Solitons and unidirectional scattering in quadratically nonlinear optical coupler with parity-time symmetry - Umarov, Bakhram (2019)
We consider the existence and stability of solitons in a χ(2) coupler. Both the fundamental and second harmonics (SHs) undergo gain in one of the coupler cores and are absorbed in the other one. The gain and loss are balanced, creating a parity-time (PT ) symmetric configuration. We present two types of families of PT -symmetric solitons having equal and different profiles of the fundamental and SHs. It is shown that the gain and loss can stabilize solitons. The interaction of stable solitons is shown. In the cascading limit, the model is reduced to the PT -symmetric coupler with effective Kerr-type nonlinearity and the balanced nonlinear gain and loss. An analysis of discrete systems is important for understanding of various physical processes, such as excitations in crystal lattices and molecular chains, the light propagation in waveguide arrays, and the dynamics of Bose-condensate droplets. In basic physical courses, usually the linear properties of discrete systems are studied. We propose a pedagogical introduction to the theory of nonlinear distributed systems. The main ideas and methods are illustrated using a universal model for different physical applications, the discrete nonlinear Schrödinger (DNLS) equation. We consider solutions of the DNLS equation and analyze their linear stability. The notions of nonlinear plane waves, modulational instability, discrete solitons and the anti-continuum limit are introduced and thoroughly discussed. A Mathematica program is provided for better comprehension of results and further exploration. Also, a few problems, extending the topic of the paper, for independent solution are given. The study of Nonlinear Schrödinger Equation has been wide focus from many researchers especially analysing the result of collision as it describes the soliton propagation. This paper considers the soliton scattering of cubic-quintic Nonlinear Schrödinger Equation on localized Gaussian potential. By applying Super Gaussian ansatz as the trial function for variational approximation (VA) method, the soliton interaction may acquire flat-top shape with appropriate parameters. The result of VA will be compared to numerical analysis to check the accuracy of analytical predictions. We consider the extended discrete nonlinear Schrödinger (EDNLSE) equation which includes the nearest neighbor nonlinear interaction in addition to the on-site cubic and quintic nonlinearities. This equation describes nonlinear excitations in dipolar Bose-Einstein condensate in a periodic optical lattice. We are particularly interested with the existence and stability conditions of localized nonlinear excitations of different types. The problem is tackled numerically, by application of Newton methods and by solving the eigenvalue problem for linearized system near the exact solution. Also the modulational instability of plane wave solution is discussed. Spatial solitons are the solutions of nonlinear partial differential equations describing the propagation of optical beams in nonlinear medium. This paper studies the scattering of a spatial solitons of the Cubic-Quintic Nonlinear Schrödinger Equation (C-Q NLSE) on an interface between two nonlinear media. The scattering process will be investigated by variational approximation method and by direct numerical solution of C-Q NLSE. This variational approximation method has been used to analyse the dynamic of the width and center of mass position of a soliton during the scattering process. Meanwhile, a direct numerical simulation of C-Q NLSE was done to check the accuracy of the approximation by using the same range of parameters and initial condition. The results for direct numerical simulation of CQNLSE for soliton parameters are quite similar with the variational equation. The studies showed that soliton can be reflected by or transmitted through the interface, also the nonlinear surface wave can be formed depending on the parameters of interface and initial soliton.