General rogue waves in the nonlocal PT-symmetric nonlinear Schrodinger equation

Abstract

Rogue waves in the nonlocal PT-symmetric nonlinear Schrodinger (NLS) equation are studied by Darboux transformation. Three types of rogue waves are derived, and their explicit expressions in terms of Schur polynomials are presented. These rogue waves show a much wider variety than those in the local NLS equation. For instance, the polynomial degrees of their denominators can be not only $n(n+1)$, but also $n(n-1)+1$ and $n^2$, where $n$ is an arbitrary positive integer. Dynamics of these rogue waves is also examined. It is shown that these rogue waves can be bounded for all space and time, or develop collapsing singularities, depending on their types as well as values of their free parameters. In addition, the solution dynamics exhibits rich patterns, most of which have no counterparts in the local NLS equation.

24 pages, 9 figures

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• Long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation with step-like initial data
• Integrable nonlocal asymptotic reductions of physically significant nonlinear equations
• Solutions to integrable space-time shifted nonlocal equations
• General stationary solutions of the nonlocal nonlinear Schrödinger equation and their relevance to the PT-symmetric system
• Rogue Waves in (2+1)-Dimensional Three-Wave Resonant Interactions
• Curved wedges in the long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation
• Asymptotic dynamics of higher-order lumps in the Davey-Stewartson II equation
• Rogue Waves and Periodic Solutions of a Nonlocal Nonlinear Schrödinger Model
• Asymptotic stage of modulation instability for the nonlocal nonlinear Schrödinger equation
• Defocusing nonlocal nonlinear Schrödinger equation with step-like boundary conditions: long-time behavior for shifted initial data
• Characteristics of rogue waves in the scalar and vector nonlocal nonlinear Schrödinger equations