Instability in large bounded domains -- branched versus unbranched resonances
aa r X i v : . [ n li n . PS ] S e p Instability in large bounded domains — branched versus unbranched resonances
Montie Avery , Cedric Dedina , Aislinn Smith , and Arnd Scheel . University of Minnesota, School of Mathematics, 206 Church St. S.E., Minneapolis, MN 55455, USA University of Texas at Austin, Department of Mathematics, 1 University Station, Austin, TX 78712, USA
Abstract
We study instabilities in large bounded domains for prototypical model problems in the presence of transportand negative nonlinear feedback. In the most common scenario, bifurcation diagrams are steep, with a jumpto finite amplitude at the transition from convective to absolute instability. We identify and analyze a genericalternate scenario, where this transition is smooth, with a gradual increase. Transitions in both cases are mediatedby propagating fronts whose speed is determined by the linear dispersion relation. In the most common scenario,front speeds are determined by branched resonances, while in an alternate scenario studied here, front speeds aredetermined by unbranched resonances and typically involve long-range interaction with boundaries.
We are interested in instabilities in dissipative dynamics in large bounded domains. Four key features to our setup are:translational invariance of the system away from the boundary; unidirectional transport; negative nonlinear feedback;and generic boundary conditions suppressing upstream instabilities. Absent boundary conditions, in the limit of anunbounded system, this scenario leads to instabilities that are at onset convective in nature, that is, perturbationsgrow in amplitude, but are transported downstream, decaying pointwise in space. A second transition, usuallyfor parameter values when the amplitude of nonlinear convective states is already of order one, leads to absoluteinstabilities, where perturbations also grow at fixed locations in space. Phenomenologically, such instabilities areoften mediated by fronts or interfaces propagating through the domain. Near onset of the convective instability,an interface between the upstream trivial state and the downstream bifurcated state is advected downstream. Nearthe absolute instability threshold, the speed of this front changes sign and the nontrivial state propagates upstreamto occupy the entire domain. In finite domains, fronts ultimately interact with boundary conditions, leading toboundary layers; see Fig. 1.1 for an illustration.A common feature of such instabilities is the “hard” transition near onset. First, at onset of absolute instability,the pointwise transition from a trivial state to the nontrivial bifurcated state is of finite, not small amplitude, sincethe nontrivial state bifurcates from the trivial state at onset of convective instability and has usually grown to finiteamplitude at the onset of absolute instability. Second, the region in parameter space occupied by the nontrivialstate increases rapidly when the direction of propagation of the front between nontrivial and trivial state reverses,so that in the limit of infinite domain size the fraction of the domain occupied by the nontrivial state jumps from0 to 1 upon crossing the threshold. Such hard transitions and steep bifurcation branches have been both observedand computed numerically, with a partial list including Langmuir-Blodgett transfer [13], convection patterns withthrough flow [14], Couette-Taylor flows with axial through flow [2]; see also [19], [3], and [20] for further references.In the absence of boundary conditions, or when the boundary conditions admit a trivial constant solution, onecan analyze instabilities through somewhat explicit computations of spectra using Fourier transform. The onset ofinstability in unbounded domains is determined by continuous spectra, characterized by a dispersion relation. In largebounded domains, spectra form clusters that accumulate on absolute spectra [15], which in our case of unidirectionaltransport are strictly to the left of the continuous spectrum in the complex plane. In between continuous and absolutespectra lies a region of pseudo spectrum that indicates sensitivity of the system to small disturbances; see for instance[19, 18] for an analysis of the dynamics near such transitions.A more descriptive analysis relies on analyzing the speed of propagation of the front between nontrivial and trivialstates. In our case of nonlinearity with negative feedback, one can try to predict the speed of such fronts based again1n the linear dispersion relation, allowing however for complex wavenumbers, that is, exponentially decaying andgrowing eigenmodes. In particular, the reversal of the direction of propagation is predicted as the transition fromlinear pointwise decay to pointwise growth, associated with a pinched double root of the complex dispersion relationon the imaginary axis [1, 20, 12]. This linear pointwise instability is accompanied by the emergence of nonlinearglobal modes [4, 5], which, near onset, resemble the nonlinear invasion front arrested at a finite distance from theupstream boundary. -500 -400 -300 -200 -100 00500100015002000 -500 -400 -300 -200 -100 0-0.400.4 -400 -200 00200040006000800010000 -500 -400 -300 -200 -100 00500100015002000 -500 -400 -300 -200 -100 0-0.400.4
Figure 1.1: Space-time plots for CGL (1.3), α = 0 . , γ = 0 . , d = 1 , R = . , k = 0 .
5, with µ below (left, Re ( A )), close to (center, | A | ),and above (right, Re ( A )) the critical value. Left and right figures show front positions diverging to left and right border, Ξ( µ ) = 0 , µ ) ∼ . Recently [12, 6, 7], selected front speeds were characterized more generally. In particular, pinched double roots areviewed in [6] as quite particular resonances, where two complex spatial modes ν collide and unfold with √ λ − λ br ,that is, on a branched surface. We refer to these as branched resonances . They were identified as part of a largerfamily of more general resonances, including unbranched 1:1-resonances, or higher, say 2:1-resonances.The implications of this view point on front propagation and the role of nonlinear terms mediating such resonanceswere analyzed in a series of papers [12, 6, 7, 11, 9, 10]. As a result, focusing on pointwise decay versus growth innonlinear systems in an unbounded domain, one may find an onset of instability, determined by nontrivial unbranchedresonances, that precedes the onset predicted from the classical pinched double root criterion. Our goal here is toinvestigate instabilities in large bounded domains from this new perspective: Predict universal features of bifurcation diagrams in large domains for branched and unbranched resonances!
We therefore introduce a class of model problems, next, and then state our main results informally in terms of asolution measure that tracks the portion of the domain occupied by the nontrivial state.
Model problems.
