Related papers: Non-Nearest-Neighbor Interactions in Nonlinear Dyn…
The question of whether features and behaviors that are characteristic to completely integrable systems persist in the transition to non-integrable settings is a central one in the field of nonlinear dispersive equations. In this work, we…
Considering the static solutions of the D-dimensional nonlinear Schroedinger equation with trap and attractive two-body interactions, the existence of stable solutions is limited to a maximum critical number of particles, when D is greater…
We consider a nonlinear chain of coupled oscillators, which is a direct generalization of the classical FPU lattice and exhibits, besides the usual nearest neighbor interaction, also next-to-nearest neighbor interaction. For the case of…
Using similarity transformations we construct explicit solutions of the nonlinear Schrodinger equation with linear and nonlinear periodic potentials. We present explicit forms of spatially localized and periodic solutions, and study their…
In recent years, there has been considerable interest in the study of wave propagation in nonlinear photonic lattices. The interplay between nonlinearity and periodicity has led researchers to manipulate light and discover new and…
We elaborate a fractional discrete nonlinear Schr\"{o}dinger (FDNLS) equation based on an appropriately modified definition of the Riesz fractional derivative, which is characterized by its L\'{e}vy index (LI). This FDNLS equation…
We consider the modulational instability of nonlinearly interacting two-dimensional waves in deep water, which are described by a pair of two-dimensional coupled nonlinear Schroedinger equations. We derive a nonlinear dispersion relation.…
In the present work we examine both the linear and nonlinear properties of two related PT-symmetric systems of the discrete nonlinear Schrodinger (dNLS) type. First, we examine the parameter range for which the finite PT-dNLS chains have…
From the mathematical side, nonlinear Schr\"odinger equations are usually investigated via variational methods, that cease to work in higher dimensions. This thesis tries to overcome this problem by focusing on spherically symmetric…
An overview is given of basic models combining discreteness in their linear parts (i.e. the models are built as dynamical lattices) and nonlinearity acting at sites of the lattices or between the sites. The considered systems include the…
We address the issue of mobility of localized modes in two-dimensional nonlinear Schr\"odinger lattices with saturable nonlinearity. This describes e.g. discrete spatial solitons in a tight-binding approximation of two-dimensional optical…
We introduce a versatile platform for studying nonlinear out-of-equilibrium physics. The platform is based on a slow light setup where an optical waveguide is interfaced with cold atoms to realize the driven nonlinear Schr\"odinger equation…
We consider a one-dimensional discrete nonlinear Schr{\"o}dinger (dNLS) model featuring interactions beyond nearest neighbors. We are interested in the existence (or nonexistence) of phase-shift discrete solitons, which correspond to…
Discrete solitons of the discrete nonlinear Schr\"odinger (dNLS) equation become compactly supported in the anti-continuum limit of the zero coupling between lattice sites. Eigenvalues of the linearization of the dNLS equation at the…
We study the occurrence of modulational instabilities in lattices with non-local, power-law hoppings and interactions. Choosing as a case study the discrete nonlinear Schr\"odinger equation, we consider one-dimensional chains with power-law…
We consider nonlinear dynamics in a finite parity-time-symmetric chain of the discrete nonlinear Schr{\"o}dinger (dNLS) type. We work in the range of the gain and loss coefficient when the zero equilibrium state is neutrally stable. We…
One dimensional nonlinear lattices with harmonic long range interaction potentials (LRIP) having an inverse power kernel type, are studied. For the nearest neighbour nonlinear interaction we consider the anharmonic potential of the…
While the Ablowitz-Ladik lattice is integrable, the Discrete Nonlinear Schr\"odinger equation, which is more significant for physical applications, is not. We prove closeness of the solutions of both systems in the sense of a "continuous…
We study the effect of long-range interactions in non-convex one-dimensional lattice systems in the simplified yet meaningful assumption that the relevant long-range interactions are between $M$-neighbours for some $M\ge 2$ and are convex.…
In this work, we investigate the existence and orbital (in)stability of several branches of standing--wave solutions for the cubic nonlinear Schr\"odinger equation (NLS) posed on a looping--edge graph $\mathcal{G}$, consisting of a circle…