Control of transversal instabilities in reaction-diffusion systems

In two-dimensional reaction-diffusion systems, local curvature perturbations in the shape of traveling waves are typically damped out and disappear in the course of time. If, however, the inhibitor diffuses much faster than the activator, transversal instabilities can arise, leading from flat to folded, spatio-temporally modulated wave shapes and to spreading spiral turbulence. For experimentally relevant parameter values, the photosensitive Belousov-Zhabotinsky reaction (PBZR) does not exhibit transversal wave instabilities. Here, we propose a mechanism to artificially induce these instabilities via a wave shape dependent spatio-temporal feedback loop, and study the emerging wave patterns. In numerical simulations with the modified Oregonator model for the PBZR using experimentally realistic parameter values we demonstrate the feasibility of this control scheme. Conversely, in a piecewise-linear version of the FitzHugh-Nagumo model transversal instabilities and spiral turbulence in the uncontrolled system are shown to be suppressed in the presence of control, thereby stabilising flat wave propagation.


Introduction
A large variety of pattern forming processes can be understood in terms of the advancement of an interface between two or more spatial domains. An interface that becomes unstable to diffusion possibly causes intricate spatio-temporal dynamics. Well known examples include the Mullins-Sekerka instability during crystal growth and formation of snow flakes [1,2], and the Saffman-Taylor instability leading to viscous fingering in multiphase flow and porous media [3,4,5]. Other phenomena affected by interfacial instabilities are flame fronts [6,7], Marangoni convection [8], and growing cell monolayers [9]. Traveling plane waves in excitable media exhibit interfacial instabilities as well. Here, an effective interface separates the excited state from the excitable rest state. A straight isoconcentration line of a two-dimensional flat wave can suffer an instability leading to stationary or time dependent modulations orthogonal to the propagation direction. Further away from the instability threshold, rotating wave segments and spreading spiral turbulence are observed [10,11]. For standard activator-inhibitor kinetics, these so-called transversal or lateral wave instabilities typically occur if the inhibitor diffuses much faster than the activator. This result was analytically predicted first by Kuramoto for piecewise-linear reaction kinetics [12,13]. Later, it was confirmed numerically by Horváth et al. for autocatalytic reaction-diffusion fronts with cubic reaction kinetics [14] as well as in experiments with the iodate-arsenous acid reaction [15] and the acid-catalyzed chlorite-tetrathionate reaction [16]. The experimental workhorse of chemical pattern formation, the Belousov-Zhabotinsky (BZ) reaction, does typically not display transversal wave instabilities. Dispersing the reagents of the BZ reaction in nanodroplets of a water-in-oil microemulsion allows to increase the inhibitor diffusivity considerably [17] and leads, for example, to segmented spiral waves as reported by Vanag and Epstein [18]. Even in the presence of an electrical field applied to enhance transversal instabilities in cubic autocatalytic reaction-diffusion fronts, the inhibitor diffusion coefficient is always required to be sufficiently larger than that of the activator [19,20,21]. Because of the possibility to apply spatio-temporal external forcing or feedbackmediated control loops by exploiting the dependence of the local excitation threshold on the intensity of applied illumination, the photosensitive variant of the BZ reaction has been widely used as a paradigm of an experimentally well controllable RD system. So far, unstable wave propagation has been stabilised by global feedback [22]. Two feedback loops were used to stabilise unstable wave segments and to guide their propagation along pre-given trajectories [23]. Also, spiral wave drift in response to resonant external forcing and various feedback-mediated control loops has been extensively studied experimentally in PBZR systems, compare for example [24,25,26,27,28]. In this paper, we design a curvature-dependent spatio-temporal feedback loop in order to destabilise a stable propagating planar reaction-diffusion wave by inducing transversal instabilities. In numerical simulations with the modified Oregonator model for the PBZR, we study the wave patterns emerging beyond the instability threshold, and demonstrate the capability to actively select wave patterns by modifying feedback parameters accessible to the experimenter. Conversely, under conditions where planar wave propagation fails due to transversal instabilities, using the same feedback mechanism we suppress ongoing breakup and segmentation of waves, thereby stabilising unstable propagating planar waves.

