ( x,
0) is small in some appropriate sense (so weare focusing on local, rather than global, stability),will the perturbation decay to zero, or at least remainsmall, for all t ≥ v is small, at least ini-tially, one could expect the linear term L v to domi-nate the nonlinear one N ( v ) in determining the dy-namics, simply because | v | p < | v | if p > | v | < L plays a key role.This relies on the fact that the linear operator is nice:it generates an analytic semigroup, and so there isa clear connection between spectrum and dynamics.Unstable (positive real part) spectrum leads to expo-nential growth, stable (negative real part) spectrumleads to exponential decay, and if there is spectrumon the imaginary axis then one must take the nonlin-earity into account. Here the focus will be on detecting spectral insta-bilities. The spectrum of L can be divided into twoparts: the essential spectrum and the point spectrum,or eigenvalues. At the moment the details of this de-composition are not so important; what is importantis the fact that the essential spectrum is relativelyeasy to compute, whereas the point spectrum is typ-ically difficult to compute. Thus, if one calculatesthe essential spectrum and it lies in the right halfplane, then an instability has been detected. Themore interesting case is therefore when the essentialspectrum is stable, and one needs to understand thepoint spectrum. Thus, the question of detecting aninstability is reduced to determining whether or notthere are any eigenvalues of the linearized operatorthat have positive real part.The simplest case is a scalar equation in one spacedimension: n = d = 1. If Ω = ( a, b ) and we considerzero Dirichlet boundary conditions, then we are inthe classical setting of a Sturm-Liouville eigenvalueproblem: λv = v xx + df ( ϕ ( x )) v, x ∈ ( a, b ) v ( a ) = v ( b ) = 0 . Note the linear operator is self-adjoint, so the spec-trum is real . Consider the Pr¨ufer coordinates v = r sin θ, v x = r cos θ, which in this setting are essentially just polar coor-dinates in the phase plane. By differentiating therelations r = v + v x and tan θ = v/v x and solvingfor r x and θ x , we find the dynamics of r and θ to begoverned by r x = r (1 + λ − df ( ϕ ( x ))) cos θ sin θ,θ x = cos θ + ( df ( ϕ ( x )) − λ ) sin θ. One can now make three key observations: the dy-namics for θ have decoupled from those for r ; the set On the bounded domain considered here, one could at-tribute the realness of the spectrum to the fact that the oper-ator is second-order and scalar, since any second-order scalaroperator can be put into self-adjoint form by means of an ap-propriate integrating factor. Later, however, we will consideroperators on the entire real line that act on vector-valued func-tions, in which case the realness of the spectrum will resultfrom the self-adjointness of the operator. r = 0 } is invariant; and therefore a solution thatis not identically zero can satisfy the boundary con-dition only if θ ( a ; λ ) , θ ( b ; λ ) ∈ { jπ } j ∈ Z . Thus, thesecond order eigenvalue problem has been reducedto the study of the first order equation for θ : if for agiven λ there exists a solution θ satisfying the bound-ary condition, then λ is an eigenvalue of L .Let’s shift our perspective slightly and, rather thanthinking of x as a spatial variable, let us view it asa time-like variable. (This is an example of spatialdynamics.) If θ ( a ; λ ) / ∈ { jπ } j ∈ Z , then θ cannot be aneigenfunction; therefore to determine if λ is an eigen-value, by periodicity we can assume θ ( a ; λ ) = 0. Be-cause of the structure of the equation, for λ large andnegative we expect θ to oscillate and to find eigenval-ues. Suppose we have found one, and we label it λ k to indicate θ ( b ; λ k ) = ( k + 1) π . If we continuouslyincrease λ , we continuously decrease θ ( b ; λ ), and thenext eigenvalue occurs when we reach the point where θ ( b ; λ k − ) = kπ . Expanding on this argument, onecan prove there is a sequence of simple eigenvalues λ > λ > . . . and