Publications Arnd Scheel

Publications - Arnd Scheel

Preprints

Refereed Journals

Review Articles

Proceedings

Theses

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Preprints

A. Scheel, Q. Wu.

Pinning and dipole asymptotics of locally deformed striped phases. (PDF)

M. Avery, P. Carter, B. de Rijk, A. Scheel.

Stability of coherent pattern formation through invasion in the FitzHugh-Nagumo system. (PDF)

M. Avery, M. Holzer, A. Scheel.

Pushed and pulled fronts in a logistic Keller-Segel model with chemorepulsion. (PDF)

Refereed Journals

R. Goh, T. Kaper, A. Scheel.

Pitchfork bifurcation along a slow parameter ramp: coherent structures in the critical scaling. (PDF)
Stud. Appl. Math. 153 (2024).

N. Li, A. Scheel.

Anchored spirals in the driven curvature flow approximation. (PDF)
London Math. Soc. Lecture Note Ser., to appear

A. Asch, M. Avery, A. Cortez, A. Scheel.

Slow passage through the Busse balloon -- predicting steps on the Eckhaus staircase. (PDF)
European J. App. Math., to appear.

Th. Gallay, A. Scheel.

Viscous shocks and long-time behavior of scalar conservation laws. (PDF)
Comm. Pure Appl. Anal. 23 (2024), 1448-1382.

M. Avery, M. Holzer, A. Scheel

Pushed-to-pulled front transitions: continuation, speed scalings, and hidden monotonicity (PDF)
J. Nonlinear Science 33 (2023) 102.

R. Goh, A. Scheel.

Growing Patterns (PDF)
Nonlinearity 36 (2023), R1-R51.

O. Cannon, T. Bondurant, M. Whyte, A. Scheel

Shifting consensus in a biased compromise model (PDF)
SIAM J. Appl. Dyn. Sys. 23 (2024), 297-324.

M. Avery, A. Scheel

Sharp decay rates for localized perturbations to the critical front in the Ginzburg-Landau equation (PDF)
J. Dyn. Diff. Eqns. 36 (2024), S287-S322.

R. Goh, T. Kaper, A. Scheel, T. Vo

Fronts in the wake of a parameter ramp: slow passage through pitchfork and fold bifurcations (PDF)
SIAM J. Appl. Dyn. Sys. 22 (2023), 2312-2356.

O. Cannon, A. Scheel

Coherent structures in nonlocal systems --- functional analytic tools (PDF)
J. Dyn. Diff. Eqns., to appear.

A. Scheel

Nonlinear eigenvalue methods for linear pointwise stability of nonlinear waves (PDF)
SIAM J. Numer. Anal. 61 (2023), 592-616.

B. Sandstede, A. Scheel

Spiral waves: linear and nonlinear theory (PDF)
Mem. Amer. Math. Soc. 1413 (2023).

G. Faye, M. Holzer, L. Siemer, A. Scheel

Invasion into remnant instability: a case study of front dynamics (PDF)
Indiana Univ. Math. J. 71 (2022), 1819-1896

M. Avery, A. Scheel

Universal selection of pulled fronts (PDF)
Comm. Amer. Math. Soc. 2 (2022), 172-231.

M. Avery, C. Dedina, A. Smith, A. Scheel

Instability in large bounded domains — branched versus unbranched resonances (PDF)
Nonlinearity 34 (2021), 7916-7937

A. Scheel, A. Stevens, C. Tenbrock

Signaling gradients in surface dynamics as basis for planarian regeneration (PDF, Supplementary movies Cutting, Oscillations, Grafting)
J. Math. Biol. 83,6 (2021)

K. Chen, Z. Deiman, R. Goh, S. Jankovic, A. Scheel

Strain and defects in oblique stripe growth (PDF)
Multiscale Model. Simul. 19 (2021), 1236-1260

M. Avery, A. Scheel

Asymptotic stability of critical pulled fronts via resolvent expansions near the essential spectrum (PDF)
SIAM J. Math. Anal. 53 (2021), 2206-2242.

