221 research outputs found
Stability of Travelling Waves for Reaction-Diffusion Equations with Multiplicative Noise
We consider reaction-diffusion equations that are stochastically forced by a
small multiplicative noise term. We show that spectrally stable travelling wave
solutions to the deterministic system retain their orbital stability if the
amplitude of the noise is sufficiently small.
By applying a stochastic phase-shift together with a time-transform, we
obtain a semilinear sPDE that describes the fluctuations from the primary wave.
We subsequently develop a semigroup approach to handle the nonlinear stability
question in a fashion that is closely related to modern deterministic methods
Do Language Models Understand <em>Anything</em>?:On the Ability of LSTMs to Understand Negative Polarity Items
Nonlinear stability of pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions
We establish the existence and nonlinear stability of travelling pulse
solutions for the discrete FitzHugh-Nagumo equation with infinite-range
interactions close to the continuum limit. For the verification of the spectral
properties, we need to study a functional differential equation of mixed type
(MFDE) with unbounded shifts. We avoid the use of exponential dichotomies and
phase spaces, by building on a technique developed by Bates, Chen and Chmaj for
the discrete Nagumo equation. This allows us to transfer several crucial
Fredholm properties from the PDE setting to our discrete setting
Negative Diffusion and Traveling Waves in High Dimensional Lattice Systems
This is the publisher's version, also available electronically from http://epubs.siam.org/doi/abs/10.1137/120880628We consider bistable reaction diffusion systems posed on rectangular lattices in two or more spatial dimensions. The discrete diffusion term is allowed to have positive spatially periodic coefficients, and the two spatially periodic equilibria are required to be well ordered. We establish the existence of traveling wave solutions to such pure lattice systems that connect the two stable equilibria. In addition, we show that these waves can be approximated by traveling wave solutions to systems that incorporate both local and nonlocal diffusion. In certain special situations our results can also be applied to reaction diffusion systems that include (potentially large) negative coefficients. Indeed, upon splitting the lattice suitably and applying separate coordinate transformations to each sublattice, such systems can sometimes be transformed into a periodic diffusion problem that fits within our framework. In such cases, the resulting traveling structure for the original system has a separate wave profile for each sublattice and connects spatially periodic patterns that need not be well ordered. There is no direct analogue of this procedure that can be applied to reaction diffusion systems with continuous spatial variables
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