Bifurcations of front motion in passive and active Allen-Cahn-type equations
BBifurcations of front motion in passive and active Allen-Cahn-type equations
Fenna Stegemerten, Svetlana Gurevich,
1, 2, a) and Uwe Thiele
1, 2, b) Institut f¨ur Theoretische Physik, Westf¨alische Wilhelms-Universit¨at M¨unster, Wilhelm-Klemm-Str. 9,48149 M¨unster, Germany Center of Nonlinear Science (CeNoS), Westf¨alische Wilhelms-Universit¨at M¨unster, Corrensstr. 40, 48149 M¨unster,Germany (Dated: 4 February 2020)
The well-known cubic Allen-Cahn (AC) equation is a simple gradient dynamics (or variational) model fora nonconserved order parameter field. After revising main literature results for the occuring different typesof moving fronts, we employ path continuation to determine their bifurcation diagram in dependence of theexternal field strength or chemical potential. We then employ the same methodology to systematically analysefronts for more involved AC-type models. In particular, we consider a cubic-quintic variational AC modeland two different nonvariational generalisations. We determine and compare the bifurcation diagrams of frontsolutions in the four considered models.PACS numbers: Valid PACS appear here
The problem of front propagation has a very longhistory with essential contributions coming fromdifferent fields. One of the simplest variationalmodel possessing front solutions is a so-called cu-bic Allen-Cahn (AC) equation for a nonconservedorder parameter field. In this paper, we system-atically analyse possible phase transitions in theAC equation employing analytical results givenin the literature and compare them to resultsobtained with numerical path continuation tech-niques. Furthermore, we apply the same method-ology to fronts occurring in more involved AC-type models, including a cubic-quintic variationalAC model and two different nonvariational gener-alisations, where the AC equation is emended bya nonequilibrium chemical potential or is coupledto a polarisation field.
I. INTRODUCTION
Many spatially extended systems can exist in differentspatially homogeneous states that depend on various am-bient parameters. In the absence of out-of-equilibriumdriving forces, energy arguments hold and the homoge-neous states may be (globally) stable, metastable (non-linearly unstable) or linearly unstable. With other wordsthey represent global minima, local minima and maxima(or saddles), respectively, of an underlying energy func-tional. In general, states of higher energy can be replacedthrough a moving front by states of lower energy. A sim-ple well studied deterministic continuum model for suchprocesses is the Allen-Cahn (AC) equation that readsin nondimensional form in one spatial dimension [their a) ORCID: 0000-0002-5101-4686 b) Eq. (11) with Eqs. (3) and (4)], ∂φ∂t = − δ F δφ with F = (cid:90) (cid:20) | ∂ x φ | + f ( φ ) − µφ (cid:21) dx. (1)It corresponds to a nonconserved gradient dynamics onan underlying energy functional F [ φ ] that contains asquare gradient term and a local energy f ( φ ). Theequation arises in many different contexts and is, some-times for specific, often quartic choices of f ( φ ), knownas Fisher-Kolmogorov , Fife-Greenlee , Schl¨ogl or Zel-dovich–Frank–Kamenetsky equation . It is studied asa model that describes dynamics in multistable systemsclose to phase transitions of a nonconserved order pa-rameter field φ ( x, t ). By “nonconserved” we refer to adynamics like Eq. (1) that does not conserve the total’mass’ (cid:82) φ ( x, t ) dx in the system. In its most commonform, the AC equation features a double-well potential,i.e., a quartic f ( φ ). It may be symmetric with minimaof equal energy at φ − and φ + = − φ − or be tilted by achemical potential or external field µ . Note that some-times Eq. (1) with this specific f ( φ ) is referred to asAllen-Cahn equation .For example, the behavior of the magnetisation densityin a ferromagnet can be described by a coarse-grainedfree energy density that is identical to such an energyfunctional. Moreover, the time evolution of this densitycan indeed be modeled by the AC equation . Althoughthe global minimum for a symmetric double-well poten-tial corresponds to homogeneous steady states of either φ − or φ + , steady states consisting of arrangements ofpatches of φ − and φ + do also exist. Here, we call theinterfaces between the patches “fronts”. In the literaturethe notions “kink” and “anti-kink” are also used. In afully symmetric situation the fronts are at rest. Other-wise they move, e.g., driven by µ (cid:54) = 0. For small driving,one can analytically determine the velocity of these frontsthat connect two linearly stable states corresponding tolocal minima of f ( φ ), see e.g. Refs. 8 and 9.Via another type of moving front, a linearly stable state a r X i v : . [ n li n . PS ] J a n invades a linearly unstable state . Such fronts occurwhen systems are suddenly quenched into an initiallyhomogeneous unstable state. Small local perturbationsthen grow and develop into patches of stable states thatspread out over the whole domain. In general, the stablestate may correspond to a pattern. This case can, forexample, be found in Taylor-Couette flow . Fronts alsoappear in Rayleigh-B´enard systems when the heat flux issuddenly increased. Then a convective vortex front prop-agates into the unstable conductive state . Moreover,front propagation into unstable states is studied in crys-tal growth and in the context of chemical reactions .Another distinction is between pulled and pushed fronts,where the velocity of the former is controlled by linear ef-fects while in the latter case nonlinear effects dominate .Here, we investigate the dependence of the motion offronts on the strength of the external field or chemicalpotential µ (cid:54) = 0. On the one hand, we consider thewell-known case of a simple AC system with a doublewell potential , and use it to introduce our methodol-ogy that is based on numerical continuation . On theother hand, we consider AC models with more compli-cated local energy that allows for a larger numberof front types and an active AC model that can not bewritten as a gradient dynamics, i.e., in the variationalform (1). We employ two ways to render the AC modelnonvariational. First, we add a nonvariational term anal-ogously to Ref. 18, i.e., we incorporate a nonequilibriumchemical potential as also frequently done for mass-conserving Cahn-Hilliard-type dynamics . Such anamended AC equation may be employed to model, e.g.,front propagation in a liquid crystal light valve . Sec-ond, we couple the AC equation for the order parameterto an evolution equation of a polarisation field P in asimilar spirit as in an active phase-field-crystal (PFC)model . For other models including an active Swift-Hohenberg equation see Ref. 26, a tentative systematicsin the case of single scalar fields is given in the introduc-tion of Ref. 15. Taking the second option for an active ACmodel in one spatial dimension results in