A. Pauthier, A. Scheel

Advection-diffusion dynamics with nonlinear boundary flux as a model for crystal growth (PDF)
Math. Nachr. 293 (2020), 1565-1590

P. Flynn, Q. Neville, A. Scheel

Self-organized clusters in run-and-tumble processes (PDF)
DCDS-S 13 (2020), 1187-1208

A. Scheel, T. Tao

Bifurcation to coherent structures in nonlocally coupled systems (PDF)
J. Dyn. Diff. Eqns. 31 (2019), 1107-1127

M. Avery, R. Goh, O. Goodloe, A. Milewski, A. Scheel

Growing stripes, with and without wrinkles (PDF) [Moduli space movie with delicate arch, Supplementary materials]
SIAM J. Appl. Dyn. Sys. 18 (2019), 1078-1117

K. Kang, A. Scheel, A. Stevens

Global phase diagrams of run-and-tumble dynamics: equidistribution, waves, and blowup (PDF)
Nonlinearity 32 (2019), 2128-2146

G. Jaramillo, A. Scheel, Q. Wu

The effect of impurities on striped phases (PDF)
Proc. Roy. Soc. Edinb. 149 (2019), 131-168

N. Ankney, M. Avery, T. Khain, A. Scheel

Pinning and Depinning: from periodic to chaotic and random media (PDF)
Chaos 29, 013127 (2019)

R. Samuelson, Z. Singer, J. Weinburd, A. Scheel

Advection and autocatalysis as organizing principles for banded vegetation patterns (PDF)
J. Nonlinear Science 29 (2019), 255-285

B. Bakker, A. Scheel

Spatial Hamiltonian identities for nonlocally coupled systems (PDF)
Forum Sigma 6 (2018), E22, 55p

P. Carter, A. Scheel

Wave train selection by invasion fronts in the FitzHugh--Nagumo equation (PDF)
Nonlinearity 31 (2018), 5536-5572

R. Goh, A. Scheel

Pattern forming fronts in a Swift-Hohenberg equation with directional quenching - parallel and oblique stripes (PDF)
J. London Math. Soc. 98 (2018), 104-128

R. Monteiro, A. Scheel

Contact angle selection for interfaces in growing domains (PDF)
Z. Angew. Math. Mech. 98 (2018), 1086-1102

G. Faye, A. Scheel

Center manifolds without a phase space (PDF)
Trans. Amer. Math. Soc. 370 (2018), 5843-5885

A. Scheel, J. Weinburd

Wavenumber selection via spatial parameter jump (PDF)
Philos. Trans. Royal Soc. A, 376 (2017), 20170191

A. Scheel, A. Stevens

Wavenumber selection in coupled transport equations (PDF)
J. Math. Biol. 75 (2017), 1047-1073

R. Monteiro, A. Scheel

Phase separation patterns from directional quenching (PDF)
J. Nonlinear Science 27 (2017), 1339-1378 .

G. Faye, M. Holzer, A. Scheel

Linear spreading speeds from nonlinear resonant interaction (PDF)
Nonlinearity 30 (2017), 2403-2442

A. Pogan, A. Scheel

Traveling fronts bifurcating from stable layers in the presence of conservation laws (PDF)
Discr. Cont. Dyn. Sys. A 37 (2017), 2619-2651

D. Lloyd, A. Scheel

Continuation and bifurcation of grain boundaries in the Swift-Hohenberg equation (PDF)
SIAM J. Appl. Dyn. Sys. 16 (2017), 252-293

Z. Wang, R. Goh, K. Bazargan, A. Scheel, N. Saraf

Stochastic implementation and analysis of dynamical system similar to the logistic map
IEEE Trans. VLSI Sys. 25 (2017), 747-759