corresponding sequence of solu-tions θ such that θ ( b ; λ k ) = ( k + 1) π . This in turnimplies that the corresponding eigenfunction v ( x ; λ k )has exactly k simple zeros in the interval ( a, b ).From the perspective of stability, this is an ex-tremely powerful result. This is classically illustratedby considering a scalar reaction-diffusion equation onthe entire real line that has a pulse as a stationarysolution; see Figure 1. This is a natural example toconsider for at least two reasons. First, in the con-text of applications reaction-diffusion equations areoften posed on the entire real line so as to avoidany potential complications arising from the bound-ary while still capturing the experimentally observedbehavior. Second, pulses are among the simplest andmost common type of coherent structures found insuch models. The relevant elements of the above the-ory remain when we replace the interval ( a, b ) withthe real line R , as long as we work in an appropriatefunction space, such as L ( R ). Because ϕ satisfies(2), if we take an x -derivative of this equation wefind that 0 = L ϕ x , and so ϕ x is an eigenfunction of L with eigenvalue zero. As illustrated in Figure 1, ϕ x has exactly one zero. This implies that 0 = λ ,and so there must be a positive eigenvalue, λ > xx' ( x ) ' ( x ) Figure 1: A pulse and its derivative.As a result, any stationary pulse solution of a scalarreaction-diffusion equation on the real line must beunstable. The details of the function f are not rel-evant, other than that the resulting equation has apulse solution, nor are the details of ϕ , other thanthat it is a pulse (or more generally has at least onelocal extrema). A complementary result holds if ϕ is a monotonic front, in which case ϕ x has no zeros,and so the largest eigenvalue is zero: λ = 0.In this example, the zeros of the eigenfunction arebeing used as a proxy for the eigenvalues. This sug-gests the alternative perspective of conjugate points,which can be described as follows. Above, the domain( a, b ) was kept fixed, λ was allowed to vary, and thevalues of λ where the solution satisfied the boundarycondition were recorded. Instead, let’s fix λ and al-low the domain to vary: x ∈ ( a, s ) with s ∈ [ a, b ].The number s is defined to be a conjugate point for λ if λ is an eigenvalue of the Dirichlet problem posedon the domain [ a, s ]. We can play a similar gameif we fix λ = λ k . We therefore know that if s = b ,then θ ( b ; λ k ) = ( k + 1) π . We can now continuouslydecrease s from b , so that θ has less time to oscil-late (that’s the spatial dynamics perspective again),and record the values s j where θ ( s j ; λ k ) = ( j + 1) π .In this way, we get a sequence of conjugate points s k = b > s k − > s k − > · · · > s > a that are inone-to-one correspondence with the eigenvalues thatare strictly bigger than λ k .This result is illustrated using the “square” de-picted in Figure 2. To complete the picture, oneneeds to show that for λ = λ ∞ sufficiently large thereare no conjugate points, and note that for s = a thereare no eigenvalues simply because there are no dy-namics. To detect instabilities, one can fix λ ∗ = 0,and then the number of conjugate points must beequal to the number of unstable eigenvalues. In theexample above regarding pulse instability, by count-3 s ba ⇤ s s s Figure 2: The square illustrating that the number ofconjugate points for λ = λ ∗ is equal to the numberof eigenvalues λ > λ ∗ .ing zeros of ϕ x we were effectively counting conjugatepoints to prove the existence of an unstable eigen-value. This is a simple case of what’s often called theMorse Index Theorem, and it goes back to the workof Morse [Mor96], Bott [Bot56], and others.The idea of counting unstable eigenvalues by in-stead counting conjugate points seems nice, but itappears to be restricted to the scalar case, wherewe can use polar coordinates to define the angle θ .However, Arnol’d [Ad85, Ad67] realized that a gen-eralization of this angle to the system case ( n > d = 1. To most