a system whichis similar to the FitzHugh-Nagumo model describ-ing spike generation in squid axons. There, the evolutionequation of the membrane potential corresponds to anAC equation with cubic nonlinearity. As the AC modelwith double-well local energy is arguably the simplestnonlinear gradient dynamics model for a parity symmet-ric spatially extended system with φ → − φ symmetry, itsactive generalisations are likely the simplest such activemodels. Their detailed understanding will help to ex-tend our knowledge of the collective behaviour of activesystems consisting of a large number of active particleswhich are able to transform different types of energy intomotion. Here, we focus on front motion, however, therange of phenomena in such systems is much richer. Ingeneral, the microscale constituents interact in such away that on a macroscopic (collective) level directed col-lective motion and clustering phenomena may occur .Different forms of interaction result in different phenom- ena, e.g., a purely repulsive interaction may give rise tomotility-induced phase separation , whereas a combi-nation of repulsive and attractive interaction allows forthe formation of swarms, e.g., of fish or birds , bacteriacolonies or cell motion . The occurring collectivestructures may consist of disordered or well ordered ar-rangements that are referred to as active clusters andactive crystals , respectively. Our work is structured asfollows: First, we focus in section II on the passive sys-tems, i.e., AC systems that evolve towards equilibrium.In particular, section II A introduces the cubic AC equa-tion, briefly reviews linear and nonlinear marginal sta-bility analyses, and determines different front types andtheir velocities as a function of driving strength µ . In sec-tion II C, we consider the quintic AC equation that hasalready attracted much interest, e.g., in Refs. 16, 17, 35–37. In contrast to the literature, we focus on the transi-tions occuring in the behaviour of fronts when the driv-ing strength µ is changed. Second, section III considersthe two nonvariational amendments of the AC equation.Thereby, section III A considers the first option, namely,the case of a nonequilibrium chemical potential. We dis-cuss the analytical expression for the shift in front veloc-ity close to the transition from the variational to the non-variational case and compare this to our fully nonlinearnumerical results. Section III B considers the second op-tion namely the coupling of a cubic AC equation with thedynamics of a polarisation field. It is shown that a condi-tion for onset of motion can be determined analytically,similar to the case of the active PFC model in Ref. 24. Bi-furcation diagrams are presented for both cases of activeAC equations that summarise the behaviour of all occur-ring front solutions when the driving strength is varied.Finally, in section IV we conclude and give an outlook tofuture work. II. PASSIVE SYSTEMSA. General form of Allen-Cahn-type models
We start our analysis with the well-known standard pas-sive version of the AC equation that describes the timeevolution of an one-dimensional nonconserved order pa-rameter field φ = φ (˜ x, t ). The general form is obtainedby introducing the energy F in Eq. (1) into the generalgradient dynamics. It reads ∂φ∂t = ∂ φ∂ ˜ x − f (cid:48) ( φ ) + µ , (2)where µ is the chemical potential, f (cid:48) ( φ ) is the first deriva-tive of the purely nonlinear local energy density f withrespect to φ , and ˜ x is the coordinate in the laboratoryframe. In order to find the velocity v of steadily travel-ling fronts propagating between two homogeneous steadystates φ s given by − f (cid:48) ( φ s ) + µ = 0, we transform the sys-tem (2) into a co-moving frame x = ˜ x − vt and get − v d φ d x = d φ d x − f (cid:48) ( φ ) + µ . (3)Multiplying Eq. (3) by d φ d x and integrating (employingNeumann boundary conditions) yields the explicit ex-pression for the velocity v = f (cid:0) φ s1 (cid:1) − f (cid:0) φ s2 (cid:1) + µ ( φ s − φ s ) (cid:82) ∞−∞ (cid:16) d φ d x (cid:17) d x , (4)for a front propagating from φ s2 at x → −∞ into φ s1 at x → ∞ .In the following, we consider two specific passive sys-tems in sections II B and II C below, namely, AC equa-tions with cubic and cubic-quintic nonlinearities, respec-tively. By employing µ as main control parameter wefocus on the influence of a physically most relevant quan-tity, that can be easily controlled externally. Note,that this differs from most employed and well-knownparametrisations . B. The passive cubic Allen-Cahn equation1. Model
In the case of the cubic Allen-Cahn equation, f ( φ ) = − φ + φ and hence, in the co-moving frame we obtain: − v d φ d x = d φ d x + φ − φ + µ . (5)As mentioned above, this equation is extensively stud-ied in the literature. Here, we review some main ap-proaches to introduce the methodology for our analy-sis of the more involved models, c.f. e.g., the textbooksRefs. 8 and 9. For the cubic nonlinearity, three homo-geneous steady state solutions φ ( µ ), φ + ( µ ) and φ − ( µ )exist for − µ c < µ < µ c with µ c = √ ≈ . φ + and φ − are linearly stable states while φ is unsta-ble. Equation (5) can be seen as a mechanical system,where φ corresponds to the position and x is time. Inthis framework, Eq. (5) describes a particle moving in adouble well potential V ( φ ) = − f ( φ ) + µφ with friction v .As for µ = 0 the maxima of V ( φ ) are of equal height, wefind that for any friction v > φ = 0, which cor-responds to the unstable homogeneous state. However,in reality one observes, that such fronts moving into theunstable state φ , have the certain specific velocity ,namely, v = 2. Hence, a dynamical selection of the frontspeed occurs (for details see, e.g., Ref. 42), that we brieflydiscuss next.
2. Linear marginal stability analysis
The selection problem is tackled employing a linearmarginal stability analysis: We consider the linearisationof the fully time-dependent AC equation about the un-stable homogeneous state φ , i.e., we consider the leadingedge of the front (cf. Ref. 42). Note that at the marginalstability point, i.e., the point where the resulting eigen-values are zero, the group velocity of perturbations at theleading edge of the front, v g = d ω r ( k )d k r (6)equals the velocity of the front v f = ω r ( k ) k r . (7)Here, k and ω are the wavenumber and frequency, re-spectively, and the superscript r denotes the real part.In this way one obtains a linear marginal velocity v l := v f = v g = ± (cid:112) − f (cid:48)(cid:48) ( φ ) . (8)Thus, at the linear marginal stability point perturbationscan not grow above the moving front profile and any frontwith v < v l is unstable. Hence, Eq. (8) provides a crite-rion to determine the dynamically selected front velocity.In Ref. 42 it is shown that the linear marginal velocityis an attractive fixed point, such that any front movingwith v > v l eventually converges to one with v = v l . Thisis referred to as a pulled front.