Models
The first model we investigate is the three component modified Oregonator model [29] Here, the parameters ǫ andǫ characterise the time scales for the dynamics of the activator u and inhibitor w, respectively, and the stoichiometric parameters q and f depend on the temperature and chemical composition. All parameter values used in numerical simulations are listed in table A1 in Appendix A. The modified Oregonator model describes the light-sensitive Belousov-Zhabotinsky (BZ) reaction. In experiments, the catalyst v can be immobilised in a gel and therefore the corresponding diffusion coefficient is set to zero. The activator u and inhibitor w diffuse with diffusion coefficients D u and D w , whose ratio for typical BZ recipes is approximately D w /D u ≈ 1.2. This value is too low to support transversal instabilities such that plane waves are stable for typical BZ recipes. The parameter Φ in equation (3) is proportional to the applied light intensity and measures the local excitation threshold. In experiments, spatio-temporal modulations of Φ can be applied to control wave propagation in the BZ reaction [23,27,28,22].
Because the modified Oregonator model does not exhibit transversal instabilities in the parameter regime relevant for experiments, we investigate a second model. The piecewise-linear caricature of the FitzHugh-Nagumo (FHN) model [30,11] received some attention in the context of transversal instabilities [11]. It is a two component model of standard activator-inhibitor type, with u being the activator and v the inhibitor. The reaction kinetics are a piecewiselinear caricature of the FHN model where The parameters k 1 and k 2 are chosen such that f (u) is continuous at u = δ and u = 1 − δ, which leads to The remaining parameters for the function f are chosen in such a way that f resembles the cubic shape of the FHN activator nullcline. All parameter values used in numerical simulations are listed in table A2 in Appendix A. The parameter a is a measure for the excitation threshold and used as the feedback parameter. For numerical simulations, we assume an elongated two-dimensional channel of width L in the y-direction with waves propagating in the x-direction. The boundary conditions in the x-direction are periodic while we assume periodic or Neumann boundary conditions in the y-direction. For both models, we use a box-like initial condition of width b for the vector of components u, where u max is the initial height of the pulse and u 0 is the stationary point of the reaction kinetics. The box function is defined as The initial shape of the wave is given by where A denotes the amplitude of deviation of the shape from a plane wave and d is an offset. For numerical simulations in two spatial dimensions, we use Euler forward for the time evolution and a five point stencil for the Laplacian. A phase diagram for the occurrence of transversal instabilities in the ǫ-σ-parameter plane of the piecewise-linear FHN model was presented by Zykov et al. in [11]. Increasing the inhibitor diffusion coefficient σ crosses the threshold for transversal instabilities. Shortly beyond the onset of transversal instabilities, a plane wave develops a fold which is stationary in a comoving frame of reference, see figure 1 for a time sequence of snapshots and the supplemental material [31] for a movie. Further away from the instability threshold, a plane wave breaks into segments which undergo selfsustained rotatory motion accompanied by permanent merging and annihilation of segments. This regime is also known as spreading spiral turbulence [11], see figure 2 for a time sequence of snapshots.

