Role of Homoclinic Breathers in the Interpretation of Experimental Measurements, With Emphasis on the Peregrine Breather

Abstract

A class of generalized homoclinic solutions of the nonlinear Schrödinger (NLS) equation in 1+1 dimensions is studied. These are homoclinic breathers that are shown to be derivable from the ratio of Riemann theta functions for the genus-2 solutions of the nonlinear Schrödinger equation. We discuss how these solutions behave in the homoclinic limit for which a fundamental parameter ε goes to zero, ε→0 (such that two points of simple spectrum converge to double points at some particular lambda-plane eigenvalue). The homoclinic solutions cover the entire lambda plane (the Riemann surface of the NLS equation) and are given in terms of simple trigonometric functions. When the spectral eigenvalues converge to the carrier amplitude in the lambda plane we have the Peregrine breather. While the Peregrine solution is often called a soliton, it is in reality a breather, albeit occurring at the “singular point” corresponding to the carrier eigenvalue in the lambda plane and consequently “breathes” only once in its lifetime. The Peregrine breather separates small-amplitude modulations below the carrier from large amplitude modulation above the carrier. This fact means that the Peregrine breather has a “central” role in the lambda plane characterization of the NLS nonlinear spectrum. The Akhmediev breather occurs somewhat below the carrier (and is therefore a small-amplitude modulation) and the Kuznetsov-Ma breather occurs above the carrier (and is therefore a large-amplitude modulation). The general homoclinic solutions can be constructed everywhere in the lambda plane and are shown to be a useful tool to interpret the nonlinear Fourier spectrum of space and time series recorded in the laboratory and ocean environment. Nonlinear filtering is suggested as a way to extract breather trains from experimental time series. The generalized homoclinic breathers can be thought of as “extreme wave packets” or “rogue wave” solutions of water waves for scientific and engineering applications in various fields of physics including physical oceanography and nonlinear optics.