A. Scheel: Spinodal decomposition and coarsening fronts in the Cahn-Hilliard equation (PDF)
J. Dyn. Diff. Eqns. 29 (2017), 431-464
R. Goh, R. Beekie, D. Matthias, J. Nunley, A.Scheel: Universal wavenumber selection laws in apical growth (PDF)
Phys. Rev. E 94 (2016), 022219
R. Goh, A. Scheel: Pattern formation in the wake of triggered pushed fronts (PDF)
Nonlinearity 29 (2016), 2196-2237
T. Anderson, G. Faye, A. Scheel, D. Stauffer: Pinning and unpinning in nonlocal systems (PDF)
J. Dyn. Diff. Eqns.28 (2016), 897-923
G. Jaramillo, A. Scheel: Pacemakers in large arrays of oscillators with nonlocal coupling (PDF)
J. Diff. Eqns. 260 (2016), 2060-2090
E. Ben-Naim, A. Scheel: Pattern selection and super-patterns in the bounded confidence model (PDF)
Europhys. Lett. 112 (2015) 18002
D. Morrissey, A. Scheel: Characterizing the effect of boundary conditions on striped phases (PDF)
SIAM J. Appl. Dyn. Sys. 14 (2015), 1387-1417
R. Goh, A. Scheel: Hopf bifurcation from fronts in the Cahn-Hilliard equation (PDF)
Arch. Rational Mech. Anal. 217 (2015), 1219-1263
G. Jaramillo, A. Scheel: Deformation of striped patterns by inhomogeneities (PDF)
Math. Meth. Appl. Sci. 38 (2015), 51-65
A. Scheel, Q. Wu: Diffusive stability of Turing patterns via normal forms (PDF)
J. Dyn. Diff. Eqns. 27 (2015), 1027-1076
C. Browne, A. Dickerson. Mentors: G. Faye, A. Scheel.: Coherent structures in scalar feed-forward chains (PDF)
SIURO 7 (2014), 306-329
G. Faye, A. Scheel: Existence of pulses in excitable media with nonlocal coupling (PDF)
Adv. Math. 270 (2014), 400-456
M. Holzer, A. Scheel: Criteria for pointwise growth and their role in invasion processes (PDF)
J. Nonlinear Science 24 (2014), 661-709
M. Holzer, A. Scheel: Accelerated fronts in a two stage invasion process (PDF)
SIAM J. Math. Anal. 46 (2014), 397-427
G. Faye, A. Scheel: Fredholm properties of nonlocal differential operators via spectral flow (PDF)
Indiana Univ. Math. J. 63 (2014), 1311-1348
R. Goh, A. Scheel: Triggered fronts in the complex Ginzburg-Landau equation (PDF)
J. Nonlinear Science 24 (2014), 117-144
A. Scheel, Q. Wu: Small-amplitude grain boundaries of arbitrary angle in the Swift-Hohenberg equation (PDF)
Z. Angew. Math. Mech. 94 (2014), 203-232
K. Bose, T. Cox, S. Silvestri, P. Varin. Mentors: M. Holzer, A. Scheel.: Invasion fronts and pattern formation in a model of chemotaxis in one and two dimensions. (PDF)
SIURO 6 (2013), 228-245
A. Pogan, A. Scheel, K. Zumbrun: Quasi-gradient systems, modulational dichotomies, and stability of spatially periodic patterns
Diff. Int. Eqns. 26 (2013), 383-432
M. Holzer, A. Scheel: A slow pushed front in a Lotka-Volterra competition model (PDF)
Nonlinearity 25 (2012), 2151-2179
M. Haragus, A. Scheel: Dislocations in an anisotropic Swift-Hohenberg equation (PDF)
Comm. Math. Phys. 315 (2012), 311-335
M. Haragus, A. Scheel: Grain boundaries in the Swift-Hohenberg equation (PDF)
European J. Appl. Math. 23 (2012), 737-759
A. Pogan, A. Scheel: Layers in the presence of conservation laws (PDF)
J. Dyn. Diff. Eqns. 24 (2012), 249-287
B. Sandstede, A. Scheel, G. Schneider, H. Uecker: Diffusive mixing of periodic wave trains in reaction-diffusion systems (PDF)
J. Differential Eqns. 252 (2012), 3541-3574
A. Pogan, A. Scheel: Fredholm properties of radially symmetric, second order differential operators (PDF)
Int. J. Dyn. Sys. Diff. Eqns. 3 (2011), 289-327
T. Gallay, A. Scheel: Diffusive stability of oscillations in reaction-diffusion systems (PDF)
Trans. Amer. Math. Soc. 363 (2011), 2571-2598
R. Goh, S. Mesuro, A. Scheel: Spatial wavenumber selection in recurrent precipitation (PDF)
SIAM J. Appl. Dyn. Sys. 10 (2011), 360-402
J. Bellay, A. Scheel: Coherent structures near the boundary between excitable and oscillatory media (PDF)
Dyn. Syst. 25 (2010), 111-132