3. Numerical path continuation
Next, we determine fronts numerically and compare themto the analytical results. In particular, we employpath continuation techniques bundled in the
Mat-lab toolbox pde2path or in the continuation pack-age auto07p to determine front profiles and veloci-ties in dependence of the parameter µ that acts as thestrength of a driving caused by an external field or chem-ical potential .The resulting bifurcation diagram is shown in terms ofthe front velocity in Fig. 1 (a). Interestingly, threebranches exist corresponding to three different fronttypes that are illustrated by the examples in panels (b)to (d). The overall symmetry of the bifurcation di-agram reflects the symmetries ( x, v ) → ( − x, − v ) and( φ, µ ) → ( − φ, − µ ) of Eq. (5). The three front types are afront between the two linearly stable homogeneous states φ − and φ + indicated by the red dashed lines, and the tworespective fronts corresponding to the two different lin-early stable states invading the unstable state indicatedby green dotted and blue solid lines. The velocity v ofthe front between stable states is determined as explainedin section II A. For the front that moves to the right at φ − /φ φ + (e)(d)(c)(b) φ − φ -2-1012 -0.2 0 0.2 µ v φ − φ φ φ + φ + φ − µ c − µ c (a) µ = µ c φ + FIG. 1. The central panel (a) presents the bifurcation diagram of front states described by the cubic AC equation (5) in termsof the front velocity v as a function of the driving strength µ . The red dashed lines correspond to fronts between the twolinearly stable homogeneous states φ − and φ + with an example given in panel (b). The two respective fronts correspondingto the two different linearly stable states invading the unstable states are indicated by green dotted and blue solid lines withexamples in panels (c) and (d), respectively. Panel (e) gives the equivalent mechanical potential V ( φ ) at the critical value µ = µ c where φ − and φ annihilate. positive µ (red dashed line in the upper right quadrantof Fig. 1 (a)), v first linearly than faster increases withincreasing µ . The curve becomes vertical when µ reachesthe critical value µ c where the lower maximum of the me-chanical potential ( φ − ) and the minimum ( φ ) annihilate(see Fig. 1 (e)). This implies that the bifurcation curvepasses a saddle-node bifurcation and folds back. Thenit continues towards lower µ as solid blue line represent-ing fronts where the globally stable state (linearly stableand of lowest energy) invades the unstable state φ . Tounderstand their origin, in the next section we introducethe nonlinear marginal analysis. Decreasing µ further,when crossing µ = 0 the fronts where the globally sta-ble state φ + invades the unstable state φ become frontswhere the metastable state φ − (linearly stable, but ofhigher energy than the globally stable state) invades φ (green dotted lines).Note, finally, that at the value of µ where the red dashedand green dotted line cross, fronts from φ to φ − andfrom φ − to φ + have the same velocity.This implies that in the vicinity of this point one mayexpect moving structures consisting of two fronts as de-picted exemplarily in Fig. 2 (a). To the left of this point φ φ xx φ φ + φ − φ + v s v l v nl (a) (b) FIG. 2. Snapshots from a time simulation of Eq. (5) at µ =0 .
377 for a front initially composed of a front between the twostable states φ + and φ − moving at v s and the linear marginalfront between φ − and φ moving at v l < v s [see (a)]. At a latertime the two fronts merge into a single nonlinear marginalfront between φ + and φ moving at v nl > v s [see (b)]. the φ − to φ + front stays behind the φ to φ − front(cf. Fig. 3 (a)) while to the right of this point the formercatches up with the latter. Then they merge as indicatedin Fig. 3 (b) and create the faster φ to φ + front shownin Fig. 2 (b). x t tv l v nl v l v s v s (a) (b) FIG. 3. The motion of the fronts illustrated in Fig. 2 isillustrated in space-time plots of the front position (a) at µ = 0 .
350 where v l > v s , i.e., the fronts do not merge, and (b)at µ = 0 .
375 where v l < v s , i.e., the fronts eventually mergeand move with v nl . .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 .
25 0 .
30 0 . . . . . k r k rnl + k rnl − k rl (a)0 .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 .
25 0 .
30 0 . µ . . . . . v v nl + v nl − v l (b) FIG. 4. Shown are the (a) real part of wave numbersRe( k ) = k r and (b) velocities v of linear (subscript “l”) andnonlinear (subscript “nl”) marginal fronts as a function ondriving strength µ . The nonlinear marginal results are givenfor fronts where state φ + invades state φ (subscript “+”) aswell as for fronts where state φ − invades state φ (subscript“ − ”). The implication of the dependencies for the selectionof front velocities is discussed in the main text.
4. Nonlinear marginal stability analysis
Using the mechanical analogy we can also understand thesecond occurring front moving into the unstable state: Aparticle moving in this potential may start at either of themaxima corresponding to φ + or φ − . In Ref. 35 and 42 itis shown, that solving the fully nonlinear equation abovea threshold µ one additionally finds a so-called invasionfront, where φ + invades φ , that is marginally stable andhas a nonlinear velocity v nl > v l .Therefore, we study (5) to find the front solution corre-sponding to the blue solid line in Fig. 1 (a) following theideas given in Refs. 8, 35, 38, and 42. We denote the twopossible linearly stable states by φ ± . Because the front is monotonic we can define h ( φ ) = d φ d x (9)as a function of φ . As the front connects two homoge-neous states we request h (cid:0) φ (cid:1) = h (cid:0) φ ± (cid:1) = 0 . (10)Deriving (9) with respect to x yieldsd φ d x d h d φ = h ( φ ) d h d φ = d φ d x . (11)Thus, inserting (9) and (11) into (5) we get: − vh ( φ ) = h ( φ ) d h d φ + φ − φ + µ . (12)Finally, with a power series ansatz up to second order for h ( φ ) we obtain v nl ( µ ) = 3 √ (cid:0) φ + φ ± (cid:1) (13)for the nonlinear marginal velocity. Note, that consid-ering an ansatz of higher order, the comparison of co-efficients always results in a second order polynomial.Therefore, the obtained expression is an exact solutionto the cubic AC equation.Figure 4 (a) shows the linear marginal wave number k rl (solid green line, obtained in section II B 2) and the hereobtained nonlinear marginal wave numbers k rnl + and k rnl − (dashed blue and red line, respectively) as a function of µ , whereas Fig. 4 (b) gives the corresponding velocities v l , v nl + , and v nl − . Subscripts “+” and “ − ” refer tofronts where states φ + and φ − invade state φ , respec-tively. The dynamical selection always implies that thesteeper (larger wave number) and slower front is cho-sen. Considering the front connecting φ − and φ , wesee that k rl > k rnl − for all µ , i.e., at all µ where thefront exists, the linear marginal velocity is selected (solidline in Fig. 4 (b), corresponding to the numerically ob-tained green dotted line in Fig. 1). In contrast, for thefront connecting φ + and φ , the condition k rl > k rnl + only holds for µ < . µ one has k rnl + > k rl . This implies that for µ < . µ > . µ < . µ two distinct velocities are found, i.e., above µ = 0 . v ( µ ) curves in Fig. 4 are tangentialto each other where the two corresponding k r ( µ ) curvescross. This implies that a clear transition in the cor-responding numerically obtained curve can be best seenwhen inspecting the second derivative of v with respectto µ .Note, that all described front types are analytically stud-ied in Ref. 8. We have seen that the numerical resultsobtained by path continuation well agree with the ana-lytical results. The given brief review and comparisonprovides the starting point for our analysis of more com-plex models. C. The passive cubic-quintic Allen-Cahn equation
