相关论文: Stagnation, Creation, Breaking
We consider Liouville-type and partial regularity results for the nonlinear fourth-order problem $$ \Delta^2 u=|u|^{p-1}u\ \{in} \ \R^n,$$ where $ p>1$ and $n\ge1$. We give a complete classification of stable and finite Morse index…
In this note, we show that there exist solutions of the Muskat problem that shift stability regimes: they start unstable, then become stable, and finally return to the unstable regime. We also exhibit numerical evidence of solutions with…
It is known that inhomogeneous second-order macroscopic traffic models can reproduce the phantom traffic jam phenomenon: whenever the sub-characteristic condition is violated, uniform traffic flow is unstable, and small perturbations grow…
We study the existence of monotone heteroclinic traveling waves for the $1$-dimensional reaction-diffusion equation $$ u_t = (| u_x |^{p-2} u_x + | u_x |^{q-2} u_x)_x + f(u), $$ where the non-homogeneous operator appearing on the right-hand…
The Lugiato-Lefever equation is a cubic nonlinear Schr\"odinger equation, including damping, detuning and driving, which arises as a model in nonlinear optics. We study the existence of stationary waves which are found as solutions of a…
This paper has two parts. In the first part, we review stable pairs and triples on curves, leading up to Thaddeus' diagram of flips and contractions starting from the blow-up of projective space along a curve embedded by a complete linear…
We study the Euler equations on a rotating unit sphere, focusing on the dynamics of vortex caps. Leveraging the $L^1$-stability of monotone, longitude-independent profiles, we demonstrate that certain ill-prepared initial data within the…
We study the stable behaviour of discrete dynamical systems where the map is convex and monotone with respect to the standard positive cone. The notion of tangential stability for fixed points and periodic points is introduced, which is…
We review recent developments in structural stability as applied to key topics in general relativity. For a nonlinear dynamical system arising from the Einstein equations by a symmetry reduction, bifurcation theory fully characterizes the…
A brief overview of dimensional reductions for diffeomorphism invariant theories is given. The distinction between the physical idea of compactification and the mathematical problem of a consistent truncation is discussed, and the typical…
This paper investigates the stability of different regions in the $(k,\gamma)$-plane for a class of fractional delay differential equations given by \begin{equation} D^{\alpha} x(t) = -\gamma x(t) + g\big(x(t - \tau_1)\big) - e^{-\gamma…
The phase diagram of a simple area-preserving map, which was motivated by the quantum dynamics of cold atoms, is explored analytically and numerically. Periodic orbits of a given winding ratio are found to exist within wedge-shaped regions…
We obtain a local stable manifold theorem for perturbations of nonautonomous linear difference equations possessing a very general type of nonuniform dichotomy, possibly with different growth rates in the uniform and nonuniform parts. We…
The primary objective of this paper is to introduce Hyers-Ulam-type stability results for monotone, subadditive, and convex graphs. We consider their standard definitions in an approximate sense and demonstrate the existence of a…
We show that a discrete harmonic function which is bounded on a large portion of a periodic planar graph is constant. A key ingredient is a new unique continuation result for the weighted graph Laplacian. The proof relies on the structure…
This paper is a basis of a part of my set of lectures, subject:`Formal methods in solving differential equations and constructing models of physical phenomena'. Addressed, mainly: postgraduates and related readers. Content: a discussion of…
The study of the dynamics of a surface homeomorphism in the neighbourhood of an isolated fixed point leads us to the following results. If the fixed point index is greater than 1, a family of attractive and repulsive petals is constructed,…
This paper concerns piecewise-smooth maps on $\mathbb{R}^d$ that are continuous but not differentiable on switching manifolds (where the functional form of the map changes). The stability of fixed points on switching manifolds is…
Given a positive function F on S n satisfying an appropriate con-vexity assumption, we consider hypersurfaces for which a linear combination of some higher order anisotropic curvatures is constant. We define the varia-tional problem for…
The formation of unstaggered localized modes in dynamical lattices can be supported by the interplay of discreteness and nonlinearity with a finite relaxation time. In rapidly responding nonlinear media, on-site discrete solitons are…