The simplest model of a transition from convective to absolute instability is the scalar amplitudeequation, in this context usually referred to as the Fisher-Kolmogorov-Petrovsky-Piscounov equation (KPP), u t = u xx + u x + µu − u , − L < x < . (1.1)We choose Dirichlet boundary conditions, suppressing an instability upstream and modeling a downstream distur-bance, u = u , x = − L, u = 0 , x = 0 . (1.2)We will discuss later how results are largely independent of boundary conditions and in particular the size of u = 0.More realistic models for instabilities in fluid flows assume oscillatory instabilities captured by a complex amplitudesolving the complex Ginzburg-Landau equation, A t = (1 + i α ) A xx + A x + ( µ + i β ) A − (1 + i γ ) A | A | , − L < x < . (1.3)2hich we equip similarly with Dirichlet or gauge-invariant boundary conditions: A = A , x = − L, A = 0 , x = 0 , or (1.4) | A | x = R , arg( A ) x = k , x = − L, A = 0 , x = 0 . (1.5)In both cases, the transition from convective to absolute instability is caused by a branched resonance, that is, apinched double root that crosses the imaginary axis at µ br = , (1.1), and µ br = α ) , (1.3).The somewhat peculiar choice of boundary conditions in (1.5) simplifies numerical continuation since the boundaryconditions preserve the gauge invariance A e i ϕ A . As a consequence, one can study the invasion process byinvestigating solutions of the form A ( t, x ) = A ∗ ( x )e i ωt , which in turn solve an ODE boundary value problem. Inparticular, in this case the parameter β is redundant and can be set to zero.Unbranched resonances typically arise when more than one mode is close to onset, an effect which is usually capturedthrough coupled amplitude equations. Our main motivating example therefore is the system u t = u xx + u x + µu − u + | A | , − L < x < ,A t = d (1 + i α ) A xx + A x + ( ρ + i β ) A − (1 + i γ ) A | A | , − L < x < . (1.6)with boundary conditions | A | x = R , arg( A ) x = k , u = u , x = − L, A = 0 , u = 0 , x = 0 . (1.7)We will also study boundary conditions that break the gauge symmetry in A , (1.4), and more generic coupling terms.To further an analytical understanding, we also consider the case where α = γ = ω = 0, and A ∈ R , which reducesto a system of coupled scalar amplitude equations, u t = u xx + u x + µu − u + v κ , − L < x < ,v t = dv xx + v x + ρv − v , − L < x < . (1.8)with boundary conditions v = v , u = u , x = − L, v = 0 , u = 0 , x = 0 . (1.9)The exponent κ determines the type of resonance that is relevant: clearly, κ = 2 – and therefore 2:1-resonances –would be obtained from (1.6), but we shall also study κ = 1, which leads to 1:1-unbranched resonances.Lastly, motivated by the exposition in [12], we also study the simplest model problem that produces nontrivialresonant instabilities, u t = (1 − µ ) u x + u − u + v, − L < x < ,v t = − v x − v, − L < x < . (1.10)with µ < v = v , x = − L, u = 0 , x = 0 . (1.11) Main result — informal.
We measure amplitudes through the support of the solution, or, more precisely, theregion where the amplitude of the solution exceeds some fixed, small threshold, within the interval [ − L, µ increases past aninstability threshold µ c , determined by the fact that the speed of an invasion front in the unbounded domain changessign, that is, the front invades the domain for µ > µ c . Writing X ( µ ) for the position of the interface, that is, theright boundary of the region where | u ( x ) | > δ for some small, positive δ , we define Ξ( µ ) = ( X ( µ ) + L ) /L , the fractionof the domain occupied by the nontrivial state, as our solution measure.A brief summary of our main results is shown in Fig. 1.2 and reads as follows.3 igure 1.2: Typical bifurcation diagrams for branched (left) and unbranched (right) resonances, as described in our main results. • branched resonance, (1.1) or (1.3) : Ξ( µ ) ∼ (cid:26) , µ < µ br , µ > µ br , in the limit L → ∞ . For any finite L , Ξ( µ ) is smooth with an increase from ε >
0, small, to 1 − ε on aparameter interval of width O ( L − / ) as L → ∞ . • unbranched resonance, (1.6) , (1.8) , or (1.10) : Ξ( µ ) = ν u0 ( µ ) ν u0 ( µ ) + ν s1 ( µ ) , in the limit L → ∞ , a smooth function for µ ≥ µ res . Here, ν u0 ( µ ) characterizes the distance of the spatialmodes from resonance and ν s1 ( µ ) is a decay rate of boundary layers. For finite L , Ξ( µ res ) = O ( L − log L ). Outline.
We demonstrate the hard onset of instability in the case of branched resonances, §
2, and turn to un-branched resonances in §
3. We conclude with a discussion in § Acknowledgments.
A.S., C.D., and A.S. were supported through grant NSF DMS-1907391. M. A. was supportedthrough the NSF GRFP, Award 00074041.
The transition from convective to absolute instability in these model problems is fairly well studied and variousasymptotics have been previously derived in the literature; see in particular [19]. We are however not aware of asomewhat rigorous derivation of the Ξ-asymptotics, particularly in the generic setting with the presence of boundarylayers. The analysis here is also closely mimicked in the next section on unbranched resonances.
