Related papers: Some case example exact solutions for quadraticall…
Multipolaron solutions were studied in the framework of the Holstein one-dimensional molecular crystal model. The study was performed in the continuous limit where the crystal model maps into the nonlinear Schr\"odinger equation for which a…
We propose an integrable system of coupled nonlinear Schrodinger equations with cubic-quintic terms describing the effects of quintic nonlinearity on the ultra-short optical soliton pulse propagation in non-Kerr media. Lax pair, conserved…
Convergence to stationary solutions in fully nonlinear parabolic systems with general nonlinear boundary conditions is shown in situations where the set of stationary solutions creates a $C^2$-manifold of finite dimension which is normally…
We generalize a finite parity-time (${\cal PT}$-) symmetric network of the discrete nonlinear Schr\"odinger type and obtain general results on linear stability of the zero equilibrium, on the nonlinear dynamics of the dimer model, as well…
We consider the nonlinear Schr\"odinger equation with a focusing cubic term and a defocusing quintic nonlinearity in dimensions two and three. The core of this article is the notion of stability of solitary waves. We recall the two standard…
We propose a lattice model, in both one- and multidimensional versions, which may give rise to matching conditions necessary for the generation of solitons through the second-harmonic generation. The model describes an array of linearly…
We report that defocusing cubic media with spatially inhomogeneous nonlinearity, whose strength increases rapidly enough toward the periphery, can support stable bright localized modes. Such nonlinearity landscapes give rise to a variety of…
We consider a $\mathcal{PT}$-symmetric chain (ladder-shaped) system governed by the discrete nonlinear Schr\"odinger equation where the cubic nonlinearity is carried solely by two central "rungs" of the ladder. Two branches of scattering…
We study extensions of N-wave systems with PT-symmetry. The types of (nonlocal) reductions leading to integrable equations invariant with respect to P- (spatial reflection) and T- (time reversal) symmetries is described. The corresponding…
Management of solitons in media with competing quadratic and cubic nonlinearities is investigated. Two schemes, using rapid modulations of a mismatch parameter, and of the Kerr nonlinearity parameter are studied. For both cases, the…
We present a unified approach for qualitative and quantitative analysis of stability and instability dynamics of positive bright solitons in multi-dimensional focusing nonlinear media with a potential (lattice), which can be periodic,…
We provide a systematic analysis of a prototypical nonlinear oscillator system respecting PT-symmetry i.e., one of them has gain and the other an equal and opposite amount of loss. Starting from the linear limit of the system, we extend…
We study stability and dynamics of the single cylindrically symmetric solitary structures and dipolar solitonic molecules in spatially nonlocal media. The main properties of the solitons, vortex solitons, and dipolar solitons are…
The time-independent nonlinear Schr\"odinger equation is solved for two attractive delta-function shaped potential wells where an imaginary loss term is added in one well, and a gain term of the same size but with opposite sign in the…
It has been recently demonstrated that self-defocusing (SDF) media with the cubic nonlinearity, whose local coefficient grows from the center to periphery fast enough, support stable bright solitons, without the use of any linear potential.…
By employing supersymmetric quantum mechanics, we present a general algorithm to construct supersymmetric partner potentials and hence derive exact stationary solutions of the inhomogeneous nonlinear Schr\"odinger equation (INLSE). This is…
We study the integrable nonlocal nonlinear Schr\"odinger equation proposed by Ablowitz and Musslimani, that is considered as a particular example of equations with parity-time ($\mathcal{PT}$) symmetric self-induced potential. We consider…
We prove the existence of travelling-wave solutions for a system of coupled nonlinear Schr\"{o}dinger equations arising in nonlinear optics. Such a system describes second-harmonic generation in optical materials with $\chi^{(2)}$…
We introduce a novel class of parametric optical solitons supported simultaneously by two second-order nonlinear cascading processes, second-harmonic generation and sum-frequency mixing. We obtain, analytically and numerically, the…
We consider the nonlinear Schr{\"o}dinger equation (NLSE) in 1+1 dimension with scalar-scalar self interaction $\frac{g^2}{\kappa+1} (\psi^\star \psi)^{\kappa+1}$ in the presence of the external forcing terms of the form $r e^{-i(kx +…