Classification of solutions of a system of nonlinear diffusion equations in a neighborhood of a bifurcation point

Complex ordering and stochastic oscillations in a class of reaction-diffusion systems with small diffusion

“…Here the role of A in (1) is played by the family of matrices A(z) = A -zD, where z _> 0. The main difference between problem and (1), (3), and (4) is the fact that if there exists z 0 >_ 0 such that A(z o) has an eigenvalue with a zero real part (and no positive part), then infinitely many characteristic exponents of the linear problem, which correspond to (5) and (4) with F = 0, will tend to zero as e, /z -~ 0. Consequently, the dimension of the critical case must be considered as infinite.…”

“…The well-known example of a quasinormal form for a two-component reaction-diffusion system is the time-dependent generalized Ginzburg-Landau equation (TDGL), which was proposed by Kuramoto and Tsuzuki [2] and has been investigated thoroughly in [3]. The study of the TDGL played an important role in the development of nonlinear science.…”

Stability and convergence of difference scheme for nonlinear evolutionary type equations

“…Tsertsadze studied in [22] the convergence of difference schemes for the KuramotoTsuzuki equation and for systems of reaction-diffusion type. For the mixed boundary value problem of Kuramoto-Tsuzuki equation, he constructed a difference scheme, based on Crank-Nicolson method, and proved that the latter is convergent in energy norm (L 2 -norm) with the convergence rate of order O(k 2 …”

“…Reaction-diffusion type equations have been applied in the study of broad class of nonlinear processes, including a well-known synergetic model [2,14].…”

Optimal L ∞ error estimates for finite element Galerkin methods for nonlinear evolution equations

“…Reaction-diffusion type equations have been applied in the study of broad class of nonlinear processes, including a well-known synergetic model [2,11] and the nonlinear heat equation. The problem of constructing and validating difference schemes for these classes of problems has been taken in detail up in [8,9], see also [5,6,22,24] and [17][18][19].…”