Next, we analyse the cubic-quintic AC equation, i.e.,Eq. (1) with ( f ( φ ) = φ − a φ + b φ ) similar to theone studied in Ref. 17 in the context of the creationof metastable phases in crystallisation processes. It isalso studied in Ref. 49 with the additional influence ofstochastic noise. The equation for steadily moving fronts(3) is then − v d φ d x = d φ d x − φ + aφ − bφ + µ , (14)It has stable homogeneous solutions φ − s , φ s , φ + s and un-stable homogeneous solutions φ − u and φ + u , all depend-ing on µ . As in Ref. 17 we use a = 5 / b = 1 / µ c ≈ . µ c ≈ . µ can not be calculatedanalytically. Nevertheless, we can find them numerically,e.g., employing continuation. In consequence, we are ableto study the linear marginal velocity by inserting the nu-merical results into the specific form of Eq. (8), i.e., into v l + = ± (cid:113) − φ + u + 3 aφ + u − b (15) v l − = ± (cid:113) − φ − u + 3 aφ − u − b , (16)where v l + and v l − are the linear marginal velocities fora front invading the unstable states φ + u and φ − u , respec-tively. They are depicted in Fig. 6 as black dot-dashedand dashed lines, respectively. The numerical results areindicated by the grey solid lines and in particular forthe linear marginal velocities well coincide with the semi-analytical result.In this way we identify the grey dot-dashed branches andthe green solid branches for | µ | > µ c in Fig. 5(a) as lin-ear marginal, i.e., pulled fronts. We note that different parts of the structure of the bifurcation diagram for thesimple cubic AC equation presented above in Fig. 1 (a)can be recognised as substructures within the bifurcationdiagram for the quintic case in Fig. 5 (a). For instance,the red solid branches corresponding to fronts betweenthe two stable states φ + s and φ − s in Fig. 5(a) behave sim-ilarly to the red dashed branches in Fig. 1 (a) over theentire µ -range. The blue dashed and green solid branchesin Fig. 5(a) are similar to blue solid and green dottedbranches in Fig. 5 (a), however, only for | µ | > µ c .Therefore, we expect the blue dashed branch in Fig. 5 (a)to correspond to a nonlinear marginal front, as they canbe identified as invasion fronts at the bifurcation point,where it merges with the red branch. Moreover, in theinterval − µ c < µ < µ c we twice observe a structuresimilar to the one of the cubic AC equation, once fornegative and once for positive velocities. Again, this canbe explained referring to the mechanical analogy: In this µ -range we can consider the potential as being composedof two cubic AC potentials. At the critical values | µ | = µ c the metastable state merges with one of the unstablestates and thus for | µ | > µ c there is only one ’cubic’ ACpotential left.We now return to the blue dashed front in Fig. 5 (a),for a profile see Fig. 5 (e). To understand why the frontdoes not exist for | µ | < µ c we study again the mechani-cal analogy at µ = µ c illustrated in Fig. 5 (h): Considera particle starting at φ + s and just reaching φ − u withoutovershooting. Let the corresponding friction be v . Con-sidering now a particle that moves from φ + s to φ s , the cor-responding friction v needs to satisfy v > v . Moreover,a particle starting in φ − u requires more negative frictionto move up to φ + s than it does to move up to φ s . Hence,we claim v > v , where v is the friction to move from φ s to φ − u . That is, we require v < v < v , where v cor-responds to the velocity v ≈ .
33 of a front on the greydot-dashed branch at µ c . However, v is the velocity atthe bifurcation point, where the orange dotted branch be-comes the green solid branch with v ≈ . < v . Hence, v , the velocity of the blue dashed solution does not sat-isfy the condition at µ = µ c and therefore, the branchcan not exist anymore. Note, that this argument is sim-ilar to one given in Ref. 17 and we can identify the bluedashed front solutions as being similar to those foundthere, however, as our equations slightly differs they arenot identical. The question how exactly this branch endsis not easy to answer. Our calculations show that fol-lowing the branch towards smaller µ , the slight shouldervisible in Fig. 5 (e) develops into a long inclined plateauof increasing length. This indicates that even at µ > µ c the front already “feels” the presence of the two addi-tional homogeneous steady states that exist for µ ≤ µ c .With other words, the spatial dynamics slows down closeto these “ghost solutions” forming the sloped plateau.The slope gets smaller the longer the plateau becomes,i.e., the closer one approaches µ c from above. We arenot able to follow the blue dashed branch further thanshown in Fig. 5 (a) as the plateau becomes too long for -1.5-1-0.500.511.5 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 µ v φ + s φ − s φ s φ − u φ − s φ s φ − s φ − u φ + s φ − u φ − s φ + s φ + s φ s φ − s φ + s (a) (f)(e) µ c µ c − µ c − µ c φ − s /φ − u φ + s (g) µ = µ c φ − s φ − u φ s /φ + u φ + s (h) µ = µ c φ − u φ − u φ + u (d) φ + u (c) φ + u (b) FIG. 5. The central panel (a) presents the bifurcation diagram of front states described by the cubic-quintic AC equation (14)in terms of the front velocity v as a function of the driving strength µ . The red solid lines correspond to fronts between the twolinearly stable states φ + s and φ − s with an example given in panel (f) for µ = 0 .
09. At the saddle-node bifurcation at µ = µ c the states φ − s and φ − u annihilate. The green solid lines corresponds to fronts between φ − s and φ − u (panel (c) for µ = − .
02) orbetween φ + s and φ + u depending on the sign of µ . At the saddle-node bifurcation at µ = µ c the states φ s and φ + u annihilate (at µ = − µ c the states φ s and φ − u ). The orange dotted lines refer to front solutions between φ + s and φ s or between φ − s and φ s (panel (d) for µ = − . µ . The blue dashed lines refer to front solutions between φ + s and φ − u (panel (e) for µ = 0 . φ s and φ + u or between φ s and φ − u (panel (b) for µ = − . f (cid:48) ( φ ) − µ in (14). Panels (g) and (h) gives the equivalent mechanical potential − f ( φ ) + µφ at thecritical values µ c and µ c , respectively. µ v v l + v l − FIG. 6. The linear marginal velocity v l − found in Eq. (14) cor-responds to the black dashed line whereas v l + corresponds tothe black dot-dashed line. The results gained with auto-07p are illustrated as the grey solid lines. the largest of our numerical domain sizes. Our hypothe-sis is that the curve becomes vertical when approaching µ c from above where the structured front between φ + s and φ − u decays into several fronts between φ + s , φ + u , φ s and φ − u that exist for µ ≤ µ c . III. ACTIVE SYSTEMS
So far, we have examined front motion described by vari-ational AC equations, i.e., systems that can be writtenas nonconserved gradient dynamics. They describe sys-tems that ultimately approach thermodynamic equilib-rium. Next, we consider active systems where this is notthe case. In particular we consider two different examplesof nonvariational extensions of the cubic AC equation.
