Superposition of Solitons with Arbitrary Parameters for Higher-order Equations

Abstract

The way in which solitons propagate and collide is an important theme in various areas of physics. We present a systematic study of the superposition of solitons in systems governed by higher-order equations related to the nonlinear Schrödinger family. We allow for arbitrary amplitudes and relative velocities and include an infinite number of equations in our analysis of collisions and superposed solitons. The formulae we obtain can be useful in determining the influence of subtle effects like higher-order dispersion in optical fibres and small delays in the material responses to imposed impulses.

Two-breather solutions for the class I infinitely extended nonlinear Schrödinger equation and their special cases

“…The degenerate breather solution is a one-parameter family of solutions which represents the collision of two breathers with the same modulation parameter κ, or, equivalently, with equal frequencies. It can be considered a generalisation of the known 2-soliton solution for the class I extension of the nonlinear Schrödinger equation [50].…”

Modulation instability in higher-order nonlinear Schrödinger equations

“…When we go beyond the simple first-order equation, the number of parameters also increases and the dynamics of two-breather interactions becomes more complex. In an effort to understand these complexities, taking several parameters to be equal sometimes reveals explicit features of the breather dynamics, such as their collisions, superpositions and transformations [53][54][55][56].…”