Subcritical nonlinear heat equation

Singular solutions of the subcritical nonlinear Schrödinger equation

where x = (x1, . . . , xd) ∈ Rd and ∆ = ∑d j=1 ∂xjxj is the Laplacian, has been the subject of intensive study, due to its role in various areas of physics, such as nonlinear optics and Bose–Einstein condensates (BEC). The NLS is called subcritical, critical, and supercritical if σd < 2, σd = 2, and σd > 2, respectively. It is well-known that in the critical and supercritical cases, the NLS (1)...

Capillary waves in the subcritical nonlinear Schrödinger equation

In a recent article, Novoa, Michinel, and Tomasini derived a Young-Laplace equation to describe a certain class of stationary solutions of the nonlinear Schrödinger equation (NLS) with cubic and quintic nonlinearities [1]. Light beams, therefore, can sometimes take on the attribute of a liquid when propagating in nonlinear media. This occasional liquidlike nature of light was noted by several a...

Null Controllability of a Nonlinear Heat Equation

Subcritical Hopf bifurcation in dynamical systems described by a scalar nonlinear delay differential equation.

A subcritical Hopf bifurcation in a dynamical system modeled by a scalar nonlinear delay differential equation is studied theoretically and experimentally. The subcritical Hopf bifurcation leads to a significant domain of bistability where stable steady and time-periodic states coexist.

On the stability of ground states in 4D and 5D nonlinear Schrödinger equation including subcritical cases

We consider a class of nonlinear Schrödinger equation in four and five space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in L) nonlinearities. We show that the center manifold formed by localized in space periodic in time solutions (bound states) is an attractor for all solutions with a...