Asymptotic Scaling in a Model Class of Anomalous Reaction-Diffusion Equations

Abstract We analyze asymptotic scaling properties of a model class of anomalous reaction-diffusion (ARD) equations. Numerical experiments show that solutions to these have, for large t, well defined scaling properties. We suggest a general framework to analyze asymptotic symmetry properties; this provides an analytical explanation of the observed asymptotic scaling properties for the considered ARD equations.

[1]  G. Cicogna "weak" Symmetries and Adapted Variables for Differential Equations , 2004 .

[2]  Cristóbal López,et al.  Reaction-Diffusion Systems: Front Propagation and Spatial Structures , 2003 .

[3]  P. Tempesta,et al.  Weak transversality and partially invariant solutions , 2002, math-ph/0206003.

[4]  Angelo Vulpiani,et al.  Front propagation in reactive systems with anomalous diffusion , 2003 .

[5]  Giuseppe Gaeta,et al.  Symmetry and Perturbation Theory in Nonlinear Dynamics Lecture Notes in Physics - Volume m57 ISBN 3-540-65904-8 , 2002 .

[6]  G. Cicogna,et al.  Partial Lie-point symmetries of differential equations , 2001, 1309.2407.

[7]  G. Cicogna,et al.  Symmetry and Perturbation Theory in Nonlinear Dynamics , 1999 .

[8]  I. Anderson,et al.  Group Invariant Solutions Without Transversality , 1999, math-ph/9910014.

[9]  N. Riley,et al.  Scaling, self-similarity, and intermediate asymptotics , 1998 .

[10]  G. I. Barenblatt Scaling: Self-similarity and intermediate asymptotics , 1996 .

[11]  P. Olver,et al.  Nonclassical And Conditional Symmetries , 1996 .

[12]  G. Gaeta Asymptotic symmetries and asymptotically symmetric solutions of partial differential equations , 1994 .

[13]  G. Gaeta Nonlinear symmetries and nonlinear equations , 1994 .

[14]  Antti Kupiainen,et al.  Renormalization Group and the Ginzburg-Landau equation , 1992 .

[15]  Liu,et al.  Anomalous dimensions and the renormalization group in a nonlinear diffusion process. , 1990, Physical review letters.

[16]  H. Stephani Differential Equations: Their Solution Using Symmetries , 1990 .

[17]  Pierre Collet,et al.  Instabilities and fronts in extended systems , 1990 .

[18]  P. Olver Applications of Lie Groups to Differential Equations , 1986 .

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

[20]  R. Fisher THE WAVE OF ADVANCE OF ADVANTAGEOUS GENES , 1937 .