A. Pogan, A. Scheel: Instability of spikes in the presence of conservation laws (PDF)
Z. Angew. Math. Phys. 61 (2010), 979-998
A. Scheel, E. van Vleck: Lattice differential equations embedded into reaction-diffusion systems (PDF)
Proc. Royal Soc. Edinburgh A, 139A (2009), 193-207
A. Doelman, B. Sandstede, A. Scheel, G. Schneider: The dynamics of modulated wave trains (PDF)
Mem. Amer. Math. Soc. 199 (2009)
A. Scheel: Robustness of Liesegang patterns (PDF)
Nonlinearity 22 (2009), 457-483
A. Scheel, J.D. Wright: Colliding dissipative pulses - the shooting manifold (PDF)
J. Differential Eqns. 245 (2008), 59-79
B. Sandstede, A. Scheel: Hopf bifurcation from viscous shock waves (PDF)
SIAM J. Math. Anal. 39 (2008), 2033-2052
B. Sandstede, A. Scheel: Relative Morse indices, Fredholm indices, and group velocities (PDF)
Discr. Cont. Dyn. Sys. A 20 (2008), 139-158
M. Haragus, A. Scheel: Interfaces between rolls in the Swift-Hohenberg equation (PDF)
Int. J. Dyn. Sys. Diff. Eqns. 1 (2007), 89-97
J.D.M. Rademacher, A. Scheel: Instabilities of wave trains and Turing patterns in large domains (PDF )
Int. J. Bif. Chaos 16 (2007), 2679-2691
J.D. Wright, A. Scheel: Solitary waves and their linear stability in weakly coupled KdV equations (PDF)
Z. Angew. Math. Phys. 58 (2007), 535-570
B. Sandstede, A. Scheel: Period-doubling of spiral waves and defects (PDF)
SIAM J. Appl. Dyn. Syst. 6 (2007), 494-547
M. Haragus, A. Scheel: Stable viscous shocks in elliptic conservation laws (PDF)
Indiana Univ. Math. J. 56 (2007), 1261-1278
J.D.M. Rademacher, A. Scheel: The saddle-node of nearly homogeneous wave trains in reaction-diffusion systems (PDF)
J. Dyn. Diff. Eqns. 19 (2007), 479-496
J.D.M. Rademacher, B. Sandstede, A. Scheel: Computing absolute and essential spectra using continuation (PDF)
Physica D 229 (2007), 166-183
R. Kollár, A. Scheel: Coherent structures generated by inhomogeneities in oscillatory media (PDF)
SIAM J. Appl. Dyn. Syst. 6 (2007), 236-262
M. Haragus, E. Lombardi, A. Scheel: Spectral stability of wave trains in the Kawahara equation (PDF)
J. Math. Fluid Mechanics 8 (2006), 482-509
A. Scheel: Coarsening fronts (PDF)
Arch. Rational Mech. Anal. 181 (2006), 505-534
M. Haragus, A. Scheel: Almost planar waves in anisotropic media (PDF)
Comm. Partial Differential Equations 31 (2006), 791-815
M. Haragus, A. Scheel: Corner defects in almost planar interface propagation (PDF)
Ann. Inst. H. Poincare (C) Anal. Non Lineaire 23 (2006), 283-329
B. Sandstede and A. Scheel: Curvature effects on spiral spectra: Generation of point eigenvalues near branch points (PDF)
Phys. Rev. E 73 (2006), 016217
B. Sandstede, A. Scheel: Basin boundaries and bifurcations near convective instabilities: A case study (PDF)
J. Differential Eqns. 208 (2005), 176-193
B. Sandstede, A. Scheel: Absolute instabilities of standing pulses (PDF)
Nonlinearity 18 (2005), 331-378
B. Sandstede, A. Scheel: Evans function and blow-up methods in critical eigenvalue problems (PDF)
Discr. Cont. Dyn. Sys. 10 (2004), 941-964
B. Sandstede, A. Scheel: Defects in oscillatory media: toward a classification (PDF)
SIAM J. Appl. Dyn. Syst. 3 (2004), 1-68
A. Scheel: Radially symmetric patterns of reaction-diffusion systems (PDF)
Mem. Amer. Math. Soc. 165 (2003)
D.Pelinovsky, A. Scheel: Stability analysis of stationary light transmission in nonlinear photonic structures (PDF)
J. Nonlinear Science 13 (2003), 347-396
A. Doelman, B. Sandstede, A. Scheel, and G. Schneider: Propagation of hexagonal patterns near onset (PDF)
Europ. J. Appl. Math. 14 (2003), 85-110
K. Matthies, A. Scheel: Exponential veraging for Hamiltonian evolution equations (PDF)
Trans. Amer. Math. Soc. 355 (2003), 747-773
J. Härterich, B. Sandstede, A. Scheel: Exponential dichotomies for non-autonomous functional differential equations of mixed type (PDF)