Evolution equations for isoconcentration lines
Theoretically, the onset of transversal instabilities can be understood with the help of the linear eikonal equation an evolution equation for a two-dimensional curve γ (s, t) = (γ x (s, t) , γ y (s, t)) T representing an isoconcentration line parametrised by the curve parameter s. The linear eikonal equation relates the normal velocity (n is the normal vector of γ) along γ linearly to its curvature, The curvature is conventionally assumed to be positive for convex isoconcentration lines, i.e., an isoconcentration lines with a protrusion in the propagation direction. The constant c corresponds to the pulse velocity of a one-dimensional solitary wave and ν is the curvature coefficient. A rigorous derivation of the eikonal equation (13) from the reaction-diffusion system identifies the constant ν in terms of the one-dimensional pulse profile, its response function and the matrix of diffusion coefficients, see [32] for details. For a plane wave, any isoconcentration level is a straight line and therefore its curvature vanishes, κ (s, t) ≡ 0, everywhere along γ. The stability of a plane wave is determined by the sign of the curvature coefficient ν. As long as ν > 0, any point along the isoconcentration line of a convex bulge moves slower than a plane wave, while points of a concave dent move faster than a plane wave, thereby smoothing out deviations from a plane wave. If ν < 0, a convex bulge moves faster than a plane wave, protruding the bulge even further and thereby leading to an ever increasing curvature: a transversal instability arises. Patterns arising for ν < 0 cannot be described by the linear eikonal equation and terms depending nonlinearly on the curvature have to be taken into account which saturate the growth of an ever increasing curvature. At least two different nonlinear versions of equation (13) exist in the literature. Zykov et al. [33,30,34,35] renormalised ǫ and σ in equation (5) to derive a renormalised onedimensional velocity c depending on the curvature. Dierckx et al. [32] derive higher order nonlinear corrections in the curvature by a rigorous perturbation expansion with a small parameter proportional to the curvature, additionally generalising the eikonal equation to anisotropic media. Apart from nonlinear eikonal equations, which are difficult to solve numerically, patterns arising beyond the threshold of transversal instability can be described by the Kuramoto-Sivashinsky (KS) equation, Equation (16) is an evolution equation for the x-component φ (y, t) of an isoconcentration line γ parametrised in the form γ (y, t) = (φ (y, t) , y) T . See [13] and [36] for a derivation of equation (16) from a general RD system. The case of Neumann boundary conditions in the y-direction for the RD system imply that any isoconcentration line of activator and inhibitor meets the domain boundary in a right angle. This corresponds to Neumann boundary conditions for φ, Similarly, periodic boundary conditions in the y-direction of the RD system carry over to periodic boundary conditions for φ. Equation (16) was originally proposed by Sivashinsky [6] in the study of turbulent flame propagation and adapted for reactiondiffusion systems by Kuramoto [37,13]. The parameter λ can be expressed in terms of a sum over all eigenfunctions of the linear stability operator arising through a linearisation of the one-dimensional RD system around the traveling wave solution [13]. To compute λ, we use a method which avoids the virtually impossible numerical computation of all eigenfunctions, see [36] for details. The values of λ and ν for the modified Oregonator model with parameters as given in Appendix A are The Kuramoto-Sivashinsky equation (16) allows a refined investigation of the onset of transversal instabilities. For a stability analysis of a plane wave in channel of width L with Neumann boundary conditions, we apply a perturbation expansion in 0 <ǫ ≪ 1 with an ansatz in form of a Fourier series, where the term ct corresponds to a plane wave solution to the RD system traveling in the x-direction. The dispersion relation follows as Transversal instabilities occur only if ω 1 > 0, i.e., ν must be negative and the channel width must exceed Thus, in general, the onset of a transversal instability depends on the boundary conditions and can be suppressed in thin channels. It is a long-wavelength instability, i.e., the first mode which becomes unstable upon reaching the threshold is the mode with the longest possible wavelength. If ν < 0, the fourth order term in the KS equation (16) counteracts the negative diffusion term and leads to a saturation of the growth of wavefront modulations.Starting at the threshold of instability, the solution to the KS equation (16) displays a fold with a minimum located at y = L/2. Upon increasing L, this steady wave loses stability via a supercritical Hopf bifurcation [14] and the wave front starts to oscillate back and forth in a symmetrical fashion. Increasing L even further leads to a symmetry breaking bifurcation with asymmetrical oscillations followed by a period doubling cascade to fully developed spatio-temporal chaos. In this regime, the KS equation displays a strong dependence on the initial data, with small differences in the initial conditions leading to dramatically different future time evolution. This characteristic of the KS equation is also studied as an analogy for hydrodynamic turbulence [38]. As long as ν > 0, no instability can arise and the fourth order term can be safely neglected by setting λ = 0. In this case, equation (16) simplifies to the nonlinear phase diffusion equation, which in turn can be transformed to the usual diffusion equation via the Cole-Hopf transform [12]. Therefore, equation (16) with λ = 0 can be solved analytically for arbitrary initial and boundary conditions. To assess the accuracy of the Kuramoto-Sivashinsky equation (16) as an approximation for propagating reaction-diffusion waves, we compare the transition from an initially curved shape to a plane wave for ν > 0 with numerical simulations of the underlying two-dimensional modified Oregonator model Eqs.