directlyutilize the Maslov index, we’ll assume the nonlinear-ity is a gradient, f = ∇ G for some G : R n → R . Theeigenvalue problem then becomes λv = v xx + ∇ G ( ϕ ( x )) v = L v, x ∈ R , where now Ω = R and it is required that v ∈ L ( R ; R n ), in lieu of specifying boundary conditions.Note that the linear operator is again self adjoint,so λ ∈ R . To fix ideas, let’s again suppose ϕ is apulse, meaning that lim x →±∞ ϕ ( x ) = ϕ ∞ for some ϕ ∞ ∈ R n . As mentioned above, the most interest-ing case is to assume the essential spectrum of L isstable, so we can focus on detecting unstable eigen-values. It turns out this is equivalent to assumingthat ∇ G ( ϕ ∞ ) is a negative matrix; this will be uti-lized below. This second-order eigenvalue problemcan again be written as a first order system, now via ddx (cid:18) vw (cid:19) = (cid:18) −
11 0 (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) =: J (cid:18) λ − ∇ G ( ϕ ( x )) 00 − I (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) =: B ( x ; λ ) (cid:18) vw (cid:19) . (3)There’s that spatial dynamics perspective again.To understand how to associate an angle with thisfirst-order eigenvalue problem, let’s step back anddiscuss the Maslov Index. An accessible explana-tion of the topics we are about to describe can befound in [HLS17]. To begin, consider the symplecticform ω ( U, V ) := (cid:104)
U, JV (cid:105) R n , where J is defined in (3)and (cid:104)· , ·(cid:105) R n is the usual inner product in R n . Theassociated Lagrangian-Grassmanian is the set of all n -dimensional subspaces of R n on which the sym-plectic form vanishes:Λ( n ) = { (cid:96) ⊂ R n : dim( (cid:96) ) = n, ω | (cid:96) × (cid:96) = 0 } . Each Lagrangian plane has an associated frame ma-trix, defined in terms of square matrices
A, B ∈ R n such that (cid:96) = (cid:26)(cid:18) AB (cid:19) u : u ∈ R n (cid:27) . The plane is just the column space of the frame ma-trix. In fact, the above frame matrix is not unique,and each plane corresponds to an equivalence classof frame matrices. Suppose we have a path of La-grangian subspaces, (cid:96) ( t ) for t ∈ ( a, b ), and we areinterested in intersections of this path with a fixedreference Lagrangian plane, say the Dirichlet plane: D = { (0 , v ) ∈ R n : v ∈ R n } . (This is analogousto looking for conjugate points.) Associate the path (cid:96) ( t ) with frame matrices A ( t ) , B ( t ). Arnol’d showedthere is a well-defined angle θ ( t ) such that e i θ ( t ) = det[( A ( t ) − i B ( t ))( A ( t ) + i B ( t )) − (cid:124) (cid:123)(cid:122) (cid:125) =: W ( t ) ] . (4)4he reason this works is that the Lagrangian struc-ture of (cid:96) forces W to be unitary, so its spectrum lieson the unit circle. Moreover, it can be shown thatdim[ker( W ( t ) + I )] = dim( (cid:96) ( t ) ∩ D ) . Note that the quantity on the left hand side refers tothe complex dimension of the complex vector spaceker( W ( t ) + I ) ⊂ C n , whereas the quantity on theright hand side refers to the real dimension of the realsubspace of (cid:96) ( t ) ∩D ⊂ R n . To write down the defini-tion of the Maslov index in full detail would be quitelengthy; here the key fact is that the Maslov indexcounts, with multiplicity and direction, the numberof times an eigenvalue of W ( t ) crosses through − (cid:96) ( t )with the reference plane D . In this sense, the Maslovindex counts conjugate points. The Maslov index isrelated to the fact that the fundamental group of theLagrangian-Grassmanian is the integers; if (cid:96) ( t ) is aloop, its Maslov index is its equivalence class in thefundamental group [Ad67].Let’s return now to our eigenvalue problem (3).Our assumption that the essential spectrum is stable, ∇ G ( ϕ ∞ ) <