We first analyze (1.1) in unbounded and large bounded domains, before turning to (1.3).In an unbounded domain, pointwise stability is determined by the location of pinched double roots [1, 12]. Theseare readily found from the dispersion relation to the linear part, obtained in turn from an ansatz u = e λt + νx , D ( λ, ν ) = ν + ν + µ − λ. Double roots solving D = 0, ∂ ν D = 0 are given by ( λ dr , ν dr ) = ( µ − , − ) such that we have pointwise stability ofthe origin for µ ≤ µ br = . The condition ∂ ν D = 0 implies that we cannot solve for ν using the implicit functiontheorem. If, as in this case, ∂ λ D = 0 and ∂ νν D = 0, we locally find two roots ν ± ( λ ) ∼ ν dr + √ λ − λ dr . The squareroot singularity induces a branch point singularity in the dispersion relation. Homogeneous Dirichlet boundaryconditions at x = − L and x = 0 suppress the instability slightly, with a corrected onset ˜ µ br = + π L that one canreadily compute by solving the Sturm-Liouville eigenvalue problem.4 igure 2.1: Phase plane for (2.1) at µ . / µ = 1 /
4, and µ & / L . The flight time near the blown up equilibrium u = 0 decreases with ( µ − / − / , and additional flighttime is added through a long passage near the equilibrium u = 1. Rather than pursuing a bifurcation analysis from a trivial solution, which usually has limited validity and predictivestrength in large domains, we pursue a conceptual analysis based on spatial dynamics that also generalizes to systemsand resembles the point of view taken in the case of unbranched resonances. We look for solutions as orbits in thephase plane connecting subspaces determined by the boundary conditions.We therefore consider the steady-state equation u x = vv x = − v − µu + u . (2.1)At µ = 1 /
4, the equilibrium u = v = 0 possesses a double eigenvalue ν = − / u, v ) = r (cos ϕ, sin ϕ ), ( r, ϕ ) ∈ [0 , ∞ ) × S . The origin u = v = 0 becomes an invariant circle { } × S ,with equilibria given by the eigenspaces of the linearization. The corresponding saddle-node bifurcation at µ = 1 / µ > /
4. The resulting phase portraits (blowing up the origin in polarcoordinates ( r + 1)(cos( ϕ ) , sin( ϕ )) is plotted in the coordinate plane in Fig. 2.1. Less geometric but computationallymore convenient coordinates are the stereographic projection charts ( u, z ), with z = v/u , and ( v, z ), with z = u/v ,with the origin blown up to the lines { } × R and explicit equations ( u ′ = uz z ′ = − (cid:0) z + (cid:1) − (cid:0) µ − (cid:1) + u , ( v ′ = v (cid:0) − − µz + v z (cid:1) z ′ = (cid:0) z + 1 (cid:1) + (cid:0) µ − (cid:1) z − v z ; (2.2)see Fig. 2.2 for an illustration. Key ingredients to the bifurcation, schematically illustrated in Figure 2.2, are:(i) the trivial equilibrium p t , shown as the invariant circle r = 0;(ii) a saddle-node bifurcation of equilibria on the critical circle reflecting the branched resonance in a collision ofeigenspaces;(iii) the nontrivial equilibrium p nt , hyperbolic in this spatial dynamics formulation since stable in temporal dynam-ics;(iv) a heteroclinic orbit q f , the invasion front on the unbounded domain, between nontrivial equilibrium and trivialequilibrium, not contained in the strong stable manifold of the saddle-node equilibrium at criticality;(v) a boundary subspace for x = 0 that does not contain the saddle-node equilibrium at criticality;(vi) a “singular heteroclinic” q s from the saddle node to the boundary condition in the invariant circle;(vii) a boundary subspace for x = − L that intersects the stable manifold of the nontrivial equilibrium transverselyat criticality yielding a “boundary layer heteroclinic” q bl .Finding these objects is explicitly possible in the KPP equation but can be readily verified computationally in morecomplicated systems. We remark that the fact in (iv) that q f is not contained in the strong stable manifold of p t is equivalent to requiring the absence of a pole of the resolvent for the linearization at a critical pulled front. Such5 pole, reflected in a zero of an Evans function would usually indicate a transition from pulled to pushed frontpropagation. In the KPP case, it is equivalent to the fact that the front possesses asymptotics ( ax + b )e − x/ with a = 0. Also note that existence of a boundary layer (vii) can be guaranteed for all values of u , in particular forhomogeneous Dirichlet boundary conditions, as well as for Neumann or Robin boundary conditions, implying thatthe asymptotics derived below are universal. Of course the fine structure of the bifurcation at small amplitude, forinstance if and how the pitchfork symmetry is broken, depends on the precise form of the boundary conditions. Figure 2.2: Visualization of the coordinate changes from polar coordinates to projective coordinates (left): Blown up equilibrium (black)with singular eigenspaces and strong stable manifolds (red). Projection onto ( u, z )-coordinates (blue) and ( v, z )-coordinates. Alsoshown a schematic of the chain of heteroclinic orbits (right) as detailed in the itemized list of building blocks in the text. With these assumptions, one finds orbits that follow the chain of “heteroclinics”, from boundary condition at x = − L to the nontrivial equilibrium, then from the non-trivial equilibrium to the trivial equilibrium, and ultimately fromthe saddle-node equilibrium to the boundary subspace. Those trajectories spend long times near the two equilibriaincluded in the heteroclinic and times O (1) in between the equilibria. Passage times near the saddle node are readilyfound as L = π ( µ − / − / + O (1). Solutions with flight times L between the boundary conditions thereforeneed to add a plateau near the nontrivial equilibrium u = √ µ of length L − π ( µ − / − / + O (1), which gives theprediction Ξ( µ ) = 1 − πL √ µ − µ br . (2.3)As a result, we find Ξ( µ ) ∼ max (cid:18) , − πL √ µ − µ br (cid:19) , for µ & µ br . (2.4)which converges to 1 for µ > µ br , fixed, as L → ∞ . In a more general setting, for a branched double root in thedispersion relation at λ = ¯ µ ∼
0, with expansion of the dispersion relation D ( λ, ν ) = λ − d ( ν − ν ) + ¯ µ , thesaddle-node bifurcation unfolds as dz ′ = − d ( z − ν br ) − ¯ µ and we predict thatΞ( µ ) = max , − πL p ¯ µ/d ! . (2.5)We believe that the ingredients to this analysis are quite generally satisfied near onset of instability and the predictionsare therefore universally valid across a wide range of systems.For the complex Ginzburg-Landau equation, the problem is very similar. The key difference is that the pincheddouble root is complex Im λ dr = 0, Im ν dr = 0. As a consequence, the resonant subspace is 4-dimensional and thedesingularization leads to [0 , ∞ ) × S . The gauge symmetry acts however on S via the Hopf fibration and theproblem reduces to [0 , ∞ ) × S ∼ [0 , ∞ ) × ¯ C , with ¯ C the Riemann sphere, with singular heteroclinics given bysolutions to z ′ = − z ∈ ¯ C . An equivalent analysis was carried out in [8, §
3] which we refer to for more details of thechoice of coordinates.Since the reduced equation is simply the complexification of the real saddle-node, we find the same leading orderterm in passage times, Ξ( µ ) ∼ max (cid:18) , − πL √ µ − µ br (cid:19) , for µ & µ br , (2.6)6here this time µ br = α ) ; see also [19, § q f is not generally known outside of a regime where α, γ ∼
0, although the existence problem reduces to a shooting problem in a 3-dimensional ODE with good evidencefor existence across all parameters. There also does not appear to be evidence for existence of pushed fronts. Absentinstabilities of the primary front, the main limitation to universal validity of our expansion then are boundary layers.Since the pulled fronts in CGL select wave trains with group velocities pointing away from the front interface [20, 8],boundary layers q bl are merely boundary sinks and are expected to exist in a robust fashion for a family of wavetrainsand frequencies [17], in particular the ones selected by the invasion front.Lastly, we note that in both KPP and in CGL, there are multiple singular heteroclinic orbits, corresponding tomultiple (half-)rounds on the singular circle (sphere). The associated profiles are typically unstable but accessible tonumerical continuation. Scalings of positions are similar, with passage times near the saddle-node replaced simplyby multiples, leading to Ξ( µ ) ∼ max (cid:18) , − j πL √ µ − µ br (cid:19) , for µ & µ br , j = 1 , , , . . . (2.7) We discretized the boundary-value problems using second order finite differences with h = 10 − and continuedsolution branches using secant continuation. Results are shown in Fig. 2.3. Figures include only the positive branchin the KPP equation and the branch with j = 1 in CGL (2.7). The branch with plateau near −√ µ for µ > µ br undergoes a saddle-node near µ = 1 / µ to an unstable branch. For the unstable branch, Ξ( µ ) issignificantly smaller. We also studied CGL (1.3) with the same gauge-invariant boundary conditions (1.5) in directsimulations, comparing in particular with homogeneous Dirichlet boundary conditions and inhomogeneous Dirichletboundary conditions. In the latter case, the amplitude | A | is time-periodic and the interface location is oscillating intime, albeit with an extremely small amplitude. The graphs shown are instantaneous measurements in time. Sincefront dynamics are extremely slow, we continued dynamically, that is, we let the front position relax for a fixedparameter value until the measured speed was smaller than 5 · − and then decreased the parameter, using thelast simulation data as initial condition. Clearly, for u = 0, homogeneous Dirichlet boundary conditions, the system Figure 2.3: Numerical bifurcation diagrams for KPP (left, (1.1)) and CGL (center, (1.3)), and comparison with theory (2.4) and(2.6), respectively, showing excellent agreement for the front position. Parameters for CGL are α = 0 . , γ = 0 . , R = 0 . , k = − .