Indiana Univ. Math. J. 51 (2002), 1081-1109
M. Haragus, A. Scheel: Linear stability and instability of ion-acoustic plasma solitary waves (PDF)
Physica D 170 (2002), 13-30
M. Haragus, A. Scheel: Finite-wavelength stability of capillary-gravity solitary waves (PDF)
Commun. Math. Phys. 225 (2002), 487-521
B. Sandstede, A. Scheel: On the structure of spectra of modulated travelling waves (PDF)
Math. Nachr. 232 (2001), 39-93
B. Sandstede, A. Scheel: Super-spiral structures of meandering and drifting spiral waves (PDF)
Phys. Rev. Lett. 86 (2001), 171-174
B. Sandstede, A. Scheel: On the stability of periodic travelling waves with large spatial period (PDF)
J. Differential Eqns. 172 (2001), 134-188
B. Sandstede, A. Scheel: Essential instabilities of fronts: bifurcation and bifurcation failure (PDF)
Dynamical Systems 16 (2001), 1-28
B. Sandstede, A. Scheel: Absolute versus convective instability of spiral waves (PDF)
Phys. Rev. E. 62 (2000), 7708-7714
B. Sandstede, A. Scheel: Absolute and convective instabilities of waves on unbounded and large bounded domains (PDF)
Physica D 145 (2000), 233-277
B. Sandstede, A. Scheel: Gluing unstable fronts and backs together can produce stable pulses (PDF)
Nonlinearity 13 (2000), 1465-1482
B. Sandstede, A. Scheel: Spectral stability of modulated travelling waves bifurcating near essential instabilities (PDF)
Proc. Roy. Soc. Edinb. A 130 (2000), 419-448
G. J. Lord, D. Peterhof, B. Sandstede, A. Scheel: Numerical computation of solitary waves in infinite cylindrical domains (PDF)
SIAM J. Numer. Anal. 37 (2000), 1420-1454
B. Sandstede, A. Scheel: Essential instability of pulses and bifurcations to modulated travelling waves (PDF)
Proc. Roy. Soc. Edinb. A 129 (1999), 1263-1290
B. Sandstede, A. Scheel, C. Wulff: Bifurcations and dynamics of spiral waves ( PDF)
J. Nonlinear Science 9 (1999), 439-478
B. Fiedler, A. Scheel, M.I. Vishik: Large patterns of elliptic systems in infinite cylinders (PDF)
J. Math. Pures Appl. 77 (1998), 879-907
A. Scheel: Bifurcation to spiral waves in reaction-diffusion systems (PDF)
SIAM J. Math. Anal. 29 (1998), 1399-1418
A. Scheel: Subcritical bifurcation to infinitely many rotating waves (PDF)
J. Math. Anal. Appl. 215 (1997), 252-261
B. Sandstede, A. Scheel, C. Wulff: Dynamics of spiral waves on unbounded domains using center-manifold reduction (PDF)
J. Differential Eqns. 141 (1997), 122-149
D. Peterhof, B. Sandstede, A. Scheel: Exponential dichotomies for solitary wave solutions of semilinear elliptic equations on infinite cylinders (PDF)
J. Differential Eqns. 140 (1997), 266-308
B. Sandstede, A. Scheel, C. Wulff: Center manifold reduction for spiral wave dynamics (PDF)
C. R. Acad. Sci. Paris, Serie I, t.324 (1997), 153-158
P. Chossat, M. Krupa, I. Melbourne, A. Scheel: Transverse bifurcations of homoclinic cycles (PDF)
Physica D 100 (1997) 85-100
A. Scheel: Existence of fast travelling waves for some parabolic equations --- a dynamical systems approach. (PDF)
J. Dyn. Diff. Eqns. 8 (1996), 469-548
B. Fiedler, B. Sandstede, A. Scheel, C. Wulff: Bifurcation from relative equilibria of noncompact group actions: skew products, meanders, and drifts (PDF)
Documenta Mathematica 1 (1996), 479-505.
B. Sandstede, A. Scheel: Forced symmetry breaking of homoclinic cycles (PDF)
Nonlinearity 8 (1995), 333-365
A. Scheel: Fast travelling waves for convective reaction-diffusion equations
C. R. Acad. Sci. Paris, Serie I, t. 317 (1993), 347-351.
A. Scheel: Fast travelling waves for the Cahn-Hilliard equation in an infinite strip
C. R. Acad. Sci. Paris, Serie I, t. 317 (1993), 261-265.
A. Scheel, P. Chossat: Bifurcation d'orbites periodiques a partir d'un cycle homocline symetrique (PDF)
C. R. Acad. Sci. Paris, Serie I, t. 314 (1992), 49-54.

