Proving the existence of localized patterns and saddle node bifurcations in 1D activator-inhibitor type models

Abstract

In this paper, we develop a methodology for constructively proving the existence of saddle-node bifurcations on unbounded domains. In particular, we combine several previous results in order to formulate a Newton-Kantorovich argument to obtain our results. We demonstrate our approach on a class of 1D activator-inhibitor models. The first ingredient is for us to prove the existence of stationary localized 1D solutions. Our approach relies on having an approximate, compactly supported solution, $\overline{\mathbf{u}}$. We construct this solution using Fourier series and then define a well-chosen fixed point map that is contracting on a neighborhood around $\overline{\mathbf{u}}$. For this matter, we construct an approximate inverse of the linearization around $\overline{\mathbf{u}}$ and establish sufficient conditions under which the contraction is achieved. The second ingredient is to derive an augmented zero finding problem whose solutions correspond to a saddle-node bifurcations. Extending the methodology for proving the existence of localized solutions to this augmented problem, we are able to constructively obtain the existence of saddle-nodes on unbounded domains. The third ingredient is for us to verify that we obtain a non-degenerate saddle-node bifurcation. To do so, we control the spectrum of the linearization around the solution to the augmented zero finding problem. This allows us to rule out a degenerate Hopf bifurcation, as well as controlling exactly the number of unstable directions at the bifurcation. Quantitative explicit estimates as well as the requisite codes for the rigorous numerics are provided.