5. Bifurcation diagrams from direct simulations of CGL (right) with gauge-invariant boundary conditions (1.5) as in center figure,homogeneous and inhomogeneous Dirichlet boundary conditions ( R = 0 , R = 1, (1.4)). possesses a pitchfork symmetry. Nonzero u breaks this symmetry leading to an imperfect pitchfork bifurcation,where the perturbation effect is very small even for finite u due to the presence of downstream transport. Sincein a pitchfork bifurcation, secant continuation typically continues the nontrivial branch as a saddle-node, very smallstep sizes are necessary to resolve the imperfection and continue toward the trivial branch and µ < µ br . A secondnumerical difficulty is caused by the necessity to resolve the exponential tails of solutions near x = 0, limiting the7ossible allowed sizes of L due to the occurrence of underflow. In fact, the size of the layer Ξ( µ ), as can be seenfrom the analysis and Fig. 2.1, is determined by a slow passage near a fold in projective space, all for amplitudes of( u, u x ) that are O (e − L/ ), which limits domain sizes to about L = 1000.In the case of CGL, the set of solutions is significantly more complex, in particular when varying other systemparameters. Of course, the onset of instability in the complex Ginzburg-Landau equation will lead to dynamics ascomplicated as the dynamics of the Ginzburg-Landau equation, which can be difficult to describe, in particular inthe regime where coherent solutions are all unstable due to the Benjamin-Feir instability. We therefore focus onsmall values of linear and nonlinear dispersion α and γ , here. Direct numerical simulations show that typical initialconditions converge to truly periodic solutions (that is, the amplitude | A ( t, x ) | is not stationary) with oscillationslocalized near the boundary for many choices of boundary conditions and system parameters, in particular for mostchoices of Dirichlet boundary conditions | A | = R at x = 0. We did not attempt to resolve the subtle exponentiallysmall effect near the touchdown at Ξ ∼ j = 0 and good agreement with theory (2.7).We are not aware of a comprehensive study of bifurcations in CGL with advection for various types of boundarydownstream boundary conditions and a comprehensive analysis is beyond the scope of this paper. We emphasize,however, that, despite a possibly complex bifurcation structure, our predictions do give throughout an overall veryaccurate prediction of front interface locations in the domain. A more general mechanism for speed selection was uncovered in [6]. While the mechanism was shown there to bepresent in scalar equations, it is most easily illustrated in systems representing modes with different spatio-temporalfrequencies. An explicit example is (1.6), which models the interaction of stationary and oscillatory modes. We willillustrate the mechanism in § § § We first motivate the resonant instability mechanism in a simple example, and then simplify models to arrive atsimple models amenable to explicit analysis.
Resonant source terms driving pointwise instabilities.
Consider the linear KPP equation and its dispersionrelation for solutions exp( λt + νx ), u t = u xx + u x + µu, x ∈ R , D ( λ, ν ) = ν + ν + µ − λ. Solutions with initial data u ( x ) are given through the explicit “heat kernel”, which in turn can be constructedthrough inverse Laplace transform of the resolvent, u ( t, x ) = 12 π i Z λ ∈ Γ Z y e λt G λ ( x − y ) u ( y )d y d λ, where Γ is a curve in the complex plane to the right of the spectrum of L = ∂ xx + ∂ x + µ with sectorial asymptotics r e ± i θ , θ ∈ ( π/ , π ), and ( L − λ ) G λ ( x ) = − δ ( x ). The exponential growth rate of u ( t, x ) is bounded by the maximalreal part of Γ and we therefore deform the contour to the left as far as possible. With a view on restricting to acompact interval, later, we think of compactly supported initial conditions and observations x in a bounded region,8o that we may deform contours until we reach singularities of the pointwise Green’s function G λ ( ξ ), which coincidewith the pinched double roots λ dr = µ − /
4; see [12] for more details and references.The deformation of the contour requires us to be able to continue the integral into regions where G λ L , exploitingthat initial data is strongly localized. On the other hand, assuming initial data with a prescribed exponential rate u ( x ) = e νx , we find solutions e λ ( ν ) t + νx that travel at the linear envelope speed − Re λ ( ν ) / Re ν . Alternatively, sourceterms f ( t, x ) in the equation, u t = u xx + u x + µu + f, x ∈ R , (3.1)can induce instabilities even if they exhibit pointwise spatial decay. For instance, f ( t, x ) = e νx with 1 ≫ − ν > µ > λ ( ν ) = ν + ν + µ ∼ µ > νx +i t would not lead to instabilities [7].Note that the slow spatial decay is seen as slow pointwise temporal decay after slightly changing the speed of thecomoving frame, such that even temporally and spatially decaying source terms can induce instabilities.This mechanism is most obvious if the source term stems from an explicit coupling to an oscillatory amplitude,modeled for instance by the complex Ginzburg-Landau equation A t = d (1 + i α ) A xx + A x + ( ρ + i β ) A − (1 + i γ ) A | A | , where we think of ρ < d (1+ α ) , below the threshold for absolute instability, such that solutions decay pointwise.Similarly, we consider (1.1) with µ < /
4, below the threshold of absolute instability, but assume coupling of the twoequations. We are mostly interested in coupling terms that act as sources in the equation, such as terms g A ( u ) in the A -equation and g u ( A ) in the u -equation. With the discussion above, non-oscillatory terms are the most dangerouscandidates for coupling, which motivates the choice of the source term | A | in the KPP equation in (1.6). Weemphasize here that we are thinking of pointwise stability in both u - and A -equations separately, which implies, dueto the fact that the coupling is nonlinear, linear pointwise stability of the origin in the system. This linear pointwisestability is in fact robust under addition of small linear coupling in both u - and A -equations, using continuity ofpinched double roots as established in [12]. In this sense, the resonant instability is completely determined by thelinear part but enabled by the presence of a nonlinear coupling term mediating the resonance. From CGL-KPP to KPP-KPP.
