Quasiperiodic Solutions of the Fibre Optics Coupled Nonlinear Schr{\"o}dinger Equations

TL;DR

This paper investigates quasiperiodic wave solutions in birefringent optical fibers modeled by coupled nonlinear Schrödinger equations, introducing new solutions linked to integrable potentials and expressing them via theta functions.

Contribution

It provides a general method for finding elliptic solutions and introduces new quasiperiodic solutions associated with two-gap Treibich-Verdier potentials.

Findings

Derived explicit quasiperiodic solutions in terms of theta functions.

Connected solutions to integrable two-particle quartic potentials.

Discussed reduction of solutions to elliptic functions.

Abstract

We consider travelling periodical and quasiperiodical waves in single mode fibres, with weak birefringence and under the action of cross-phase modulation. The problem is reduced to the ``1:2:1" integrable case of the two-particle quartic potential. A general approach for finding elliptic solutions is given. New solutions which are associated with two-gap Treibich-Verdier potentials are found. General quasiperiodic solutions are given in terms of two dimensional theta functions with explicit expressions for frequencies in terms of theta constants. The reduction of quasiperiodic solutions to elliptic functions is discussed.