Threshold dynamics for corotational wave maps

Threshold for blowup for equivariant wave maps in higher dimensions

We consider equivariant wave maps from R + d 1 to Sd in supercritical dimensions ⩽ ⩽ d 3 6. Using mixed numerical and analytic methods, we show that the threshold of blowup is given by the codimension-one stable manifold of a self-similar solution with one instability. To probe self-similar blowup, we develop a novel numerical method, based on an adaptive rescaling of coordinates, which may be ...

Complex wave dynamics in microlasers and nonlocal maps

Complex wave dynamics in microlasers and nonlocal maps. Abstract The optical implementation of complex network is considered for laser cavity formed by active mirrors deposited on the surface of plane-parallel slice, ellipsoidal microball , whose eccentricity is close to 0.5 and equivalent optical circuit, containing amplifying and absorbing media in Fourier-conjugated planes. In this types of ...

Wave Maps

is the natural combination of these equations, describing the free motion of an ndimensional surface in a non-Euclidean space; for instance, the motion of a string that is constrained to lie on a sphere would be given (at least to first approximation) as a wave map. As such, wave maps are one of the fundamental equations used to describe geometric motion, although they are not as well understoo...

A Feti Solver for Corotational Nonlinear Problems

Threshold Behavior for Nonlinear Wave Equations

In this brief contribution, which is based on my talk at the conference, I discuss the dynamics of solutions of nonlinear wave equations near the threshold of singularity formation. The heuristic picture of threshold behavior is first presented in a general setting and then illustrated with three examples. Consider an initial value problem du dt = A(u), u(0) = φ ∈ D, (1) where A is a nonlinear ...