We can obtain a simplified model assuming α = β = γ = 0, which allows usto also assume A ∈ R and find the coupled KPP system (1.8), albeit with quadratic coupling, κ = 2. In the linearpart, since we averaged out the oscillations, α = β = 0, there now also is a strong linear, 1:1-resonance leading toinstabilities precisely when there is a linear coupling term, κ = 1. This 1:1-resonance is a double root of the dispersionrelation, solution to D ( λ, ν ) = 0 , ∂ ν D ( λ, ν ) = 0. In fact, the dispersion relation factors due to the block-triangularstructure, D ( λ, ν ) = D u ( λ, ν ) · D v ( λ, ν ) = ( ν + ν + µ − λ )( dν + ν + ρ − λ ) , and double roots can be formed through D u ( λ, ν ) = D v ( λ, ν ) = 0, with local expansion D ( λ, ν ) = ( D uλ · ( λ − λ dr ) + D uν · ( ν − ν dr )) · ( D vλ · ( λ − λ dr ) + D vν · ( ν − ν dr )) = O (cid:0) | λ − λ dr | + | ν − ν dr | (cid:1) . In contrast to the case of a branched double root, we typically find analytic solutions ν ( λ ) locally near the doubleroot using either the Newton polygon, or, more directly, the fact that the two solutions ν ± ( λ ) can be continuedseparately as solutions of D u = 0 and D v = 0 by the implicit function theorem. We therefore refer to such pincheddouble roots as unbranched resonances . The expansion above shows however that adding coupling terms to both u -and v -equations will introduce constant terms into the dispersion relation near ( λ dr , ν dr ). Solving for double rootswill then yield a unique solution ν from ∂ ν D = 0, which in turn yields locally two solutions λ from D ( λ, ν ) = 0.These solutions would then be branched resonances and produce phenomena discussed in the previous section. Sincesuch pinched double roots cannot occur through linear coupling of KPP and Ginzburg-Landau, we preserve theanalogy and also avoid these pinched double roots in this simplified setup of coupled stationary modes by restrictingto unidirectional linear coupling, that is, excluding linear u -dependence in the v -equation. We refer to [12] for a morethorough discussion of such unbranched resonances and the role of the pinching condition.9 rom KPP-KPP to CPW. Simplifying even further, we inspect the dispersion relation near the branch pointin a canonical form, eliminating ( λ − λ dr )( ν − ν dr ) terms and higher-order terms, to motivate a dispersion relation D ( λ, ν ) = ( λ − λ dr ) − ( ν − ν dr ) = (( λ − λ dr ) − ( ν − ν dr )) · (( λ − λ dr ) + ( ν − ν dr )) , which, with λ dr = 0, is realized (for instance) in the equation u t = + u x − ν dr u + βv,v t = − v x + ν dr v. (3.2)Note that we alternatively could have added a coupling term βu in the v -equation, but not both terms simultaneously.Integrating (3.2) on the real line is simple. The u component is advected to the left while decaying exponentially. Atthe same time v advects to the right while growing exponentially. Compactly supported initial conditions thereforedecay to zero in finite time in the absence of the coupling term, β = 0. With the coupling term, however, the u -equation integrates an exponentially growing source term stemming from the v -equation. To understand theinstability, we place an initial condition in v at x = 0, which is then advected to x = T /
T /
2, withan exponential growth e T/ . There, it acts as a source term in the u -equation, equivalent to an initial conditionafter some finite time, and transported back in the u -equation to x = 0 with an exponential dampening e − T/ ,yielding in summary the predicted neutral decay λ dr = 0. Note that for increasingly long times, this neutral stabilityrequires a sufficiently large domain ahead of the location of observation to enable the reentry of information from u initial conditions. This suggests that arbitrarily distant boundary conditions could eventually impede the instabilitymechanism.In our system (1.10), we set ν dr = − µ through the transport speed in the u -equation.Slower transport in the u -equation for µ > u -equation represents generic nonlinear saturation. x ∈ R . Detailed criteria for spreading speeds and onset of pointwise instability mediated by resonances were developed in[6]. We state the criteria in the relevant scenarios considered here and determine onset of pointwise instability.
Onset of pointwise instability — counter-propagating waves and linear coupling.
Double roots of thedispersion relation solve D u ( λ, ν ) = (1 − µ ) ν + 1 − λ = 0 , D v ( λ, ν ) = − ν − − λ = 0 , which gives λ dr = µ − µ , ν dr = − − µ . The double root is pinched when µ < µ > µ res = 0.
Onset of pointwise instability — counter-propagating waves and nonlinear coupling.
With a sourceterm v instead of v in the u -equation, the mode in the u -equation generated by the coupling is twice the v -mode,such that we need to solve D u ( λ , ν ) = (1 − µ ) ν + 1 − λ = 0 , D v ( λ , ν ) = − ν − − λ = 0 , λ = 2 λ , ν = 2 ν , which gives λ , res = 2 µ − − µ , ν , res = − − µ , with instability for µ > µ res = 1 /
2. The resonance is pinched when µ < nset of pointwise instability — predictions for KPP-KPP.
We consider (1.8) with d > µ > ρ , and4 dρ <
1, where the latter condition guarantees pointwise stability of the v -equation. The equation D u = 0 , D v = 0gives in this case two solutions, only one of which is relevant in the sense of [12], λ dr = ( dµ − ρ ) − p ( d − µ − ρ ) d − , with instability for µ > µ res = 12 d (cid:16) d − dρ + ( d − p (1 − dρ ) (cid:17) . Below, we will choose d = 5 which leads to resonant instabilities for ρ ∈ (0 , ) at µ res = (cid:0) √ − ρ + 10 ρ (cid:1) .Specifying further, the resonance causes an instability for µ ∈ (0 . , .
