We report a numerical observation that multiplicative random forcing (noise) significantly increases the probability of formation of extreme events in the one-dimensional, focusing nonlinear Schrödinger equation. Furthermore, this phenomenon is sensitive to the noise’s spatial correlation length. Highly correlated multiplicative noise may increase the probability of extreme events even when the...

We investigate the existence of solutions of the fourth order nonlinear elliptic boundary value problem under Dirichlet boundary condition A2u + CAU = bu+ + f in 0, where R is a bounded open set in Rn with smooth boundary and the nonlinearity bu+ crosses eigenvalues of A2 + CA. We also investigate a relation between multiplicity of solutions and source terms of the equation with the nonlinearit...

We derive a pulse propagation equation for a graphene-clad optical fiber, treating the optical response of the graphene and nonlinearity of the dielectric fiber core as perturbations in asymptotic expansion of Maxwell equations. We analyze the effective nonlinear and attenuation coefficients due to the graphene layer. Based on the recent experimental measurements of the nonlinear graphene condu...

We derive pulse propagation equation for a graphene-clad optical fibre, treating optical response of graphene and nonlinearity of the dielectric fibre core as perturbations in asymptotic expansion of Maxwell equations. We analyse the effective nonlinear and attenuation coefficients due to the graphene layer. Based on the recent experimental measurements of the nonlinear graphene conductivity, w...

Mountain pass in a suitable Orlicz space is employed to prove the existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in RN . These equations contain strongly singular nonlinearities which include derivatives of the second order. Such equations have been studied as models of several physical phenomena. The nonlinearity here corresponds to the superf...

We present exact analytical bright and dark (black and grey) solitary wave solutions of a nonlinear Schrödinger-type equation describing the propagation of spatial beams in media exhibiting a saturable nonlinearity (such as centrosymmetric photorefractive materials). A qualitative study of the stationary equation is carried out together with a discussion of the stability of the solutions.

We study the nonlinear Schrödinger equation with general nonlinearity of competing type. This equation have travelling waves with nonvanishing condition at infinity in one dimension. We give a sharp condition for the stability and instability of these solutions. This justifies the previous prediction posed in physical literature.

In this article, we improve the Strichartz estimates obtained in [12] for the Schrödinger equation with white noise dispersion in one dimension. This allows us to prove global well posedness when a quintic critical nonlinearity is added to the equation. We finally show that the white noise dispersion is the limit of smooth random dispersion.

Memory effects in transport require, for their incorporation into reaction-diffusion investigations, a generalization of traditional equations. The well-known Fisher's equation, which combines diffusion with a logistic nonlinearity, is generalized to include memory effects, and traveling wave solutions of the equation are found. Comparison is made with alternative generalization procedures.