A. Nonvariational cubic Allen-Cahn equation
In the first considered example of a nonvariational ACequation a term g nv is added to (2) that breaks the gra- -2-1012 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 µ v b = 2 b = 1 FIG. 7. Comparison of bifurcation diagrams for the varia-tional cubic AC equation with b = 1 (grey solid) and the non-variational cubic AC equation (18) with b = 2 (black dashed).The shift in velocity v for a front propagating from φ + into φ − at µ = 0 for (cid:15) = 0 . dient dynamics form (2). The resulting equation is ∂φ∂t = ∂ φ∂ ˜ x + φ − φ + µ + (cid:15)g nv (cid:18) φ, ∂φ∂ ˜ x , ∂ φ∂ ˜ x (cid:19) with g nv (cid:18) φ, ∂φ∂ ˜ x , ∂ φ∂ ˜ x (cid:19) = (cid:18) ∂φ∂ ˜ x (cid:19) + 2 bφ ∂ φ∂ ˜ x . (17)Note, that the complete equation becomes variational for b = 1.Again, we are interested in the velocities of fronts andtheir dependence on the external driving strength µ . Fol-lowing Ref. 18, we use a front solutions of the variationalEq. (5) (i.e., Eq. (17) with (cid:15) = 0) as a reference solution φ F , and employ the ansatz φ (˜ x, t ) = φ F (˜ x − vt ) + u (˜ x − vt, (cid:15)t ). Here, v ≈ v − (cid:15)v with v being the velocity ofthe reference front, i.e., φ F (˜ x − v t ) := φ F ( x ).One multiplies Eq. 17 by ∂φ/∂ ˜ x , integrates in x , employ-ing the point symmetry φ F ( x ) = − φ F ( − x ) and expandsin (cid:15) (see the Appendix for details). Then, to linear orderone can write the nonvariational contribution as v = (cid:82) ∞−∞ g nv (cid:16) φ F , ∂φ F ∂x , ∂ φ F ∂x (cid:17) ∂ x φ F d x (cid:82) ∞−∞ ( ∂ x φ F ) d x . (18)We notice that to this order v , in contrast to v does notdepend on the difference in energy densities f , but onlyon the nonvariational part g nv and the shape of the refer-ence front shape φ F ( µ ). Figure 7 gives the correspondingnumerically obtained bifurcation diagram of front solu-tions to Eq. (18) for a nonvariational case with b = 2 andrelatively small (cid:15) = 0 . b = 1 (grey solid line). Dueto g nv , the symmetry µ → − µ is broken because g nv doesnot have the φ → − φ symmetry.Inserting φ F into (18) and solving the integrals for µ = 0where v = 0 yields v = √ , i.e., in the nonvariationalcase the front moves even without external driving. The shift in velocity for the chosen (cid:15) coincides on the scaleof Fig. 7 with the numerical result. At larger strength (cid:15) of the nonvariational term, linear considerations do notsuffice anymore (not shown). Note that the point of zerovelocity for the fronts connecting φ − and φ + is with in-creasing (cid:15) shifted towards positive µ . It may be seen asthe out-of-equilibrium coexistence point of the two statesconnected by the front. B. Cubic Allen-Cahn equation coupled to polarisation field
As second and final example of a nonvariational AC equa-tion we consider a cubic AC equation coupled in a sim-ple way to the linear dynamics of a polarisation field.Here we focus on the one-dimensional case where P de-scribes the local strength of directional order and employthe same coupling between order parameter field φ and P as employed in the active phase-field-crystal (PFC)model . The system of equations then reads ∂φ∂t = − δ F δφ − α ∂P∂x ,∂P∂t = D T ∂ ∂x δ F δP − D r δ F δP − α ∂φ∂x , with F [ φ, P ] = F AC [ φ ] + F P [ P ] (19)where F AC = (cid:90) d x (cid:34) (cid:18) ∂∂x φ (cid:19) − φ + 14 φ − µφ (cid:35) (20)is the energy for the cubic passive AC equation as usedbefore, F P = (cid:90) d x P (21)is the energy of the polarisation P that favours a stateof random orientation ( P = 0) and does not allow forspontaneous polarisation. Both coupling terms have astrength α (called “activity”) and their form representsthe simplest possible form to couple the scalar φ and the’vector’ P . Furthermore, D T , and D r are positive trans-lational and rotational diffusivities, respectively. Notethat the coupling terms break the gradient dynamicsstructure.In analogy to the active PFC model we expect thatsteady states of the passsive model ( α ) remain at restuntil they undergo a drift pitchfork bifurcation at a crit-ical activity where they start to move. We determine ananalytical criterion for the onset of motion in analogy tothe derivation for the active PFC model in Ref. 24. Itreads 0 = (cid:107) ∂ x φ s (cid:107) − (cid:107) ∂ x P s (cid:107) , (22)and allows us to determine the critical values of the ac-tivity parameter α . Here, φ s and P s denote steady state . . . . k φ k (a) . . . . . . α v (b) . . . . . . FIG. 8. Bifurcation diagram for resting and moving fronts forthe active cubic AC equation (19) at µ = 0. Panels (a) and (b)show the norm || φ || and the front velocity v in dependence ofactivity α , respectively. The resting and moving fronts aregiven as black dashed and red solid lines, respectively. Theinsets focus on the region where the drift pitchfork bifurca-tion occurs. The black solid line represents the homogenoussolution branch. profiles whose L -norm is taken. Note, that the specificcriterion differs from the one in Ref. 24 as there the orderparameter field φ follows a conserved dynamics while herethe dynamics is nonconserved. This is further discussedin Ref. 52.Fig. 8 presents results of continuation runs following frontsolutions of Eqs. (19) at µ = 0 using the activity α ascontrol parameter. The black dashed line correspondsto resting fronts connecting φ − and φ + . These rest-ing fronts become shallower with increasing α and thebranch terminates on the branch of trivial homogeneousstates at about α = 36 .
