Abstract We study a nonlocal parametric problem driven by the fractional Laplacian operator combined with Kirchhoff-type coefficient and involving critical nonlinearity term in Sobolev embedding sense. Our approach is of variational topological nature. The obtained results can be viewed as nontrivial extension to setting some recent contributions already present literature.

This paper is devoted to study a fractional Choquard problem with slightly subcritical exponents on bounded domains. When the exponent of convolution type nonlinearity tends critical one in sense Hardy–Littlewood–Sobolev inequality, we obtain existence multiple positive solutions via Lusternik–Schnirelmann category and nonlocal global compactness. Moreover, prove that topology domain furnishes ...

It is shown that ambipolar diffusion as a useful model for nonlinearity leads to similar behaviour of large scale turbulent dynamos as full MHD. This is demonstrated using both direct simulations in a periodic box and a closure model for the magnetic correlation functions applicable to infinite space. Large scale fields develop via a nonlocal inverse cascade as described by the α-effect. Howeve...

We present a theoretical and experimental study of the propagation, decay, and interaction of optical vortices in media with an anisotropic nonlocal nonlinearity. The initial stage of decay of a circular vortex is characterized by charge dependent rotation, and stretching of the vortex. Our results suggest that a compact vortex of unit topological charge cannot exist in such media, but that a c...

We find self-accelerating beams in highly nonlocal nonlinear optical media, and show that their propagation dynamics is strongly affected by boundary conditions. Specifically for the thermal optical nonlinearity, the boundary conditions have a strong impact on the beam trajectory: they can increase the acceleration during propagation, or even cause beam bending in a direction opposite to the in...

We consider classical and entanglement-assisted versions of a distributed computation scheme that computes nonlinear Boolean functions of a set of input bits supplied by separated parties. Communication between the parties is restricted to take place through a specific apparatus which enforces the constraints that all nonlinear, nonlocal classical logic is performed by a single receiver, and th...

A semiglobal chaotification problem of n-dimensional polynomial continuous-time systems satisfying one special condition is examined. This problem is solved in two steps. Firstly, by using some polynomial mapping we transform the initial system to some n-th order explicit scalar ordinary differential equation with a rational nonlinearity. On the second step we apply the anticontrol feedback pro...

In contrast to an infinite degree of nonlocality, we demonstrate that vortex solitons in nonlinear media under competing self-focusing cubic and self-defocusing quintic nonlocal nonlinearities can be stabilized with a finite degree of nonlocality. Stable vortex solitons in the upper branch, bifurcated from the competing cubic-quintic nonlinearities, are found to be supported when the original d...

The Fokker–Planck equation with diffusion coefficient quadratic in space variable, linear drift coefficient, and nonlocal nonlinearity term is considered in the framework of a model of analysis of asset returns at financial markets. For special cases of such a Fokker– Planck equation we describe a construction of exact solution of the Cauchy problem. In the general case, we construct the leadin...

Existence and uniqueness of mild and classical solutions are discussed for an abstract second-order evolution problem. The nonlinearity contains a local term and a non-local term. The non-local term is an integral in the form of a convolution of a singular kernel and a regular function involving fractional derivatives. This term may be regarded also as a fractional integral of that regular func...