Dark solitons and localized defect modes against periodic backgrounds are considered in arrays of waveguides with defocusing Kerr nonlinearity constituting a nonlinear lattice. Bright supported by local increase the nonlinearity, while dark decrease nonlinearity. exist for both types defect, although case weak they feature side bright humps making total energy propagating through system larger ...

We analyze the formation of one-dimensional localized patterns in a nonlinear dissipative medium including a set of two narrow "hot spots" (HSs), which carry the linear gain, local potential, cubic self-interaction, and cubic loss, while the linear loss acts in the host medium. This system can be realized as a spatial-domain one in optics and also in Bose-Einstein condensates of quasiparticles ...

In this article, we consider a combination of local and nonlocal Laplace equation with singular nonlinearities. For such mixed problems, establish the existence at least one weak solution for parameter-dependent nonlinearity multiple solutions perturbed nonlinearity. Our argument is based on variational approximation approach.

The effective linear and third-order nonlinear susceptibility of graded metallic films with weak nonlinearity have been investigated. Due to the simple geometry, we were able to derive exactly the local field inside the graded structures having a Drude dielectric gradation profile. We calculated the effective linear dielectric constant and third-order nonlinear susceptibility. We investigated t...