Role of Homoclinic Breathers in the Interpretation of Experimental Measurements, With Emphasis on the Peregrine Breather - Osborne, Alfred R. (Frontiers in Physics, 2022)
A particular class of homoclinic solutions of the nonlinear Schrödinger (NLS) equation in 1+1 dimensions is studied. These solutions are shown to be derivable from the ratio of Riemann theta functions for the genus-2 solutions of the nonlinear Schrödinger equation in the homoclinic limit. The special cases of the well-known homoclinic solutions of the Akhmediev, Peregrine and Kuznetsov-Ma breathers are included: These are often applied as “rogue wave” solutions in scientific and engineering applications in various fields of physics including physical oceanography and nonlinear optics. We show how to determine the general breather solutions of NLS, in terms of Riemann theta functions. We further show how these solutions behave in the homoclinic limit for which a fundamental parameter goes to zero, (such that two points of simple spectrum converge to double points at some particular lambda-plane eigenvalue). The homoclinic solutions cover the entire imaginary axis of the 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. We show that the Peregrine breather separates small-amplitude modulations below the carrier from large-amplitude modulations above the carrier. 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 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). Another infinity of breathers exist along the imaginary axis. I also discuss the genus-1 solutions of NLS as simple dnoidal (Stokes) wave modulations.