Peak-height formula for higher-order breathers of the nonlinear Schrödinger equation on non-uniform backgrounds

Abstract

Given any background (or seed) solution of the nonlinear Schrödinger equation, the Darboux transformation can be used to generate higher-order breathers with much greater peak intensities. The Darboux transformation is generic in iterating a pair of generating solutions of the Lax-pair equation which preserves the solution of the nonlinear Schrödinger equation as a consistency condition. The Darboux transformation itself knows nothing about the background solution except through the initial pair of Lax solutions. Because of this, in this work, we can prove in an unified manner, and without knowing the analytical form of the background solution, that the peak-height of a high-order breather, is just a sum of peak-heights of first-order breathers plus that of the background, {\it irrespective} of the specific choice of the background. Detailed results are verified for breathers on a cnoidal background. Generalizations to more extended nonlinear Schrödinger equations are also indicated.

arXiv comment19 pages and 6 figures, revised to include "Pythagorean triplet" of rogue waves

References in corpus (3)

• Instabilities, breathers and rogue waves in optics
• Three-dimensional rogue waves in non-stationary parabolic potentials
• Maximal intensity higher-order Akhmediev breathers of the nonlinear Schrodinger equation and their systematic generation

Cited by in corpus (3)

• The height of an $n$th-order fundamental rogue wave for the nonlinear Schrödinger equation
• Breathers and solitons on two different backgrounds in a generalized coupled Hirota system with four wave mixing
• NonlinearSchrodinger: Higher-Order Algorithms and Darboux Transformations for Nonlinear Schrödinger Equations