25) at ρ = 0.In case of quadratic coupling, one needs to solve D u ( λ , ν ) = 0 , D v ( λ , ν ) = 0 , λ = 2 λ , ν = 2 ν . The resonance condition changes to, assuming d > µ > ρ , λ , res = 1 d − (cid:16) dµ + p d − µ − ρ ) − ρ (cid:17) , ν , res = s µ − ρ d − . At d = 5, ρ = 0, the onset of the 2:1-resonant instability occurs for µ > µ res = = 0 .
24, just below the value µ = 0 .
25 where the pinched double root in the u -equation becomes unstable. Onset of pointwise instability — predictions for KPP-CGL.
In this case, since double roots are complex,the full resonance condition from [6] including complex group velocities is needed. This resonance condition is D u ( λ , ν ) = 0 , D A ( λ , ν ) = 0 , D ¯ A ( λ , ν ) = 0 , λ = λ + λ , ν = ν + ν , c g , = c g , , where the condition on group velocities c g ,j = − d λ j d ν j ensures interaction and optimality [6]. We find for the resonantgrowth rates, assuming µ > ρ and (1 + α ) d > λ , res = (1 + α ) dµ − ρ − p µ − ρ )((1 + α ) d − α ) d − , ν , res = − s µ − ρ )(1 + α ) d − . Instabilities λ , res > µ > µ res = − α ) d (1 + 4 ρ ) + ((1 + α ) d − p − α ) dρ α d , where we also assumed 4(1 + α ) dρ < Ξ( µ ) The onset of instability discussed above involves interaction of evanescent modes at large distances ahead of thefront, as we discussed in § x = − L and x = 0, the trivial solution is stable in the regime where resonant fronts mediate theinstability as one can readily infer from the fact that the v -component converges to zero and, due to the couplingstructure, the u -component is then stable as well. We now present an analysis of boundary layer movement in theCPW case, predicting in particular Ξ( µ ) for large L , and adapt the results to the more complicated models.11 igure 3.1: Phase portraits for (3.3) with origin blown up to finite-size circle in polar coordinates at µ < µ = 0, and µ > L large at µ <
0, with long passage time near equilibria u = 0 and u = 1, respectively; see text for analysis offlight times in terms of eigenvalues near equilibria and first-hit maps between sections to the flow near equilibria. Solutions are composedof a heteroclinic boundary layer (green) and a trivial heteroclinic v ≡ Predicting Ξ( µ ) for counter-propagating waves with linear coupling. We look for steady-states of (1.10)solving u x = 11 − µ (cid:0) − u + u − v (cid:1) , v x = − v. (3.3)A phase plane analysis shows 3 equilibria, u ∈ {− , , } and v = 0, with eigenvalues {− − µ , − } at (0 ,
0) and { − µ , − } at ( ± , µ = 0, we use polar coordinatesas in the case of the branched resonance. The phase portrait is shown in Fig. 3.1. Total flight time near boundaryconditions is L = T + T + O (1), where T and T are the flight times between sections to the flow near the equilibria u = 0 and u = 1, respectively. The heteroclinic shown in red is found from the heteroclinic solution to u x = − u + u , u ( x ) →
1, for x → −∞ , and u ( x ) → x → ∞ . The green boundary layer is found from integrating thestable manifold of u = 1 backward in time using that v -dynamics are decoupled. A schematic representation of theheteroclinic chains is found in Fig. 3.2.Flight times can be computed from a Shilnikov passage time analysis. From the analysis near the equilibrium u = 1,we find z in0 ∼ z out1 ∼ (cid:0) z in1 (cid:1) ν s1 /ν u1 , T ∼ − ν u1 log z in1 , (3.4)where we neglected constants and higher-order terms. The eigenvalue ν u0 = µ gives dynamics in projective space andis given by the difference between the two stable eigenvalues at u = 0. Assuming ν u0 = O (1), we can use the samesimple linear analysis to compute z out1 ∼ (cid:0) z in1 (cid:1) | ν s0 | /ν u0 , T ∼ − ν u0 log z in0 ∼ − | ν s1 | ν u0 ν u1 log z in1 ∼ | ν s1 | ν u0 T (3.5)Together with L ∼ T + T , this gives Ξ( µ ) ∼ ν u0 ν u0 + | ν s1 | . (3.6)Substituting values ν u0 = − − − − µ = µ − µ , ν s1 = − µ ) ∼ µ. (3.7)For µ ∼
0, linearization at the origin does not give accurate predictions due to quadratic terms in the transcriticalbifurcation. Including the quadratic term with a coefficient β gives a time of flight T = 1 ν u0 log (cid:18) ν u0 + βz in0 ( ν u0 + β ) z in0 (cid:19) = 1 ν u0 log (cid:18) ν u0 βz in0 (cid:19) + O (1) , T = − ν u1 z in1 . (3.8)Eliminating z in0 using (3.5) and T = L − T gives L − T = 1 ν u0 log (cid:16) cν u0 e | ν s1 | T (cid:17) , igure 3.2: Illustration of the heteroclinic chains responsible for the bifurcation diagram below (left) at (center) and above (right)criticality. Below criticality, the stable manifold of the bifurcated equilibrium provides upper bounds on the passage time near p nt andthereby Ξ( µ ) = 0 for L → ∞ . At criticality, the expansion near p t is weak, algebraic, leading to long passage times near p t and thereforesmall upper bounds on the passage times near p nt . For ¯ µ >
0, passage times are balanced by eigenvalue ratios as explained in the text. for some constant c >
0. This yields an implicit equation for Ξ,Ξ − − ν u0 L log (cid:16) cν u0 e | ν s1 | L Ξ (cid:17) = 0 . (3.9)One can derive more explicit expressions in asymptotic regimes T = ( ν u0 ν u0 + | ν s1 | L, ν u0 e | ν s1 | L Ξ ≫ , | ν s1 | log L + ν u0 | ν s1 | ) L, ν u0 e | ν s1 | L Ξ ≪ , (3.10)Of course, the regime ν u0 ρ ≫ § q f , the invasion front on the unbounded domain, between nontrivial equilibrium and trivialequilibrium, contained in the strong stable manifold of the saddle-node equilibrium at criticality.Of course, since the bifurcation here is nonlocal in parameter space, extending over an interval of µ -values untilreaching a branched resonance, one needs to require existence of heteroclinics and equilibria over the entire rangeof parameter values where predictions are to be derived. Assumption (iv)’ is typical either due to the presence ofinvariant subspaces such as A = A x = 0 or v = v x = 0 in CGL-KPP and KPP-KPP, respectively. In systems withoutgauge invariance, one finds such invariant subspaces in spatial-dynamics formulations for time-periodic functions asthe subspace of trivial, constant time-dependence; see for instance [16] for an example illustrating bifurcation fromheteroclinic orbits within such subspaces. Predicting Ξ( µ ) for counter-propagating waves with quadratic coupling. The steady-state equation withquadratic coupling reads u x = 11 − µ (cid:0) − u + u − v (cid:1) , v x = − v. (3.11)One can now directly set w = v , which gives w x = − w and continue the previous analysis with ν s1 = − µ ) ∼ µ − . (3.12)In a more general approach, one would consider a desingularization near the origin using the scaling from the 2:1-resonance, setting ( u, v ) = R (cos( ϕ ) , sin( ϕ )) in { v > } , with projective coordinates v /u , in which the transcriticalbifurcation occurs at µ = 1 /
