Higher-order breathers as quasi-rogue waves on a periodic background

Abstract

We investigate higher-order breathers of the cubic nonlinear Schrödinger equation on a periodic elliptic background. We find that, beyond first order, any arbitrarily constructed breather on a disordered background generates a single-peaked solitary wave. However, on the periodic backgrounds, the so-called quasi-rogue waves are found more common. These are the quasiperiodic breathers that feature distorted side peaks. We construct such higher-order breathers out of constituent first-order breathers with commensurate periods (i.e., as higher-order harmonic waves). In addition to quasiperiodic, we also find fully periodic breathers, when their wavenumbers are harmonic multiples of the background and each other. But they are truly rare, requiring finely tuned parameters. Thus, on a periodic background, we arrive at the paradoxical conclusion that the higher-order quasi-rogue waves are rather common, while the truly periodic breathers are exceedingly rare.

NonlinearSchrodinger: Higher-Order Algorithms and Darboux Transformations for Nonlinear Schrödinger Equations

“…4.3.2 and 4.3.3, there are so-called maximal intensity families that match the constituent breathers' periods to each other. See [42] for the uniform background case and [54] for the dnoidal background case.…”

“…For the breathers on the dnoidal background, the process is similar but more involved [54]. We start by defining the following function…”

“…There is an analogous but more complicated expression for ν * given in Ref. [54]. However, one must also match the breathers to the background for a truly periodic solution.…”

Rational, semi-rational solution and self-consistent sources extension of the variable-coefficient extended modified Kadomtsev-Petviashvili equation

“…Constructing nonlinear wave solutions on periodic background can help us to understand in-depth the dynamical properties of these solutions. The periodic background has profound impact on the nonlinear waves [50,51]. Solitons, breathers and semirational solutions of integrable system on constant and periodic backgrounds have been intensively studied [52,53].…”