0, implies that the asymptotic matriceslim x →±∞ J B ( x ; λ ) are both hyperbolic, with stableand unstable subspaces of dimension n . If we let E u − ( x ; λ ) and E s + ( x ; λ ) denote the subspaces of solu-tions that are asymptotic to the unstable eigenspaceat −∞ and the stable subspace at + ∞ , respectively,then in order to have an eigenfunction v ∈ L wemust have ( v, w )( x ; λ ) ∈ E u − ( x ; λ ) ∩ E s + ( x ; λ ); oth-erwise, the solution would be growing exponentiallyfast in forward or backward time, thus preventing v from being square integrable. See Figure 3.Studying the intersection of these subspaces leadsto the now standard theory behind the Evans func-tion [San02]. So far we have made no reference to anyLagrangian structure. It turns out that our assump-tion that f = ∇ G implies that in fact both E u − ( x ; λ )and E s + ( x ; λ ) are paths of Lagrangian subspaces.With this additional structure, we can adopt a differ-ent perspective and look for conjugate points: given (cid:96) ( x ; λ ) := E u − ( x ; λ ) ∈ Λ( n ), we define a conjugatepoint to be a value of x such that (cid:96) ( x ; λ ) ∩ D (cid:54) = { } .Using this framework, in [BCJ +
18] it was shownthat the square depicted in Figure 2, suitably adapted E s + ( ) E s + ( x ; ) E u ( x ; ) E u 1 ( ) R n R n Figure 3: The subspaces of decaying solutions.to reflect the fact that the spatial domain is now allof R , holds for the eigenvalue problem (3). This re-lies on the homotopy invariance of the Maslov in-dex and the fact that the boundary of the squaremaps to a null-homotopic curve in the Lagrangian-Grassmanian. Thus, one can count unstable eigen-values by instead counting conjugate points. Fur-thermore, this result was used to prove that, in equa-tions of the form (1) with Ω = R and f = ∇ G , anygeneric pulse solution must necessarily be unstable.This is again quite powerful; no further informationis needed about the function f or the pulse ϕ thatit supports. The topology is, in a sense, forcing theexistence of a positive eigenvalue.Some remarks may be helpful here. First, the proofof the “square” relies on the Maslov index and itstopological properties, although the definition of θ given in (4) is not directly used. Instead, the resultis developed using the associated crossing form pre-sented in [RS93]. Second, a key step in the proofis proving a so-called monotonicity result. The path (cid:96) ( x ; λ ) = E u − ( x ; λ ) is a path around the entire bound-ary of the square, if one considers either x or λ tobe the path parameter on the appropriate sides, andhence a loop. After compactifying the domain, sothat x ∈ R becomes ˜ x ∈ [ − , − , × [0 , λ ∞ ] ⊂ R is contractible, its imagein the Lagrangian-Grassmannian is also contractible,and hence the Maslov index of the loop (cid:96) ( x ; λ ) mustbe zero. Showing there can be no intersections on theright side, where λ = λ ∞ sufficiently large, or on thebottom, where ˜ x = −
1, is not too difficult. One canthen show that all crossings on the top (eigenvalues)must contribute in a negative way to the index, while5n the left (conjugate points) they must contribute ina positive way; this is the monotonicity. Another wayto view this monotonicity is in terms of the matrix W , defined in (4). In this setting, W = W ( x, λ ), andthis monotonicty result means that eigenvalues of W must always pass through − λ is varied, and always in the opposite direction as x is varied. Hence, the number of eigenvalues mustequal the number of conjugate points. The fact thatthere must be at least one conjugate point when lin-earizing about a pulse comes from a symmetry argu-ment that uses the reversibility of (1) (the fact thatit is invariant under the transformation x → − x ).Not only does this result allow for the extensionto the system case of the “pulses must be unstable”result from Sturm-Liouville theory, but it also pro-vides a more efficient way, in general, for detectinginstabilities, provided one has the required symplec-tic structure, for example if f = ∇ G . To explain this,note that the Evans function, mentioned above, canbe defined by E ( λ ) := E u − (0; λ ) ∧ E s + (0; λ ) : C → C .