3. However, already at rathersmall α ≈ .
19, a drift pitchfork bifurcation occurs (seeinset of Fig. 8 (a)) where a branch of steadily movingfronts (red solid line) emerges from the branch of rest-ing fronts. The numerical results confirm the criterion(22). In the limit of large activity we find v = α , as de-picted in Fig. 8 (b). This agrees with analogous resultsin Ref. 24 where also an analytical argument is given.Next, we take a moving front at a particular value ofactivity ( α = 1 . µ to obtain a bifurcation diagram in analogy to Fig. 1 for the passive cubic AC equation. The result ispresented in Fig. 9 (a). Panels (b) to (g) show selectedfront profiles φ ( x ) in colors equal to the correspondingbranch in panel (a). The accompanying polarisation pro-files P ( x ) are given in grey. For completeness, we alsoindicate the steady homogeneous states φ − , φ and φ + as dotted horizontal lines. Note that the front location isoff center in the computational domain as the relaxationto the homogeneous states can be very asymmetric.The fronts on the red solid and black dot-dashed branchesat µ = 0 correspond to the moving and resting front inFig. 8. Increasing µ , the symmetry of the potential in φ is broken and two red branches emerge. Both of themundergo saddle-node bifurcations at respective criticalvalues 0 .
26 and µ c . The larger one is identical to µ c in section II B as the homogeneous steady states do notdepend on activity (in contrast to their stability). Thesmaller one can not be obtained from an analysis of thelocal equilibrium potential f ( φ ) − µ .Another interesting feature are fronts that do not movesteadily but are modulated in a time-periodic manner.They are found on the grey solid branches in Fig. 9 (a)and emerge with finite oscillation period of about T =12 . µ with a monotonically increasing period(up to about T = 14 . µ = 0). Typical changes inthe moving solution profiles over one period of time aredepicted in Fig. 10.Overall one can state that the coupling of a simple ACdynamics to a linear dynamics of a polarisation field in-troduces a number of additional unstable front solutionsall connecting φ + and φ − . Hence, the structure of the bi-furcation diagram strongly changes from the passive casein Fig. 1 (a) and the active case in Fig. 9 (a). Note thatdue to the coupling, also the marginal stable fronts prop-agating into the unstable state become unstable in theactive model. IV. CONCLUSION
In the present work we have investigated front solutionsin a number of passive and active Allen-Cahn (AC) equa-tions employing continuation techniques. We have fo-cused on the dependency of front velocities on an exter-nal driving µ , e.g., a chemical potential or external field.The results have been presented in the form of bifurcationdiagrams. First, we have reviewed the widely availableanalytical results for the simple cubic AC equation andhave introduced the concepts of pulled and pushed frontsemploying linear and nonlinear marginal stability anal-ysis. We have highlighted that there exist fronts thatchange their character from pulled to pushed as the driv-ing µ is increased across a threshold.Next we have extended the analysis to the cubic-quinticAC equation that allows for more homogeneous steadystates and, in consequence, for more fronts connecting0 φ + φ φ − φ φ + φ − φ − φ + φ − φ φ φ + φ + φ − φ v µ φ φ − φ + (a) (b) (c) (d)(g)(f )(e)420-2-4-0.4 -0.2 0 0.2 0.4 FIG. 9. (a) Bifurcation diagram of front states described by the active Allen-Cahn model (19). Shown is the front velocity asa function of the external driving strength µ at fixed activity α = 1 . φ − and φ + , namely, the red solid, black dot-dashed, grey solid and orange dashed lines. The blue and green dashedlines correspond to fronts propagating into the unstable state φ . The grey solid line corresponds to a branch of time periodicoscillating fronts. Panels (b) to (g) show selected front profiles φ ( x ) in colors equal to the corresponding branch in panel (a)at µ ≈ .
2. The accompanying polarisation profiles P ( x ) are given in grey. For completeness, we also indicate the steadyhomogeneous states φ − , φ and φ + as dotted horizontal lines. Note however, that the solution panels are differently scaled onthe φ, P axes as the polarisation profiles differ in magnitudes. them. We have presented a rather involved bifurcationdiagram employing again the driving µ as control param-eter. It shows the various front solutions connecting theup to five homogeneous steady states. In general, ourresults allow one to better understand how the differentfronts are related and how they transform with increasingdriving strength. To understand substructures of the di-agram it has been helpful to discuss symmetries and howparts of the cubic-quintic potential resemble the cubicpotential.The considered model is similar to the one studied inRef. 17 in the context of the creation of metastablephases in crystallisation processes. In particular, theyinvestigate how double-fronts emerge that first create ametastable phase before it is transformed into the stablephase. Their main control parameter is the order param-eter value corresponding to one of the thermodynamicphases, while here we have kept the energy functionalfixed and employed the chemical potential (external field)as control parameter. We have shown how fronts andtransitions similar to the ones discussed in Ref. 17 are em-bedded into the full bifurcation diagram of front states. As Ref. 53 extends the discussion of Ref. 17 to systemswhere two order parameter fields are coupled, in the fu-ture it may be interesting to revisit also such and morecomplicated two- or multi-field models employing contin-uation techniques.After considering the cubic and cubic-quintic AC equa-tions that represent gradient dynamics models, we haveconsidered nonvariational extensions. First, we haveanalysed a cubic AC equation with the addition of a stan-dard nonvariational chemical potential (see, e.g., classi-fication in the introduction of Ref. 15). We have foundthat main symmetries of the bifurcation diagram are bro-ken, however, occurring front types and overall topologyof the diagram remain the same as in the passive case.Our numerical results agree at small strength of the non-variational influence (cid:15) with the approximate analyticalresults of Ref. 18 but show some deviations at larger (cid:15) .Here it will be interesting in the future to employ con-tinuation techniques to investigate fronts in the nonvari-ational cubic-quintic AC equation recently introduced inRef. 54 as a model for a liquid crystal light valve experi-ment with optical feedback.1 x . − t/Tφ (a) 0 . . . t/Tx . − − − t/TP (b) 0 . . . t/T FIG. 10. Space-time plots for a periodically modulated frontsolution of Eq. (19). Shown are (top) the concentration profileand (bottom) the polarisation profile at µ = 0 . v = 0 .
