NLS breathers, rogue waves, and solutions of the Lyapunov equation for Jordan blocks

2016/09/30 by Oleksandr Chvartatskyi, Folkert Müller-Hoissen
Mathematics · Physics and Astronomy · #Advanced Fiber Laser Technologies #Breather #Eigenvalues and eigenvectors #Jordan matrix #Lyapunov equation #Lyapunov function #Mathematical analysis #Mathematical physics #Mathematics #Matrix (chemical analysis) #Nonlinear Photonic Systems #Nonlinear Schrödinger equation #Nonlinear Waves and Solitons #Nonlinear system #Physics #Quantum mechanics #Rogue wave #Schrödinger equation #Transformation (genetics) #math-ph #math.MP #nlin.SI

paper · pdf · doi:10.1088/1751-8121/aa6185

18 pages, 5 figures, second and third version: Remark 2.3 added, some minor corrections. To appear in J. Phys. A: Math. Theor

arxiv created 2017/03/07 · openalex publication_date 2017/03/14 · arxiv updated 2017/04/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Abstract The infinite families of Peregrine, Akhmediev and Kuznetsov–Ma breather solutions of the focusing nonlinear Schrödinger (NLS) equation are obtained via a matrix version of the Darboux transformation, with a spectral matrix of the form of a Jordan block. The structure of these solutions is essentially determined by the corresponding solution of the Lyapunov equation. In particular, regularity follows from properties of the Lyapunov equation.

Citations

• Soliton solutions for ABS lattice equations: I. Cauchy matrix approach
• Tenth Peregrine breather solution to the NLS equation
• Modulation instability and periodic solutions of the nonlinear Schrödinger equation
• A Power Method for Computing Square Roots of Complex Matrices
• The Perturbed Plane‐Wave Solutions of the Cubic Schrödinger Equation

Cited by

• Miura transformation in bidifferential calculus and a vectorial Darboux transformation for the Fokas-Lenells equation