Curvature-dependent feedback control
The feedback law proposed in this article requires that the velocity c of a onedimensional wave depends sufficiently strongly on a parameter which can be controlled in experiments. For the modified Oregonator model, we use the parameter Φ proportional to the applied light intensity as the feedback parameter. A numerical computation of the dependence of c on Φ is shown in figure 4. With relatively good accuracy, the dependence can be assumed to be linear, with parameters c 1 = −90.191, c 0 = 9.013 obtained from a least square fit. Solitary waves exist only in the excitable regime of Φ values indicated by the dashed lines in figure 4. For Φ 0.045, the rest state is unstable and the medium becomes oscillatory. For Φ 0.068, the solitary pulse profile becomes unstable and decays to the stable rest state. A successful feedback control is possible if Φ is restricted to lie between these two values. We introduce a feedback law for Φ depending linearly on the curvature, The parameters α and β are accessible to an experimenter. In general, these parameters can be adjusted with time to achieve a better performance of the control. Together with equation (23) and equation (24), the linear eikonal equation (13) becomes with the effective curvature coefficient Depending on the sign ofν, the control will have very different effects. If a plane wave is stable with respect to transversal perturbations because ν > 0, we can excite transversal instabilities ifν = ν − c 1 β < 0. Conversely, if ν > 0 such that plane waves are unstable with respect to transversal modulations, patterns can be stabilised ifν = ν − c 1 β > 0. An appropriate choice of the parameters α and β in the feedback law (24) allows to control transversal instabilities.
To apply the feedback law (24) it is necessary to compute the curvature of a chosen isoconcentration line of a chosen component with sufficient accuracy, which raises considerable difficulties.

Computation of curvature by Level Set Methods
The curvature κ (s, t) of an isoconcentration line γ (s, t), equation (15), is proportional to the second derivative of the isoconcentration line with respect to the curve parameter s. Computations of isoconcentration lines from numerical simulations or experiments are affected by noise due to the discretised nature of the computed or measured concentration field u, respectively. Numerical differentiation is an ill-posed mathematical operation and typically amplifies the noise. A variety of methods to compute the curvature κ directly from a numerically determined isoconcentration line were tested and discarded due to insufficient performance. An indirect method which avoids the differentiation of an isoconcentration line is to compute the curvature fieldκ as According to the formula of Bonnet [39], evaluatingκ at an isoconcentration line r = γ (s, t) of u yields the curvature κ of γ, i.e., κ (s, t) =κ (γ (s, t) , t) .
