Ill posed linear and nonlinear initial value problems may be stabilized, that it converted to to well posed initial value problems, by the addition of purely nonscalar linear dispersive terms. This is a stability analog of the Turing instability. This idea applies to systems of quasilinear Schrödinger equations from nonlinear optics.

Long-wavelength instabilities of steady patterns, spatially periodic in three dimensions, are studied. All potentially stable patterns with the symmetries of the simple-, face-centered- and body-centered-cubic lattices are considered. The results generalize the well-known Eckhaus, zigzag, and skew-varicose instabilities to three-dimensional patterns and are applied to two-species reaction-diffu...

We consider a one-dimensional Swift-Hohenberg equation coupled to conservation law, where both equations contain additional dispersive terms breaking the reflection symmetry x↦−x. This system exhibits Turing instability and we study dynamics close onset of this instability. First, show that periodic traveling waves bifurcate from homogeneous ground state. Second, fixing bifurcation parameter in...

The concept of cross diffusion is applied to some biological systems. The conditions for persistence and Turing instability in the presence of cross diffusion are derived. Many examples including: predatorprey, epidemics (with and without delay), hawk-dove-retaliate and prisoner’s dilemma games are given.

We consider the following Gierer-Meinhardt system with a precursor μ(x) for the activator A in R:    At = 2A ′′ − μ(x)A + A2 H in (−1, 1), τHt = DH ′′ −H + A in (−1, 1), A′(−1) = A′(1) = H ′(−1) = H ′(1) = 0. Such an equation exhibits a typical Turing bifurcation of the second kind, i.e., homogeneous uniform steady states do not exist in the system. We establish the existence and stabili...

The Turing instability in the reaction-diffusion system is a widely recognized mechanism of the morphogen gradient self-organization during the embryonic development. One of the essential conditions for such self-organization is sharp difference in the diffusion rates of the reacting substances (morphogens). In classical models this condition is satisfied only for significantly different values...

We introduce diffusively coupled networks where the dynamical system at each vertex is planar Hamiltonian. The problems we address are synchronisation and an analogue of diffusion-driven Turing instability for time-dependent homogeneous states. As a consequence of the underlying Hamiltonian structure there exist unusual behaviours compared with networks of coupled limit cycle oscillators or act...

Abstract This paper is devoted to investigating the pattern dynamics of Lotka–Volterra cooperative system with nonlocal effect and finding some new phenomena. Firstly, by discussing Turing bifurcation, we build conditions instability, which indicates emergence patterns in this system. Then, using multiple scale analysis, obtain amplitude equations about different patterns. Furthermore, all poss...

Cellular gene expression is a complex process involving many steps, including the transcription of DNA and translation of mRNA; hence the synthesis of proteins requires a considerable amount of time, from ten minutes to several hours. Since diffusion-driven instability has been observed to be sensitive to perturbations in kinetic delays, the application of Turing patterning mechanisms to the pr...