On the variation of of bi-periodic waves in the transverse direction

TL;DR

This study investigates bi-periodic wave patterns with amplitude variation in the transverse direction using nonlinear Schrödinger equations, showing that elliptic function solutions provide more accurate predictions than Stokes-like solutions.

Contribution

It compares vector and scalar NLSE solutions for bi-periodic wave patterns and demonstrates the superior accuracy of elliptic function solutions in predicting amplitude variation.

Findings

Elliptic function solutions have less error than Stokes-like solutions.

No evidence of instability growth in the x-direction was observed.

Including a third harmonic term improves the vNLSE solution accuracy.

Abstract

Bi-periodic patterns of waves that propagate in the x direction with amplitude variation in the y direction are generated in a laboratory. The amplitude variation in the y direction is studied within the framework of the vector (vNLSE) and scalar (sNLSE) nonlinear Schrodinger equations using the uniform-amplitude, Stokes-like solution of the vNLSE and the Jacobi elliptic sine function solution of the sNLSE. The wavetrains are generated using the Stokes-like solution of vNLSE; however, a comparison of both predictions shows that while they both do a reasonably good job of predicting the observed amplitude variation in y, the comparison with the elliptic function solution of the sNLSE has significantly less error. Additionally, for agreement with the vNLSE solution, a third harmonic in y term from a Stokes-type expansion of interacting, symmetric wavetrains must be included. There is no evidence of instability growth in the x-direction, consistent with the work of Segur and colleagues, who showed that dissipation stabilizes the modulational instability. There is some extra amplitude variation in y, which is examined via a qualitative stability calculation that allows symmetry breaking in that direction.