(The choice to look for intersections of the subspacesof decaying solutions at x = 0 is arbitrary; any point x ∈ R could be chosen here.) Zeros of the Evansfunction correspond, with multiplicity, to eigenval-ues. In general, to detect instabilities using the Evansfunction, one must prove that any unstable eigenval-ues must lie in some compact ball and then computethe winding number of E around the boundary of thisball. On the other hand, to count conjugate points,one must only do analysis for a single value λ = 0.Thus, if one were to use validated numerics to pro-duce a proof of (in)stability via such a detection pro-cedure, the computation would be much faster usingconjugate points than using the Evans function. Thisis the subject of current work.It is interesting to note that this connection be-tween the Maslov index and stability, including theabove demonstration of pulse instability, is not theonly connection between topology and dynamic sta-bility. It is also know that in some systems that sup-port traveling waves, the wave can be constructedas the intersection of appropriate stable and unsta-ble manifolds. This intersection typically occurs fora unique wavespeed, and the direction in which thosemanifolds cross as the wavespeed parameter is varied can be connected with E (cid:48) (0) and hence the parity ofthe number of unstable eigenvalues; if this numberis odd, there must be at least one, and the wave isunstable [Jon95].So far, everything that has been discussed for (1)has been restricted to the case of one spatial domain, d = 1. It turns out, however, that these ideas canalso be expanded to cover the multidimensional case[DJ11, CJM15] In this case, the eigenvalue problemtakes the form λv = ∆ v + ∇ G ( ϕ ( x )) v, x ∈ Ω ⊂ R d v | ∂ Ω = 0 . To create the above theory in this setting, we needa notion of a conjugate point. This can be definedusing a one-parameter family of domains, { Ω s : s ∈ [0 , , Ω = Ω , Ω = { x }} , that shrinks the originaldomain down to a point [Sma65]. One can then con-struct the path of subspaces (cid:96) ( s ; λ ) = (cid:26)(cid:18) u, ∂u∂n (cid:19) (cid:12)(cid:12)(cid:12) ∂ Ω s : u ∈ H (Ω s ) , ∆ u + V ( x ) u = λu, x ∈ Ω s } determined by weak solutions on Ω s , but with noreference yet to the boundary data. By consideringthe Hilbert space H = H / ( ∂ Ω) × H − / ( ∂ Ω)and the symplectic form ω (( f , g ) , ( f , g )) = (cid:104) g , f (cid:105)−(cid:104) g , f (cid:105) , where (cid:104)· , ·(cid:105) denotes the dual pairing,once can show that both the path (cid:96) and the Dirichletsubpace D = (cid:26)(cid:18) u, ∂u∂n (cid:19) = (cid:18) , ∂u∂n (cid:19)(cid:27) ⊂ H , lie in the associated Fredholm-Lagrangian-Grassmanian, a generalization of the LagrangianGrassmannian Λ( n ) to the infinite-dimensionalsetting. This Dirichlet subspace is now the fixedreference space, and a conjugate point is a valueof s such that (cid:96) ( s ; λ ) ∩ D (cid:54) = { } . Note that theterm “Dirichlet subspace” in this context referencesthe fact that this subspace corresponds to the zero6irichlet boundary condition in the above eigenvalueproblem. This perspective was pioneered in [DJ11]and allows for much of the above theory to work forthe multi-dimensional eigenvalue problem, includingthe system case v ∈ R n and a variety of boundaryconditions other than Dirichlet.These multidimensional results are particularly ex-citing because most of the results related to nonlinearwaves and coherent structures, not just their stability,apply only in one dimension. This is largely becausemany of the techniques rely on the perspective of spa-tial dynamics, which, for the most part, only appliesto systems in one space dimension, or on cylindri-cal domains with a single distinguished spatial vari-able. Interestingly, the above procedure of using ashrinking family of domains, { Ω s } , suggests a way todevelop spatial dynamics in higher dimensions. Spatial Dynamics
