We prove that for compatible weakly nonlocal Hamiltonian and symplectic operators, hierarchies of infinitely many commuting local symmetries and conservation laws can be generated under some easily verified conditions no matter whether the generating Nijenhuis operators are weakly nonlocal or not. We construct a recursion operator of the two dimensional periodic Volterra chain from its Lax repr...

The Born–Kirkwood–Onsager ~BKO! model of solvation, where a solute molecule is positioned inside a cavity cut into a solvent, which is considered as a dielectric continuum, is studied within the bounds of nonlocal electrostatics. The nonlocal cavity model is explicitly formulated and the corresponding nonlocal Poisson equation is reduced to an integral equation describing the behavior of the ch...

Direct algebraic method of obtaining exact solutions to nonlinear PDE’s is applied to certain set of nonlinear nonlocal evolutionary equations, including nonlinear telegraph equation, hyperbolic generalization of Burgers equation and some spatially nonlocal hydrodynamic-type model. Special attention is paid to the construction of the kink-like and soliton-like solutions.

This paper is concerned with existence of viscosity solutions of non-translation invariant nonlocal fully nonlinear equations. We construct a discontinuous viscosity solution of such nonlocal equation by Perron’s method. If the equation is uniformly elliptic, we prove the discontinuous viscosity solution is Hölder continuous and thus it is a viscosity solution.

We introduce a nonlocal transformation to generate exact solutions of the constant astigmatism equation zyy + (1/z)xx + 2 = 0. The transformation is related to the special case of the famous Bäcklund transformation of the sine-Gordon equation with the Bäcklund parameter λ = ±1. It is also a nonlocal symmetry.

We deal with symmetry properties for solutions of nonlocal equations of the type (−∆) s v = f (v) in R n , where s ∈ (0, 1) and the operator (−∆) s is the so-called fractional Laplacian. The study of this nonlocal equation is made via a careful analysis of the following degenerate elliptic equation −div (x α ∇u) = 0 on R

This paper deals with a numerical method for the solution of the heat equation with nonlinear nonlocal boundary conditions. Here nonlinear terms are approximated by Richtmyer’s linearization method. The integrals in the boundary equations are approximated by the composite Simpson rule. A difference scheme is considered for the one-dimensional heat equation. In final part, the numerical results ...

In this paper, a continuum model based on the nonlocal elasticity theory is developed for vibration analysis of embedded orthotropic circular and elliptical micro/nano-plates. The nano-plate is bounded by a Pasternak foundation. Governing vibration equation of the nonlocal nano-plate is derived using Nonlocal Classical Plate Theory (NCPT). The weighted residual statement and the Galerkin method...

In this paper, an adaptation of the infinity Laplacian equation to weighted graphs is proposed. This adaptation leads to a nonlocal artial difference equation on graphs, which is an extension of the well-known approximations of the infinity Laplacian equation. o do so, we study the limit as p tends to infinity of minimizers of p-harmonic function on graphs. We also prove the existence and nique...

Two new integrable nonlocal Davey–Stewartson equations are introduced. These equations provide two-spatial dimensional analogues of the integrable, nonlocal nonlinear Schrö-dinger equation introduced in Ablowitz and Musslimani (2013 Phys. Rev. Lett. 110 064105). Furthermore, like the latter equation, they also possess a PT symmetry and, as it is well known, this symmetry is important for the ...