Nonlinear PT-symmetric models bearing exact solutions

Staggered $\mathcal{PT}$-symmetric ladders with cubic nonlinearity

We introduce a ladder-shaped chain with each rung carrying a $\mathcal{PT}$ -symmetric gain-loss dipole. The polarity of the dipoles is staggered along the chain, meaning that a rung bearing gain-loss is followed by one bearing loss-gain. This renders the system $\mathcal{PT}$-symmetric in both horizontal and vertical directions. The system is governed by a pair of linearly coupled discrete nonlinear Schrödinger (DNLS) equations with self-focusing or defocusing cubic onsite nonlinearity. Starting from the analytically tractable anti-continuum limit of uncoupled rungs and using the Newton's met…

量子液滴承载系统中的孤立波

Solitary waves in a quantum droplet-bearing system

本研究通过相平面分析和朗道动力学，揭示了同核对称玻色混合物中量子液滴系统的暗孤子存在性与稳定性，发现单孤子稳定而多孤子存在振荡不稳定窗口。

Solitons and their ghosts in PT-symmetric systems with defocusing nonlinearities

We examine a prototypical nonlinear Schrödinger model bearing a defocusing nonlinearity and Parity-Time (PT) symmetry. For such a model, the solutions can be identified numerically and characterized in the perturbative limit of small gain/loss. There we find two fundamental phenomena. First, the dark solitons that persist in the presence of the PT-symmetric potential are destabilized via a symmetry breaking (pitchfork) bifurcation. Second, the ground state and the dark soliton die hand-in-hand in a saddle-center bifurcation (a nonlinear analogue of the PT-phase transition) at a second critical…

Exactly solvable Wadati potentials in the PT-symmetric Gross-Pitaevskii equation

This note examines Gross-Pitaevskii equations with PT-symmetric potentials of the Wadati type: $V=-W^2+iW_x$. We formulate a recipe for the construction of Wadati potentials supporting exact localised solutions. The general procedure is exemplified by equations with attractive and repulsive cubic nonlinearity bearing a variety of bright and dark solitons.

Some case example exact solutions for quadratically nonlinear optical media with $\mathcal{PT}$-symmetric potentials

In the present paper we consider an optical system with a $χ^{(2)}$-type nonlinearity and unspecified $\mathcal{PT}$-symmetric potential functions. Considering this as an inverse problem and positing a family of exact solutions in terms of cnoidal functions, we solve for the resulting potential functions in a way that ensures the potentials obey the requirements of $\mathcal{PT}$-symmetry. We then focus on case examples of soliton and periodic solutions for which we present a stability analysis as a function of their amplitude parameters. Finally, we numerically explore the nonlinear dynamics …

Localized exact solutions of $\mathcal{PT}$ symmetric nonlinear Schrödinger equation with space and time modulated nonlinearities

Using canonical transformations we obtain localized (in space) exact solutions of the nonlinear Schrödinger equation (NLSE) with space and time modulated nonlinearity and in the presence of an external potential depending on space and time. In particular we obtain exact solutions of NLSE in the presence of a number of non Hermitian $\mathcal{PT}$ symmetric external potentials.