See Appendix B for a proof of Bonnet's formula. Equation (27) involves the determination of the second derivative of u with respect to x and y. These expressions are readily available from the finite difference algorithm used to solve the RD system numerically. The problem is now that the concentration u of a pulse solution typically varies very fast in a small spatial region while it is constant everywhere else, leading to an ill-defined denominator in equation (27). This difficulty can be addressed with the help of a level set method, which, however, is numerically quite expensive. Originally, level set methods were developed by Osher and Sethian to compute and track the motion of interfaces. These methods have since been successfully applied in such diverse areas of applications as computer graphics, medical image segmentation and crystal growth [40,41,42]. We introduce a second field variable χ (r, τ ) which evolves in (virtual) time τ according to the so-called reinitialisation equation [43,42,44] Equation (29) is solved with the initial condition where u c is the activator value along the isoconcentration line γ for which we want to determine the curvature κ, i.e., u (γ (s, t) , t) = u c . Note that χ (γ (s, t) , τ ) = χ 0 (γ (s, t)) = 0 for all times τ such that the position of the level set γ is not changed by equation (29). However, equation (29) transforms the neighbourhood of χ = 0 such that, after sufficiently many time steps τ , The curvature κ of γ, equation (15) can now readily be computed in terms of the Laplacian of χ as κ (s, t) =κ (γ (s, t) , t) = lim τ →∞ ∆χ (γ (s, t) , τ ) .
Numerically, the evolution of χ up to the final time τ = 0.01 is sufficient to obtain a very accurate smooth result for the curvature of γ. The reinitialisation equation (29) has to be solved at every real time step t. However, because the time evolution of the RD system is slow enough, we recompute the curvature κ only every 200th time step t.

Excitation of transversal instabilities in the modified Oregonator model
We study the possibility to excite transversal instabilities by curvature-dependent feedback in the modified Oregonator model. The feedback law (24) is realised via the parameter Φ proportional to the illumination in the BZ reaction. For the parameters of the feedback law we set α = Φ max and β = − (Φ max − Φ min ) /κ norm such that the effective curvature coefficient is The values of Φ max and Φ min can be chosen arbitrarily as long as Φ min < Φ max and both values lie in the regime of an excitable medium, see figure (4). The curvature κ is determined for the activator isoconcentration line γ with u (γ (s, t) , t) = u c = 0.2. An area of fixed size in front of and behind γ is illuminated with the same value Φ (κ (s, t)), while within the remaining medium Φ attains its background value Φ = Φ 0 . Before the feedback is switched on at time t 1 = 0.4, the wave evolves uncontrolled. The value of κ norm = 1.2 is an estimate of the largest value which the curvature attains during the overall time evolution. For simplicity, we choose a constant value of κ norm , but in principle this value can be set to the maximum curvature every time the curvature is recomputed. Figure 5 shows  wave length of the modulations, this type of pattern appears similar to the patterns arising in the uncontrolled FHN model slightly beyond the threshold of transversal instabilities, see figure 1. Figure 6 displays the effects of moderate feedback with an effective curvature coefficient ν = −1.059. V-shaped patterns arise which travel much faster than a corresponding one-dimensional solitary pulse. In a frame of reference comoving with the centre of mass, the V-shaped patterns appear stationary apart from modulations traveling along the isoconcentration line. The V-patterns observed under feedback are longtime stable and do not decay or break up. A solitary V-pattern in an unbounded domain can be explained analytically as a solution to the linear and nonlinear eikonal equations [45,46]. A V with opening angle α has a mean velocity V given by where c is the one-dimensional velocity. Because |sin (α)| < 1, all V-patterns are moving faster than a plane wave. Experimentally, these patterns were observed in homogeneous [47] and stratified [48] BZ media. Figure 7 shows the effect of strong feedback with an effective curvature coefficient ν = −2.337. Similar as for moderate feedback, V-shaped patterns appear. However, their shape is non-stationary but oscillating. The V-shape is segmented in an irregular and non-stationary way, with segments either merging again or breaking off and serving as the nucleation centre for new waves. These new waves propagate as segmented circles and occasionally start to rotate until they annihilate upon collision with other waves. Qualitatively, the segmentation and occurrence of rotating segments is similar to the spreading spiral turbulence observed for the uncontrolled FHN model deep in the regime of transversal instabilities, see figure 2.