15. The different colorsof the contour indicate time scaled by the period T ≈ . x − t plane in grey-scale. Changes with respect to the passive case are more dra-matic in the second nonvariational model that couples acubic AC equation with a linear equation of a polarisationfield, similar to such couplings in more involved modelsof active matter. In this case we have encountered frontsthat move due to activity even at zero external driving.They emerge in a drift pitchfork bifurcation similar tothe onset of motion in active phase-field-crystal andactive Cahn-Hilliard models . This then implies a muchricher bifurcation diagram that even contains oscillatingfront states that emerge at a Hopf bifurcation of steadyfronts. We believe future comparative studies that anal-yse front motion and its emergence in a larger class ofsystems would be highly valuable. Appendix A: Nonvariational cubic Allen-Cahn equation
As in the literature the derivation of Eq. 18 is onlysketched, and we find it instructive, here we reproduceit in greater detail. The aim is to determine an analyti-cal expression for the velocity of moving fronts that aresolutions of the nonvariational AC equation (17) and todiscuss its dependence on the strength of the nonvaria-tional influence.We introduce the ansatz φ (˜ x, t ) = φ F (˜ x − vt ) + u (˜ x − vt, (cid:15)t ) v = v − (cid:15)v (A1)where φ F is the solution to (5) with velocity v , i.e., φ F (˜ x − v t ) := φ F ( x ) as the velocity changes due to thenonvariational term. Moreover, we add a small adjust-ment function u also of order (cid:15) . For the ansatz (A1) weintroduce ξ = ˜ x − vt = x + (cid:15)v t . Linearizing φ F ( ξ ) around x yields φ F ( ξ ) = φ F ( x ) + (cid:15)v ∂φ F ∂ξ (cid:12)(cid:12)(cid:12)(cid:12) ξ = x + O ( (cid:15) ) (A2)Inserting the ansatz (A1) into (17) using (A2) and lin-earizing in (cid:15) yields − (cid:15)v v ∂ φ F ∂ξ (cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ + (cid:15)v ∂φ F ∂ξ (cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ − v ∂u∂ξ = (cid:15)v ∂ φ F ∂ξ (cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ + (cid:15)v ∂φ F ∂ξ (cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ − (cid:15)v φ F ∂φ F ∂ξ (cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ + ∂ u∂ξ + u − φ F ( ξ ) u + (cid:15)g nv (cid:18) φ F ( ξ ) , ∂φ F ∂ξ , ∂ φ F ∂ξ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ . (A3)Deriving (5) with respect to x − v ∂∂x ∂φ F ( ξ ) ∂x = ∂∂x ∂ φ F ( x ) ∂x + ∂φ F ( x ) ∂x − φ F ( x ) ∂φ F ( x ) ∂x , (A4)and introducing the linear operator L = − v ∂∂x − ∂ ∂x − φ F ( x ) , (A5)we obtain L † (cid:18) ∂∂x φ F ( − x ) (cid:19) = 0 , (A6)where L † is the adjoint of L . Next, we identify the linearoperator (A5) in (A3) (cid:15)v ∂φ F ∂ξ (cid:12)(cid:12)(cid:12)(cid:12) ξ = x + (cid:15)v L (cid:18) ∂∂x φ F (cid:19) + Lu = (cid:15)g nv (cid:12)(cid:12)(cid:12)(cid:12) ξ = x . (A7)With (A6), (A7) simplifies to Lu = (cid:15)g nv (cid:18) φ F , ∂φ F ∂x , ∂ φ F ∂x (cid:19) − (cid:15)v ∂φ F ∂x (A8)2According to the Fredholm alternative (A8) is onlysolvable if (cid:28) (cid:15)g nv (cid:18) φ F , ∂φ F ∂x , ∂ φ F ∂x (cid:19) − (cid:15)v ∂φ F ∂x (cid:12)(cid:12)(cid:12)(cid:12) ∂∂x φ F (cid:29) = 0 , (A9)because (A6) has a nontrivial solution. S. M. Allen and J. W. Cahn, “Microscopic theory for antiphaseboundary motion and its application to antiphase domain coars-ening,” Acta Metall. Mater. , 1085–1095 (1979). A. Kolmogorov, I. Petrovsky, and N. Piskunov, “Investigationof the equation of diffusion combined with increasing of the sub-stance and its application to a biology problem,” Bull. MoscowState Univ. Ser. A: Math. Mech , 1–25 (1937). P. C. Fife and W. M. Greenlee, “Interior Transition Layers for El-liptic Boundary Value Problems with a Small Parameter,” Russ.Math. Surv. , 103–131 (1974). F. Schl¨ogl, “Chemical reaction models for non-equilibrium phasetransitions,” Z. Phys. , 147–161 (1972). Y. B. Zeldovich and D. Frank-Kamenetsky, “K teoriiravnomernogo rasprostranenia plameni (Toward a theory of uni-formly propagating flames),”
Dokl. Akad. Nauk SSSR , , 693–697 (1938). R. Monteiro and A. Scheel, “Phase separation patterns from di-rectional quenching,” J. Nonlinear Sci. , 1339–1378 (2017). J. S. Langer, “An introduction to the kinetics of first-orderphase transitions,” in
Solids far from Equilibrium , edited byC. Godr`eche (Cambridge University Press, 1992) Chap. 3, pp.297–363. L. Pismen,
Patterns and Interfaces in Dissipative Dynamics- (Springer Verlag, Berlin Heidelberg, 2006) pp. 83–139. C. Misbah,
Complex Dynamics and Morphogenesis: An Intro-duction to Nonlinear Science (Springer Netherlands, 2016). W. van Saarloos, “Front propagation into unstable states,” Phys.Rep.-Rev. Sec. Phys. Lett. , 29–222 (2003). G. Ahlers and D. S. Cannell, “Vortex-front propagation in rotat-ing Couette-Taylor flow,” Phys. Rev. Lett. , 1583–1586 (1983). J. Fineberg and V. Steinberg, “Vortex-front propagation inRayleigh-B´enard convection,” Phys. Rev. Lett. , 1332–1335(1987). J. S. Langer, “Instabilities and pattern formation in crystalgrowth,” Rev. Mod. Phys. , 1–28 (1980). M. Burger,
Oscillations and Traveling Waves in Chemical Sys-tems - (Wiley, New York, 1985). S. Engelnkemper, S. V. Gurevich, H. Uecker, D. Wetzel, andU. Thiele, “Continuation for thin film hydrodynamics and relatedscalar problems,” in
