Related papers: Explicit breather solution of the nonlinear Schr\"…
The spectrum of the Quantum Discrete Nonlinear Schr\"odinger equation on a periodic 1D lattice shows some interesting detailed band structure which may be interpreted as the quantum signature of a two-breather interaction in the classical…
We consider the cubic nonlinear Schr\"odinger equation with long-range linear potentials in one space dimension, and prove the modified scattering in the energy space for the associated final state problem with a prescribed small asymptotic…
In this paper, we present a simple analytical method for obtaining a nonspreading solution of the time-dependent Schr\"odinger equation, which is given by the Airy function. The solution is derived by imposing a restriction on the phase…
It is shown that the tight-binding approximation of the nonlinear Schr\"odinger equation with a periodic linear potential and periodic in space nonlinearity coefficient gives rise to a number of nonlinear lattices with complex, both linear…
We consider the question of existence of periodic solutions (called breather solutions or discrete solitons) for the Discrete Nonlinear Schr\"odinger Equation with saturable and power nonlinearity. Theoretical and numerical results are…
Using similarity transformations we construct explicit nontrivial solutions of nonlinear Schr\"odinger equations with potentials and nonlinearities depending on time and on the spatial coordinates. We present the general theory and use it…
Nonlinear lattice models can support "discrete breather" excitations that stay localized in space for all time. By contrast, the localized Wannier states of linear lattice models are dynamically unstable. Nevertheless, symmetric and…
We point out that the nonlinear Schr{\"o}dinger lattice with a saturable nonlinearity also admits staggered periodic as well as localized pulse-like solutions. Further, the same model also admits solutions with a short period. We examine…
We report the first experimental observation of breather gases (BGs) in optics, realized in a recirculating fiber loop enabling virtually lossless propagation over $1200$ km. Initiated by a slowly modulated optical background perturbed by…
In many physical contexts, notably including deep water waves, modulation instability in one space dimension is often studied using the nonlinear Schr\"odinger equation. The principal solutions of interest are solitons and breathers which…
We consider a derivative nonlinear Schr\"odinger equation with a general nonlinearity. This equation has a two parameter family of solitary wave solutions. We prove orbital stability/instability results that depend on the strength of the…
We consider the $(1+1)$-dimensional quasilinear wave equation $g(x)w_{tt}-w_{xx}+h(x) (w_t^3)_t=0$ on $\mathbb{R}\times\mathbb{R}$ which arises in the study of localized electromagnetic waves modeled by Kerr-nonlinear Maxwell equations. We…
We consider a two-dimensional Fermi-Pasta-Ulam (FPU) lattice with hexagonal symmetry. Using asymptotic methods based on small amplitude ansatz, at third order we obtain a reduction to a cubic nonlinear Schrodinger equation (NLS) for the…
We consider the defocusing nonlinear Schr{\"o}dinger equation in several space dimensions, in the presence of an external potential depending on only one space vari-able. This potential is bounded from below, and may grow arbitrarily fast…
We prove the existence of a class of time-localized and space-periodic breathers (called q-gap breathers) in nonlinear lattices with time-periodic coefficients. These q-gap breathers are the counterparts to the classical space-localized and…
In this chapter, we discuss experiments that realize the discrete nonlinear Schr\"odinger (DNLS) equations. The relevance of such descriptions arises from the competition of three common features: nonlinearity, dispersion, and a medium to…
We obtain exact moving and stationary, spatially periodic and localized solutions of a generalized discrete nonlinear Schr\"odinger equation. More specifically, we find two different moving periodic wave solutions and a localized moving…
We consider travelling periodic and quasiperiodic wave solutions of a set of coupled nonlinear Schr\"odimger equations. In fibre optics these equations can be used to model single mode fibers with strong birefringence and two-mode optical…
In this paper the one-dimensional nonparaxial nonlinear Schr\"odinger equation is considered. This was proposed as an alternative to the classical nonlinear Schr\"odinger equation in those situations where the assumption of paraxiality may…
We consider the cubic nonlinear Schrodinger equation with a potential in one space dimension. Under the assumptions that the potential is generic, sufficiently localized, and does not have bound states, we obtain the long time asymptotic…