Peak-height formula for higher-order breathers of the nonlinear Schr\"odinger equation on non-uniform backgrounds

TL;DR

This paper derives a universal peak-height formula for higher-order breathers of the nonlinear Schrödinger equation on arbitrary backgrounds, demonstrating that peak heights are additive and independent of background specifics.

Contribution

It proves a universal, background-independent peak-height formula for higher-order breathers generated via Darboux transformation, applicable to various backgrounds without explicit solutions.

Findings

Peak-height of high-order breathers equals sum of first-order peaks plus background.

The formula holds for arbitrary backgrounds, verified on cnoidal solutions.

Generalizations to extended nonlinear Schrödinger equations are suggested.

Abstract

Given any background (or seed) solution of the nonlinear Schr\"odinger 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 Schr\"odinger 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 Schr\"odinger equations are also indicated.