Stability of the Static Spike Autosolitons in the Gray--Scott Model

We performed an asymptotic linear stability analysis of the static spike autosolitons (ASs)---self-sustained solitary inhomogeneous states---in the Gray--Scott model of an autocatalytic chemical reaction. We found that in one dimension these ASs destabilize with respect to pulsations or the onset of traveling motion when the inhibitor is slow enough. In higher dimensions, the one-dimensional static spike ASs are always unstable with respect to corrugation and wriggling. The higher-dimensional radially symmetric static spike ASs may destabilize with respect to the radially nonsymmetric fluctuations leading to their splitting when the inhibitor is fast or with respect to pulsations when the inhibitor is slow.

[1]  B. S. Kerner,et al.  Stochastic inhomogeneous structures in nonequilibrium systems , 1980 .

[2]  Cyrill B. Muratov,et al.  Traveling spike autosolitons in the Gray-Scott model , 2001 .

[3]  J. E. Pearson.  Complex Patterns in a Simple System , 1993, Science.

[4]  E. M. Lifshitz,et al.  Course in Theoretical Physics , 2013 .

[5]  Stephen K. Scott,et al.  Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Isolas and other forms of multistability , 1983 .

[6]  V. V. Osipov,et al.  Theory of self-replication and granulation of spike autosolitons , 1996 .

[7]  B. S. Kerner,et al.  Autosolitons in a hot semiconductor plasma , 1985 .

[8]  B. S. Kerner,et al.  Nonlinear theory of stationary strata in dissipative systems , 1978 .

[9]  B. S. Kerner,et al.  Autosolitons in Active Systems with Diffusion , 1986 .

[10]  James P. Keener,et al.  Diffusive effects on dispersion in excitable media , 1989 .

[11]  P. Mikhailov,et al.  Foundations of Synergetics II , 1996, Springer Series in Synergetics.

[12]  Robert Gardner,et al.  Stability analysis of singular patterns in the 1-D Gray-Scott model I: a matched asymptotics approach , 1998 .

[13]  B. S. Kerner,et al.  Autosolitons: A New Approach to Problems of Self-Organization and Turbulence , 1994 .

[14]  V. V. Osipov.  Criteria of spontaneous interconversions of traveling and static arbitrary dimensional dissipative structures , 1996 .

[15]  Alexander S. Mikhailov,et al.  Foundations of Synergetics II , 1990 .

[16]  Cyrill B. Muratov,et al.  Spike autosolitons in the Gray-Scott model , 1998, patt-sol/9804001.

[17]  R. J. Field,et al.  Oscillations and Traveling Waves in Chemical Systems , 1985 .

[18]  Gregory A. Kriegsmann,et al.  Stability of localized structures in non-local reaction-diffusion equations , 1998 .

[19]  Kenneth Showalter,et al.  Chemical waves and patterns , 1995 .

[20]  Osipov,et al.  Scenarios of domain pattern formation in a reaction-diffusion system. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[21]  Cyrill B. Muratov,et al.  Static spike autosolitons in the Gray-Scott model , 2000 .

[22]  J. K. Hale,et al.  Exact Homoclinic and Heteroclinic Solutions of the Gray-Scott Model for Autocatalysis , 2000, SIAM J. Appl. Math..

[23]  Arjen Doelman,et al.  Pattern formation in the one-dimensional Gray - Scott model , 1997 .

[24]  Cyrill B. Muratov,et al.  Spike autosolitons and pattern formation scenarios in the two-dimensional Gray-Scott model , 2001 .

[25]  John E. Pearson,et al.  Self-replicating spots in reaction-diffusion systems , 1997 .

[26]  Kerner Bs,et al.  Properties of stationary dissipative structures in mathematical models of morphogenesis , 1982 .

[27]  Reynolds,et al.  Dynamics of self-replicating patterns in reaction diffusion systems. , 1994, Physical review letters.

[28]  M. Cross,et al.  Pattern formation outside of equilibrium , 1993 .

[29]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

[30]  Osipov,et al.  General theory of instabilities for patterns with sharp interfaces in reaction-diffusion systems. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[31]  B S Kerner,et al.  [Properties of stationary dissipative structures in mathematical models of morphogenesis]. , 1982, Biofizika.

[32]  Boris S. Kerner,et al.  Highly nonequilibrium localized states in systems slightly away from thermodynamic equilibrium , 1985 .

[33]  V. V. Osipov,et al.  Transverse instability of spike autosolitons , 1997 .

[34]  Muratov.  Self-replication and splitting of domain patterns in reaction-diffusion systems with the fast inhibitor. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.