These results show that the proposed feedback law is not only able to excite transversal instabilities but allows, to some extent, the selection of the patterns beyond the instability threshold by tuning the feedback parameters Φ max and Φ min accessible to an experimenter. We display a phase diagram with a classification of the observed patterns in the Φ max − Φ min plane in figure 8. Note that according to the KS equation (16), the observed patterns should only depend on the effective curvature coefficient ν given by equation (34). However, numerical simulations show that the type of patterns depends not only on the difference of Φ max and Φ min , but also display a slight dependence on their absolute values. This dependence is due to nonlinear corrections in the relation for the one-dimensional velocity c over Φ and higher order effects neglected by the KS equation (16). By adjusting the effective curvature coefficientν, we are able to validate the predicted onset of transversal instabilities equation (21), and its dependence on the channel width L. We perform numerical simulations of the controlled Oregonator model in a channel with width L and Neumann boundary conditions in the y-direction. Starting with a plane box-like initial condition equation (10) with noisy box width b, we change the effective curvature coefficientν until a plane wave becomes unstable, i.e., the curvature along the isoconcentration line is different from zero. Figure 9 shows that both numerical simulations and analytical prediction yield a linear relation between channel width L and 1/ √ −ν over a large range of effective curvature coefficientsν. The slopes differ due to higher order corrections for the KS equation (16) and nonlinear corrections for the velocity c over Φ, equation (23), used for the feedback law. Beyond the onset of transversal instabilities, the emerging patterns can in principle be described by the KS equation (16). We compare the time evolution of the modified Oregonator model with the solution of the KS equation for an effective curvature coefficient ofν = −0.02. Because the centre of mass velocity is incorrectly predicted by the KS equation, figure 10 shows a sequence of snapshots of isoconcentration lines in a frame of reference comoving with the centre of mass. Due to the strong dependence on the initial data, any initial agreement between the two curves is vanishing fast during the time evolution.

Suppression of transversal instabilities
The curvature-dependent feedback law (24) is used to suppress transversal instabilities occurring in the uncontrolled piecewise linear FHN model given by equations (4), (5). Here, the excitation threshold a is used as the feedback parameter. First, we linearly approximate the velocity -excitation threshold relation as The coefficient β (t) is adjusted in time such that the maximum value of a (κ) along the isoconcentration line does not exceed or undershoot the range of existence of solitary pulses. Every 100 time steps, we determine the maximum curvature κ max (t) along the isoconcentration line and set κ max to this value, The background value of a is set to a 0 = 0.1 everywhere before the feedback control is switched on at time t = t 1 = 215, and outside the region affected by the feedback control. Figure 11 displays the suppression of a transversal instability slightly beyond the threshold. For the same parameter values as in figure 1, the initially sinusoidally shaped wave relaxes back to a plane wave and no fold appears, see also the video in the supplemental material [31]. Patterns deep in the regime of transversal instabilities are characterised by a continuing segmentation of waves and spreading spiral turbulence as shown in figure 2. For the same parameter values, patterns stop to segment after the feedback is switched on, giving rise to a persistent plane wave and two counter rotating spiral waves, see figure 12. The wave front of rotating patterns has a nonbinding positive curvature. According to the linear eikonal equation (13), it advances slower than a plane wave if the effective curvature coefficientν is positive. Therefore, the plane wave has a tendency to annihilate rotating waves, finally leading to a solitary plane wave.

Conclusions
In this article, we present a feedback loop to induce, control, and suppress transversal instabilities of reaction-diffusion waves. The control signal is calculated from the local curvature of the isoconcentration line of the wave. We show that the curvature dependent control can amplify or quench small curvature perturbations in the wave shape. Simultaneously, the feedback allows to study a large variety of artificially produced wave patterns associated with transversal instabilities. Often these patterns are non-stationary and sensitively depending on small changes in the initial conditions characteristic for chaotic dynamics. Mathematically, the onset of transversal instabilities can be understood with the help of the linear eikonal equation which relates the wave velocity normal to an isoconcentration line to its local curvature. The coefficient ν in front of the curvature determines the stability of a flat wave. For positive values of ν, convex wave segments slow down while concave wave segments propagate at a higher velocity. Under these conditions a perturbed flat traveling wave recovers its flat shape. In the case of negative ν, a small positive curvature causes an increase of the wave velocity which in turn results in an increase of the local curvature. Now, a flat wave is unstable with respect to small curvature perturbations. The proposed feedback loop is able to change the sign of the coefficient ν.