In order to more precisely characterize what is meantby the term “spatial dynamics,” let’s recall the mostbasic setting in which spatial dynamics has beenused, second order ordinary differential equations(ODEs) of the form u xx + F ( u ) = 0. By writingthis as the first order system u x = v, v x = − F ( u ) , one can study the behavior of solutions using tech-niques from dynamical systems, such as phase planeanalysis and exponential dichotomies. Here the spa-tial domain is Ω = R , and the phase space of thespatial dynamical system is R (or R n if u ∈ R n ).The above system is a spatial dynamical system, orequivalently it is the second order ODE viewed fromthe perspective of spatial dynamics, because in it thespatial variable x is viewed as a time-like evolutionvariable, and techniques from the theory of dynami-cal systems can be used to study an equation that wasnot originally formulated as an evolutionary equa-tion.On a cylindrical domain, Ω = R × Ω (cid:48) with Ω (cid:48) ⊂ R d − compact, the PDE ∆ u + F ( u ) = 0 can be writ-ten u x = v, v x = − ∆ Ω (cid:48) v − F ( u ) , (5) where ∆ Ω (cid:48) is the Laplacian on the cross section Ω (cid:48) .The phase space is now infinite-dimensional, for ex-ample ( u, v )( x ) ∈ H (Ω (cid:48) ) × L (Ω (cid:48) ) for each x ∈ R ,and so one must be more careful in analyzing the dy-namics. This can be seen explicitly if Ω (cid:48) = [0 , π ]with periodic boundary conditions, in which case thelinear part of (5) coming from the Laplacian, (cid:18) − ∂ y (cid:19) , has spectrum equal to the integers. This canbe seen by using the Fourier expansion u ( x, y ) = (cid:80) k ˆ u k ( x ) e i ky , v ( x, y ) = (cid:80) k ˆ v k ( x ) e i ky , in which case − ∂ y → k and the eigenvalues can be explicitly com-puted. The fact that there are arbitrary large positiveand negative eigenvalues means that, in general, so-lutions to (5) will grow arbitrarily fast both forwardsand backwards in time. In other words, the system(5) is ill-posed. Nevertheless, applying techniquesfrom dynamical systems to analyze the behavior ofsolutions is extremely useful.For example, in many cases one can construct anexponential dichotomy associated with the linear partof (5), and also construct stable and unstable (orpossibly center-stable and center-unstable) manifoldsassociated with the nonlinear system. This allowsfor the analysis of subspaces, in the case of the di-chotomy, or more generally manifolds of solutionsthat exist in forwards or backwards time, respec-tively. As a result, one can study bifurcations bylooking at intersections of the relevant manifolds assystem parameters are varied. One can also studystability, both at the spectral level using a general-ization of the Evans function, at the linear level usingpointwise Green’s function estimates, and at the non-linear level by combining these estimates with a rep-resentation of solutions to the full nonlinear equation,for example via Duhamel’s formula. This infinite-dimensional spatial dynamics perspective began withthe work of Kirchgassner [Kir82], and subsequentcontributions include [Mie86, PSS97].The perspective of spatial dynamics has proven tobe quite useful, and it has allowed for an extensivevariety of interesting and beautiful results to be ob-tained for PDEs on either one-dimensional or cylin-drical domains. It has not, however, been utilized7n multidimensional domains that do not have thiscylindrical structure, and this is arguably the mainreason why there are many fewer results available inhigher space dimensions. The hope is that recent re-sults, motivated by the above stability theory andwhich I will now describe, will change this.Consider the PDE∆ u + F ( x, u ) = 0 (6)with x ∈ Ω ⊂ R d , and recall Smale’s idea of shrinkingthe domain Ω via a one-parameter family { Ω s } s ∈ [0 , .Suppose that this family is parameterized by a familyof diffeomorphisms ψ s : Ω → Ω s . This allows for anice definition of the boundary data on ∂ Ω s : f ( s ; y ) = u ( ψ s ( y )) , g ( s ; y ) = ∂u∂n ( ψ s ( y )) , for s ∈ [0 ,