2. 13 redicting Ξ( µ ) for the coupled KPP system. The analysis in this case completely parallels the analysis in theprevious case, although the 4-dimensional phase portraits are less accessible. One still finds two relevant equilibria, u = √ µ and u = 0, v = 0, and heteroclinic orbits connecting the boundary conditions at x = − L to u = √ µ , then u = √ µ to u = 0 within v = 0, and a heteroclinic in the singular sphere of the blown up origin. One finds ν s1 = κ − − √ − dρ d , ν u0 = 1 − √ − µ ν s1 , where κ denotes the power of the coupling terms and determines the order of the resonances. With these adaptations,one then recovers all formulas from the previous case. Predicting Ξ( µ ) for the coupled CGL-KPP system. With the particular skew-product structure, this situationis very similar to the coupled KPP system. For now, assume that | A | is stationary, that is, A is stationary up tothe gauge symmetry, and let ν be the exponential rate of decay of | A | . We then recover all formulas from the caseof counter-propagating waves, with ν u0 = −√ − µ + ν s1 . It turns out that the value of ν s1 depends on the boundarycondition through the frequency β in a nontrivial fashion. Varying the boundary conditions, we found stationaryprofiles for | A | with zero and nonzero frequencies β , but also, in direct simulations, time-periodic solutions. Wesaw exponential decay rates varying significantly changing only boundary conditions. We measured the exponentialdecay rate numerically and also compared with rates obtained from the dispersion relation when the boundary layeris stationary up to the gauge symmetry and used the result to obtain ν s1 and predictions for the bifurcation diagram. -0.2 0 0.2 0.4 0.6 0.800.10.20.30.40.50.60.70.8 -0.1 0 0.100.050.1 Figure 3.3: Comparison of Ξ( µ ) from numerical continuation to theory. Counter-propagating wave system ((1.10), left) for L = 50 , , ρ = 20 . − , κ = 2, u = 0 . v = 0 .
1) for L = 200 , , c = 1); approximation at µ = µ res from (3.9) marked with “ ∗ ”. Bifurcation curves with L = 1000, d = 10, varying ρ , same parameters as before (right). We computed bifurcation diagrams numerically via secant continuation and in direct simulations. Comparisons withtheoretical predictions are shown in Fig. 3.3. Exponential decay is stronger in (1.10) compared to the numericalexperiments for KPP and CGL, such that domain sizes are comparatively smaller in the plots. Bifurcation diagramsdepend little on the boundary conditions but change depending on parameters such as ρ . Note that one eventuallyalways sees the branched resonance set in and dominate the eventual climb to Ξ = 1 in Fig. 3.3 for the KPP-KPP andCGL-KPP system (but of course not for the CPW system, which does not have a branched resonance but becomesill-posed when µ = Ξ( µ ) = 1), consistent with the fact that Ξ < R = 0 . , k = 0 . α = 0 . , γ = 0 . − . As a consequence, fits are less good for finite L . Roughly adjusting the constant c in (3.9)14 .18 0.2 0.22 0.24 0.26 0.28 0.300.20.40.60.81 Figure 3.4: Comparison of Ξ( µ ) from numerical continuation to theory in the CGL-KPP system (1.6), parameters α = 0 . , β = 0 , γ =0 . , d = 5 , ρ = 0 . , R = 0 . , k = 0 . L = 500, same parameter values, varying β (center), andintroducing back coupling (3.13) for β = 1, varying ε , L = 100, with predictions for ε = 0. to c = 10 for L = 200 ,
500 and c = 10 for L = 1000 does yield quite good fits, nevertheless; see left panel in Fig.3.4. We used the fact that the boundary layer was stationary in time to obtain the exponential decay rate from thedispersion relation with λ = 0, with excellent agreement to measured exponential decay.We also compared predictions with direct simulations and Dirichlet boundary conditions A = 0 . β = 0 , . , β , one can calculate an exponential decay rate of a stationary boundary layerfrom the stationary ODE, which yields ν s1 and thereby predictions for the position Ξ( µ ). The exponential decay ratematches numerical observations and predictions give good fits for moderate values of β . For large β , the boundarylayer oscillates, and generates an effective decay similar to the one for a stationary boundary layer with frequencyclose to zero. Predictions for a boundary layer with frequency β would predict absence of a resonant instability butoscillations in the boundary layer mediate a weaker decay that drives a resonant instability, albeit not quite as strongas in the case β = 0. In fact, β = − αρ = − .
12 would yield weakest decay, only slightly less than in the case β = 0.Position measurements for β = 1 are instantaneous in time and therefore vary slightly depending on the phase of theoscillation.Lastly, we demonstrate the robustness of the phenomenon described here by introducing back coupling, that is,considering u t = u xx + u x + µu − u + | A | + ε Re A, < x < L,A t = d (1 + i α ) A xx + A x + ( ρ + i β ) A − (1 + i γ ) A | A | + εu, < x < L, (3.13)with Dirichlet boundary conditions A = 0 . u = 0 . ε = 0 . , . , . , . , . , .
0, increasing the strengthof interaction, with β = 1 , ρ = 0 .