Computational Modeling of Bifurcations andInstabilities in Fluid Mechanics , Computational Methods in Ap-plied Sciences, vol 50, edited by A. Gelfgat (Springer, 2019) pp.459–501. J. Lajzerowicz, “Domain wall near a first order phase transition:role of elastic forces,” Ferroelectrics , 219–222 (1981). J. Bechhoefer, H. L¨owen, and L. S. Tuckerman, “Dynamicalmechanism for the formation of metastable phases,” Phys. Rev.Lett. , 1266–1269 (1991). A. J. Alvarez-Socorro, M. G. Clerc, G. Gonz´alez-Cort´es, andM. Wilson, “Nonvariational mechanism of front propagation:Theory and experiments,” Phys. Rev. E , 010202 (2017). R. Wittkowski, A. Tiribocchi, J. Stenhammar, R. J. Allen,D. Marenduzzo, and M. E. Cates, “Scalar phi(4) field theory foractive-particle phase separation,” Nat. Commun. , 4351 (2014). M. E. Cates and J. Tailleur, “Motility-induced phase separation,”Annu. Rev. Condens. Matter Phys. , 219–244 (2015). J. Stenhammar, A. Tiribocchi, R. J. Allen, D. Marenduzzo, andM. E. Cates, “Continuum theory of phase separation kinetics foractive brownian particles,” Phys. Rev. Lett. , 145702 (2013). L. Rapp, F. Bergmann, and W. Zimmermann, “Systematic ex-tension of the Cahn-Hilliard model for motility-induced phaseseparation,” Eur. Phys. J. E , 57 (2019). A. M. Menzel and H. L¨owen, “Traveling and resting crystals inactive systems,” Phys. Rev. Lett. , 055702 (2013). L. Ophaus, S. V. Gurevich, and U. Thiele, “Resting and travel-ing localized states in an active phase-field-crystal model,” Phys.Rev. E , 022608 (2018). H. Emmerich, H. L¨owen, R. Wittkowski, T. Gruhn, G. I. T´oth,G. Tegze, and L. Gr´an´asy, “Phase-field-crystal models for con-densed matter dynamics on atomic length and diffusive timescales: an overview,” Adv. Phys. , 665–743 (2012). G. Kozyreff and M. Tlidi, “Nonvariational real Swift-Hohenbergequation for biological, chemical, and optical systems,” Chaos , 037103 (2007). R. FitzHugh, “Impulses and physiological states in theoreticalmodels of nerve membrane,” Biophys. J. , 445 – 466 (1961). J. Nagumo, S. Arimoto, and S. Yoshizawa, “An active pulsetransmission line simulating nerve axon,” Proceedings of the IRE , 2061–2070 (1962). P. Zhang, A. Be’er, E.-L. Florin, and H. Swinney, “Collectivemotion and density fluctuations in bacterial colonies,” Proc. Natl.Acad. Sci. U.S.A. , 13626–30 (2010). D. J. T. Sumpter,
Collective Animal Behavior (Princeton Uni-versity Press, Kassel, 2010). F. Peruani, J. Starruß, V. Jakovljevic, L. Søgaard-Andersen,A. Deutsch, and M. B¨ar, “Collective motion and nonequilibriumcluster formation in colonies of gliding bacteria,” Phys. Rev. Lett. , 098102 (2012). B. Szab´o, G. J. Sz¨oll¨osi, B. G¨onci, Z. Jur´anyi, D. Selmeczi, andT. Vicsek, “Phase transition in the collective migration of tissuecells: Experiment and model,” Phys. Rev. E , 061908 (2006). N. Sep´ulveda, L. Petitjean, O. Cochet, E. Grasland-Mongrain,P. Silberzan, and V. Hakim, “Collective cell motion in an epithe-lial sheet can be quantitatively described by a stochastic inter-acting particle model,” PLoS Comput. Biol. , e1002944 (2013). J. Palacci, S. Sacanna, A. Steinberg, D. Pine, and P. Chaikin,“Living crystals of light-activated colloidal surfers,” Science ,936–940 (2013). W. van Saarloos, “Front propagation into unstable states. ii. lin-ear versus nonlinear marginal stability and rate of convergence,”Phys. Rev. A , 6367–6390 (1989). A. Couairon and J.-M. Chomaz, “Absolute and convective insta-bilities, front velocities and global modes in nonlinear systems,”Physica D , 236–276 (1997). J. A. Powell, A. C. Newell, and C. K. R. T. Jones, “Competitionbetween generic and nongeneric fronts in envelope equations,”Phys. Rev. A , 3636–3652 (1991). E. Ben-Jacob, H. Brand, G. Dee, L. Kramer, and J. Langer,“Pattern propagation in nonlinear dissipative systems,” PhysicaD , 348–364 (1985). U. Ebert and W. van Saarloos, “Front propagation into unstablestates: universal algebraic convergence towards uniformly trans-lating pulled fronts,” Physica D , 1–99 (2000). D. Aronson and H. Weinberger, “Multidimensional nonlinear dif-fusion arising in population genetics,” Adv. Math. , 33–76(1978). G. C. Paquette, L.-Y. Chen, N. Goldenfeld, and Y. Oono, “Struc-tural stability and renormalization group for propagating fronts,”Phys. Rev. Lett. , 76–79 (1994). W. van Saarloos, “Front propagation into unstable states:Marginal stability as a dynamical mechanism for velocity selec-tion,” Phys. Rev. A , 211–229 (1988). Y. A. Kuznetsov,
Elements of Applied Bifurcation Theory , 3rded. (Springer, New York, 2010). H. A. Dijkstra, F. W. Wubs, A. K. Cliffe, E. Doedel, I. F.Dragomirescu, B. Eckhardt, A. Y. Gelfgat, A. Hazel, V. Lucarini,A. G. Salinger, E. T. Phipps, J. Sanchez-Umbria, H. Schuttelaars,L. S. Tuckerman, and U. Thiele, “Numerical bifurcation meth-ods and their application to fluid dynamics: Analysis beyondsimulation,” Commun. Comput. Phys. , 1–45 (2014). H. Uecker, D. Wetzel, and J. D. M. Rademacher, “pde2path- a Matlab package for continuation and bifurcation in 2D el- liptic systems,” Numer. Math.-Theory Methods Appl. , 58–106(2014). E. Doedel, H. B. Keller, and J. P. Kernevez, “Numerical anal-ysis and control of bifurcation problems (I) Bifurcation in finitedimensions,” Int. J. Bifurcation Chaos , 493–520 (1991). E. J. Doedel, T. F. Fairgrieve, B. Sandstede, A. R. Champneys,Y. A. Kuznetsov, and X. Wang, “Auto-07p: Continuation andbifurcation software for ordinary differential equations,” Tech.Rep. (2007). If not stated otherwise we employed pde2path . C. Kuehn, “Numerical continuation and SPDE stability for the2d cubic-quintic Allen-Cahn equation,” SIAM-ASA J. Uncertain.Quantif. , 762–789 (2015). M. Menzel, T. Ohta, and H. L¨owen, “Active crystals and theirstability,” Phys. Rev. E , 022301 (2014). A. I. Chervanyov, H. Gomez, and U. Thiele, “Effect of the ori-entational relaxation on the collective motion of patterns formedby self-propelled particles,” Europhys. Lett. , 68001 (2016). L. Ophaus,
Analysis of the Active Phase-Field-Crystal Model ,Ph.D. thesis, Westf¨alische Wilhelms-Universit¨at M¨unster,M¨unster (2019). L. S. Tuckerman and J. Bechhoefer, “Dynamical mechanism forthe formation of metastable phases: The case of two noncon-served order parameters,” Phys. Rev. A , 3178–3192 (1992). A. J. ´Alvarez-Socorro, C. Castillo-Pinto, M. G. Clerc,G. Gonz´alez-Cortes, and M. Wilson, “Front propagation transi-tion induced by diffraction in a liquid crystal light valve,” Opt.Express , 12391 (2019). A. G. Ramm, “A simple proof of the fredholm alternative anda characterization of the fredholm operators,” The AmericanMathematical Monthly108