1] and y ∈ ∂ Ω. This is convenient because,even though ( f, g ) can be interpreted as the boundarydata on Ω s , the independent variable y lives in the s -independent domain ∂ Ω. One can then, at leastformally, compute an evolution equation of the form dds ( f, g ) = F ( f, g ) , (7)where the possibly nonlinear function F is definedin terms of the function F appearing in (6) and thetangential parts of the gradient and divergence oper-ators on ∂ Ω s . One can also, again at least formally,relate a solution ( f, g ) of (7) to the solution u of (6)by noting that ( f, g ) is just the function u and itsnormal derivative evaluated on the boundary on thedomain Ω s ; in other words, ( f, g ) is just the trace of( u, ∂u/∂n ) evaluated on ∂ Ω s . This has been maderigorous in [BCJ + u of the ellip-tic PDE (6) leads to a solution ( f, g ) of the spatialdynamical system (7), and vice versa.The function F is indeed quite complicated, andthe relation between u and ( f, g ) is rather techni-cal. However, for at least some domains Ω, the resultseems to be sufficiently concrete so as to be read-ily applicable. For example, if the domain is ra-dial or all of R d , one can choose to shrink the do-main using spheres: Ω s = { x ∈ R d : | x | < s } . This greatly simplifies the function F and, using thefact that in terms of generalized polar coordinates∆ = ∂ r + ( n − r − ∂ r + r − ∆ S d − , one ends up withthe spatial dynamical system dds (cid:18) fg (cid:19) = (cid:18) − s − ∆ S d − − ( d − s − (cid:19) (cid:18) fg (cid:19) + (cid:18) − F ( θ, s, f ) (cid:19) . It has been shown that the linear part of this system,after a suitable rescaling of time s = e τ and for d ≥ + d = 2 is slightly more complicated, due to theexistence of the harmonic function log r , but it couldbe similarly interpreted by allowing the dichotomy tocontain center directions.) Moreover, when d = 3 thedichotomy can be written down explicitly in termsof the spherical harmonics. This allows one to poten-tially study solutions to the original elliptic PDE thatare not necessarily radially symmetric, thus provid-ing the removal of a restriction that has been imposedon most results (at least in the spatial dynamics con-text) to date. Thus, the perspective of spatial dy-namics seems quite promising as a method for study-ing multidimensional nonlinear waves and coherentstructures. Future Directions
The theory discussed above has the potential to havea great impact, particularly for problems in multi-ple spatial dimensions. Many of the existing resultsare valid only for one-dimensional domains, or forcylindrical domains. The above results represent newtechniques that are not bound by this restriction, andthus allow for the analysis not only of stability butalso of a variety of aspects of the behavior of solu-tions to PDEs in multi-dimensional spatial domains,such as their existence and bifurcation.In the last ten years or so there have been many re-sults regarding the theory discussed above. Arguablythe only downside so far is the relative lack of appli-cations: examples of solutions, in any space dimen-sion, whose stability is determined using the conju-gate point method described above and instances of8sing the spatial dynamical system (7) to analyzemultidimensional nonlinear waves.Regarding the former, there are three existingexamples, at least where the Evans function can-not also be used to determine stability. The mostbroadly applicable is the pulse instability result inreaction-diffusion systems with gradient nonlinearity,described above. The other two examples pertain tospecific PDEs, with the first being the instability re-sult of [Jon88] for a standing wave in a nonlinearSchr¨odinger-type equation, which really began thiswhole program, and the second being the instabilityresult of [CH14] for a standing pulse in the FitzHugh-Nagumo equation, with diffusion in both variables.The development of the spatial dynamical system (7)and its relation to the elliptic PDE (6) is extremelynew, and so some time is needed for its utility to befully explored. Now that a solid foundational theoryis in place, the hope is that many more applicationswill emerge. This is an area of active, ongoing work.
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