04, and forcing oscillations in the boundary layer; see Figure 3.4, right panel. Wesee that “bifurcation” curves are roughly continuous in ε , with Ξ increasing in ε . Dynamics are chaotic, and wemeasured instantaneous values of the position, but waiting for decrease in Ξ to stabilize, leading to somewhat erraticplots with resulting upward jumps. The possibility of intermittent collapse of the interface, that is, Ξ( µ ) droppingto near-zero values for some ranges of parameter values, does appear to be robust however with respect to changesin the numerical algorithms and homotopy strategy. Since exponentially weak interactions are responsible for allphenomena here, it is however difficult to quantify the effect of round-off errors without strong theoretical predictionsas available in the case of stationary solutions. For stronger coupling, ε > .
5, a branched resonance determines theonset of instability.
We investigated instabilities in large domains in the presence of unidirectional transport. Instabilities are triggeredwhen the speed of propagation of large, nonlinear fronts reverses sign, and a front propagates into the domain fromthe downstream boundary. We focused on cases where this front propagation is determined by the dispersion relation15f the trivial state, that is, on supercritical instabilities leading to pulled fronts. The speed of such pulled fronts ismost commonly identified with a branch point of the dispersion relation, and bifurcation diagrams in large domainsuniversally exhibit a very steep onset with square root-like asymptotics for the position of the interface. Our maincontribution identifies an alternate, robust scenario, where the front speed in the unbounded domain is determinedby a nontrivial, unbranched resonance. Bifurcations in large domains are in this case gradual in that the positionof the front interface as a fraction of the domain size increases gradually. The cause of the arrest of the front faraway from boundaries is a cut-off in tails caused by the upstream boundary, which suppresses the key resonancemechanism.Our bifurcation analysis is based on a conceptual representation of boundary layers and front interfaces as chainsof heteroclinic orbits. A key step in the analysis is a geometric desingularization at the origin which reduces eigen-value resonances to saddle-node and transcritical bifurcations of eigenspaces in the branched and unbranched case,respectively, and introduces an additional singular heteroclinic orbit connecting eigenspaces and boundary conditionsinto the geometric picture of heteroclinic chains. Since such tail interaction and heteroclinic chain bifurcations arenotoriously difficult to analyze in general, relying on multiple non-degeneracy and non-resonance assumptions toquantify exponential expansions, it seems cumbersome to state general results rigorously. We do believe howeverthat the situation analyzed here is in many ways generic and hope that a more rigorous dynamical systems analysisof the heteroclinic bifurcation could clarify such genericity assumptions.In direct simulations, small initial conditions lead to fronts propagating into the domain until changing speed andsettling the position of the interface at the predicted location. We intend to study transient front dynamics in largeand semi-bounded domains in future work.In a different direction, weak tail interaction is sensitive to noise, additive or mutiplicative. It was demonstratedin [7] that multiplicative noise changes the resonance condition and absolute spectra [15] then determine changed,faster spreading speeds. It would be interesting to analyze such effects in the case of large bounded domains, as well.
References [1] A. Bers. Space-time evolution of plasma instabilities-absolute and convective. In A. A. Galeev & R. N. Sudan,editor,
Basic Plasma Physics: Selected Chapters, Handbook of Plasma Physics, Volume 1 , pages 451–517, 1984.[2] P. B¨uchel, M. L¨ucke, D. Roth, and R. Schmitz. Pattern selection in the absolutely unstable regime as a nonlineareigenvalue problem: Taylor vortices in axial flow.
Phys. Rev. E , 53:4764–4777, May 1996.[3] J.-M. Chomaz. Global instabilities in spatially developing flows: Non-normality and nonlinearity.
Annual Reviewof Fluid Mechanics , 37(1):357–392, 2005.[4] A. Couairon and J. M. Chomaz. Global instability in fully nonlinear systems.
Phys. Rev. Lett. , 77:4015–4018,Nov 1996.[5] A. Couairon and J.-M. Chomaz. Absolute and convective instabilities, front velocities and global modes innonlinear systems.
Physica D: Nonlinear Phenomena , 108(3):236 – 276, 1997.[6] G. Faye, M. Holzer, and A. Scheel. Linear spreading speeds from nonlinear resonant interaction.
Nonlinearity ,30(6):2403–2442, may 2017.[7] G. Faye, M. Holzer, A. Scheel, and L. Siemer. Invasion into remnant instability: a case study of front dynamics.
Preprint , 2020.[8] R. Goh and A. Scheel. Triggered fronts in the complex Ginzburg Landau equation.
J. Nonlinear Sci. , 24(1):117–144, 2014.[9] M. Holzer. Anomalous spreading in a system of coupled Fisher-KPP equations.
Phys. D , 270:1–10, 2014.1610] M. Holzer. A proof of anomalous invasion speeds in a system of coupled Fisher-KPP equations.
Discrete Contin.Dyn. Syst. , 36(4):2069–2084, 2016.[11] M. Holzer and A. Scheel. A slow pushed front in a Lotka–Volterra competition model.
Nonlinearity , 25(7):2151,2012.[12] M. Holzer and A. Scheel. Criteria for pointwise growth and their role in invasion processes.
J. Nonlinear Sci. ,24(4):661–709, 2014.[13] M. H. Kpf and U. Thiele. Emergence of the bifurcation structure of a Langmuir–Blodgett transfer model.
Nonlinearity , 27(11):2711–2734, oct 2014.[14] H. W. M¨uller, M. L¨ucke, and M. Kamps. Transversal convection patterns in horizontal shear flow.
Phys. Rev.A , 45:3714–3726, Mar 1992.[15] B. Sandstede and A. Scheel. Absolute and convective instabilities of waves on unbounded and large boundeddomains.
Phys. D , 145(3-4):233–277, 2000.[16] B. Sandstede and A. Scheel. Essential instabilities of fronts: bifurcation, and bifurcation failure.
Dyn. Syst. ,16(1):1–28, 2001.[17] B. Sandstede and A. Scheel. Defects in oscillatory media: toward a classification.
SIAM J. Appl. Dyn. Syst. ,3(1):1–68, 2004.[18] B. Sandstede and A. Scheel. Basin boundaries and bifurcations near convective instabilities: a case study.
J.Differential Equations , 208(1):176–193, 2005.[19] S. M. Tobias, M. R. E. Proctor, and E. Knobloch. Convective and absolute instabilities of fluid flows in finitegeometry.
Phys. D , 113(1):43–72, 1998.[20] W. van Saarloos. Front propagation into unstable states.