ﻻ يوجد ملخص باللغة العربية
Similarity solutions play an important role in many fields of science: we consider here similarity in stochastic dynamics. Important issues are not only the existence of stochastic similarity, but also whether a similarity solution is dynamically attractive, and if it is, to what particular solution does the system evolve. By recasting a class of stochastic PDEs in a form to which stochastic centre manifold theory may be applied we resolve these issues in this class. For definiteness, a first example of self-similarity of the Burgers equation driven by some stochastic forced is studied. Under suitable assumptions, a stationary solution is constructed which yields the existence of a stochastic self-similar solution for the stochastic Burgers equation. Furthermore, the asymptotic convergence to the self-similar solution is proved. Second, in more general stochastic reaction-diffusion systems stochastic centre manifold theory provides a framework to construct the similarity solution, confirm its relevance, and determines the correct solution for any compact initial condition. Third, we argue that dynamically moving the spatial origin and dynamically stretching time improves the description of the stochastic similarity. Lastly, an application to an extremely simple model of turbulent mixing shows how anomalous fluctuations may arise in eddy diffusivities. The techniques and results we discuss should be applicable to a wide range of stochastic similarity problems.
Many systems in life sciences have been modeled by reaction-diffusion equations. However, under some circumstances, these biological systems may experience instantaneous and periodic perturbations (e.g. harvest, birth, release, fire events, etc) such
A self-consistent equation to derive a discreteness-induced stochastic steady state is presented for reaction-diffusion systems. For this formalism, we use the so-called Kuramoto length, a typical distance over which a molecule diffuses in its lifeti
Under consideration is the hyperbolic relaxation of a semilinear reaction-diffusion equation on a bounded domain, subject to a dynamic boundary condition. We also consider the limit parabolic problem with the same dynamic boundary condition. Each pro
We carry out the enhanced group classification of a class of (1+1)-dimensional nonlinear diffusion-reaction equations with gradient-dependent diffusivity using the two-step version of the method of furcate splitting. For simultaneously finding the eq
Reaction-diffusion equations (RDEs) are often derived as continuum limits of lattice-based discrete models. Recently, a discrete model which allows the rates of movement, proliferation and death to depend upon whether the agents are isolated has been