Review Articles

M. Haragus, A. Scheel: A bifurcation approach to non-planar traveling waves in reaction-diffusion systems (PDF)
GAMM-Mitt. 30, (2007), 75-95.
B. Fiedler, A. Scheel: Spatio-Temporal Dynamics of Reaction-Diffusion Patterns (PDF)
In Trends in Nonlinear Analysis, M. Kirkilionis, S. Krömker, R. Rannacher, F. Tomi (Eds.), Springer-Verlag, Berlin, 2003 (145 pages).
B. Sandstede, A. Scheel, C. Wulff: Dynamical behavior of patterns with euclidean symmetry
In ``Pattern formation in continuous and coupled systems'', M. Golubitsky, D. Luss and S. Strogatz (eds.), Springer-Verlag. IMA Volumes in Mathematics and its Applications 115 (1999), 249-264.

Proceedings and book chapters

A. Scheel and S. Tikhomirov: Depinning asymptotics in ergodic media (PDF)
In P. Gurevich, J. Hell, B. Sandstede, and A. Scheel, editors, Patterns of Dynamics, 88--108, Cham, 2017. Springer International Publishing.
K. Bazargan, N. Sarag, A. Scheel, Z. Wang.: Randomness Meets Feedback: Stochastic Implementation of Logistic Map Dynamical System
In "Design Automation Conference", 2015 52nd ACM/EDAC/IEEE.
A. Scheel.: Pattern formation.
In 'The Princeton Companion to Applied Mathematics', Princeton Univ. Press, 2015.
M. Kotzagiannidis, J. Peterson, J. Redford, A. Scheel, Q. Wu: Stable pattern selection through invasion fronts in closed two-species reaction-diffusion systems (PDF)
In RIMS Kokyuroku Bessatsu B31 (2012), Far-From-Equilibrium Dynamics, eds. T. Ogawa, K. Ueda , pp 79-93.
R. Goh, S. Mesuro, A. Scheel: Coherent structures in reaction-diffusion models for precipitation (PDF)
Special volume on "Precipitation patterns in reaction-diffusion systems", Research Signpost (2010), 73-93.
A. Pogan, A. Scheel: Instability of radially-mmetric spikes in systems with a conserved quantity (PDF)
Fields Institute Communications 64 (2013), 119-140.
B. Sandstede, A. Scheel: Absolute and convective stability in large domains
In Proceedings of Equadiff 99, B. Fiedler, K. Gröoger, and J. Sprekels (eds.), World Scientific Publishing, Singapore (2000), 797-800.
M. Efendiev, A. Scheel: Upper and lower bounds for the Hausdorff dimension of the attractor for reaction-diffusion equations in R\sup n
In Proceedings of ``Nonlinear evolution equations and application'', M.Otani (ed.), RIMS, Kyoto (1999), 81-90.
P. Chossat, M. Krupa, I. Melbourne, A. Scheel: Magnetic dynamos in rotating convection --- a dynamical systems approach
Dyn. Cont. Discrete Impulsive Syst. 5 (1999), 327-340.
A. Scheel: Meandering spirals
Z. Angew. Math. Mech. 79 (1999) 163-166.

Theses

DEA INLN, Nice 1991: Bifurcation d'orbites periodiques a partir de cycles homoclines en presence de symetrie (PDF)
PhD, FU Berlin, 1994: Existence of fast travelling waves for some parabolic equations - a dynamical systems approach
Habilitation, FU Berlin, 2001: Radially symmetric patterns of reaction-diffusion systems

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