With experiments on chemical waves in the PBZR in mind, for realistic parameter values we show in numerical simulations with the Oregonator model that transversal instabilities of planar waves can be induced by the feedback. Right beyond the transversal instability of planar waves, we find nearly flat folded waves which are stationary in a comoving frame of reference. For weak feedback we observe small ripple-shaped undulations traveling along the wave front. Upon increasing the feedback strength further, V-shaped wave patterns with spatio-temporal transversal modulations appear. These V-shaped waves travel at a velocity that depends on the opening angle but is considerably faster than that of the planar wave. Far away from the instability threshold, breakup of waves causes persistent annihilation and merging of excited domains, self-sustained rotatory motion and nucleation of rotating wave segments. Qualitatively, the emerging wave patterns correspond to those observed in numerical simulations with separated activator and inhibitor diffusivity [11].
Regarding chemical wave propagation in the PBZR, we emphasise that the feedback parameters of the control law are experimentally accessible. For appropriate BZ recipes the dependence of the wave velocity on the intensity of applied light should be strong enough to induce transversal wave instabilities. The isoconcentration line of the wave can be determined by 2d spectrophotometry with sufficient spatial resolution using the contrast between the oxidised and reduced form of the catalyst. We believe that the computation of the curvature by the Level Set Method as described in Sec. 2.4 will work reliably for noisy experimental data, too. Because all chemical components share similarly shaped isoconcentration lines, the measurement of the concentration field of an arbitrary single chemical species is sufficient for setting up the control loop. Fine-tuning the feedback parameters allows to study the onset of transversal instabilities in dependence of the boundary conditions as e.g. the channel width L, as pointed out in chapter Sec. 2.2.
In the opposite case, sufficiently strong feedback changes the sign of the effective curvature coefficient from negative to positive. Consequently, naturally occurring transversal wave instabilities leading to the breakup of waves are suppressed -the feedback stabilises planar waves and spiral waves. Spreading of spiral turbulence is inhibited due to the suppression of segmentation of waves. Reaction-diffusion waves describe, at least approximately, a huge variety of wave processes in biology. Our results are potentially applicable to deliberately induce or inhibit transversal wave instabilities and to control the emerging patterns under very general conditions. The essential condition for applicability is that the propagation velocity of the wave can be externally controlled over a sufficiently large range such that the curvature coefficient of the eikonal equation switches its sign. Moreover, we expect that curvature dependent feedback might have interesting applications in interfacial pattern formation in general. For example, this feedback mechanism could be the starting point for a control strategy aimed at the purposeful selection of patterns affected by interfacial instabilities as, e.g., alloys growing into an undercooled melt. = ∂ x u (γ x (y) , y) γ ′′ x (y) + ∂ 2 x u (γ x (y) , y) (γ ′ x (y)) 2 + 2∂ y ∂ x u (γ x (y) , y) γ ′ x (y) + ∂ 2 y u (γ x (y) , y) = 0, (B.2) and generally d n dy n u (γ x (y) , y) = 0 with n ∈ N, n > 0. The curvature fieldκ, Eq. (27), expressed in Cartesian coordinates is κ (x, y) = ∂ 2 y u (x, y) (∂ x u (x, y)) 2 − 2∂ x u (x, y) ∂ y u (x, y) ∂ x,y u (x, y) (∂ x u (x, y)) 2 + (∂ y u (x, y)) 2 3/2 + ∂ 2 x u (x, y) (∂ y u (x, y)) 2 (∂ x u (x, y)) 2 + (∂ y u (x, y)) 2 3/2 .