It is shown that, in order to avoid unacceptable nonlocal effects, the free parameters of the general Doebner-Goldin equation have to be chosen such that this nonlinear Schrödinger equation becomes Galilean covariant.

It is shown that, in order to avoid unacceptable nonlocal effects, the free parameters of the general Doebner-Goldin equation have to be chosen such that this nonlinear Schrödinger equation becomes Galilean covariant.

We study existence and stability of homoclinic type solutions of a bistable integral equation. These are stationary solutions of an integrodifferential equation, which is a gradient flow for a free energy functional with general nonlocal integrals penalizing spatial nonuniformity.

In this paper we investigate a nonlocal reaction diffusion equation with potential, under Neumann boundary. We obtain the complete classification of the parameters for which the solution blows up in finite time or exists globally. Moreover, we study the blowup rate and the blowup set for the blowup solution. Key–Words: Nonlocal diffusion, Blow up, Blowup rate, Blowup set

and Applied Analysis 3 He studied the asymptotic behavior of solutions and found the influence of weight function on the existence of global and blow-up solutions. Wang et al. 10 studied porous medium equation with power form source term ut Δu u, x, t ∈ Ω × 0, ∞ , 1.8 subjected to nonlocal boundary condition 1.2 . By virtue of the method of upper-lower solutions, they obtained global existence,...

The essence of the boundary-field equation method is the reduction of the boundary value problem under consideration to an equivalent nonlocal boundary value problem in a bounded domain by using boundary integral equations. The latter can then be treated by the standard variational method including its numerical approximations. In this paper, various formulations of the nonlocal boundary value ...

We investigate the breathing of optical spatial solitons in highly nonlocal media. We use a generalization of the Ehrenfest theorem (1990 Am. J. Phys. 58 742) leading to a fourth-order ordinary differential equation, the latter ruling the beam width evolution in propagation. In actual highly nonlocal materials, the original accessible soliton model by Snyder and Mitchell (1997 Science 276 1538)...

The first and second orders of accuracy difference schemes for the approximate solutions of the nonlocal boundary value problem v ′ (t) +Av(t) = f (t) (0 ≤ t ≤ 1), v(0) = v(λ) + μ, 0 < λ ≤ 1, for differential equation in an arbitrary Banach space E with the strongly positive operator A are considered. The well-posedness of these difference schemes in difference analogues of spaces of smooth fun...