A Novel Geometric Realization of the Yajima-Oikawa Equations

TL;DR

This paper reveals that the Yajima-Oikawa equations, modeling wave interactions, can be derived from a geometric flow on curves in the 3-sphere, linking integrable systems with differential geometry.

Contribution

It introduces a new geometric interpretation of the Yajima-Oikawa equations via curves in S^3, providing explicit examples and closure conditions.

Findings

Yajima-Oikawa equations arise from geometric flows on curves in S^3.

Constructed transverse curves corresponding to periodic solutions.

Identified examples with non-trivial topology.

Abstract

We show that the Yajima-Oikawa (YO) equations, a model of short wave-long wave interaction, arise from a simple geometric flow on curves in the 3-dimensional sphere $S^3$ that are transverse to the standard contact structure. For the family of periodic plane wave solutions of the YO equations studied by Wright, we construct the associated transverse curves, derive their closure condition, and exhibit several examples with non-trivial topology.