Vector cnoidal waves in spatiotemporal propagation: exact solutions beyond slowly-varying envelopes

Christian, JM; McAteer, ÉN

Abstract

Cnoidal waves are periodic solutions to certain classes of nonlinear partial differential equations
(PDEs). They are fundamental excitations in theories of waves and their more familiar localized
counterparts––solitons––typically emerge as limits. Here, attention is focused primarily on the
context of nonlinear optical waves confined to two-dimensional waveguides.
We consider an extension of the scalar spatiotemporal propagation model with cubic nonlinearity [J.
M. Christian et al., Phys. Rev. A vol. 86, art. nos. 023838 and 023839 (2012)] into two-component
vector regimes. The model retains the double-z derivative terms, where z denotes the longitudinal
space coordinate, and waves are most conveniently described in the laboratory frame of reference
rather than in a local-time frame. It captures the spatial dispersion effect present in some novel
semiconductor materials, and it also possesses inherent bidirectionality (where both forward- and
backward-propagating solutions are supported). Our spatiotemporal formulation is thus rather general
in nature, going well beyond the slowly-varying envelope approximation (SVEA) as prescribed so
widely in the literature by nonlinear Schrödinger-type equations. Moreover, it has an instructive
geometrical representation in the space-time plane that is tightly connected to coordinate
transformations akin to those routinely encountered in special relativity.
Despite the complexity of the model––two coupled cubically-nonlinear PDEs that are fully secondorder in (longitudinal) space and time coordinates, alongside an eight-dimensional parameter space––
it is still possible to derive families of exact analytical vector cnoidal waves. These solutions involve
Jacobi elliptic functions of which there are just three types: elliptic cosine (cnoidal, or cn), elliptic sine
(snoidal, or sn), and delta amplitude (dnoidal, or dn). There are six fundamental combinations that
need to be considered: three composed of similar elements (cn-cn, dn-dn, sn-sn), and three composed
of mixed elements (cn-dn, cn-sn, dn-sn).
The presentation will survey our new spatiotemporal vector cnoidal wave solutions (obtained by
deploying an ansatz method) and show that their existence is contingent upon two complementary
conditions being satisfied simultaneously. Those auxiliary criteria demand that the components have
equal velocities and that their overlap in the space-time plane be optimized. The double-z derivatives
endow the solutions with a set of corrections to the conventional predictions for propagation
constants, velocities, and pulse widths. As required physically and mathematically, such corrections
vanish in a multiple limit indicating that predictions made under the ‘SVEA + local-time’ paradigm
are, in essence, special cases. We will conclude by demonstrating the asymptotic recovery of
corresponding vector solitons.

Citation

Christian, J., & McAteer, É. (2023, April). Vector cnoidal waves in spatiotemporal propagation: exact solutions beyond slowly-varying envelopes. Presented at 64th British Applied Mathematics Colloquium (BAMC 2023), University of the West of England and University of Bristol, UK

• Presentation Conference Type: Lecture
• Conference Name: 64th British Applied Mathematics Colloquium (BAMC 2023)
• Conference Location: University of the West of England and University of Bristol, UK
• Start Date: Apr 3, 2023
• End Date: Apr 5, 2023
• Acceptance Date: Feb 23, 2023
• Publication Date: Apr 5, 2023
• Deposit Date: Apr 11, 2023
• Publicly Available Date: Apr 11, 2023
• Publisher URL: https://rise.articulate.com/share/m_8PV5egFfp51rbRhApK6GtC3ZkRGmpN#/
• Related Public URLs: https://www.uwe.ac.uk/events/british-applied-mathematics-colloquium-bristol-2023
• Additional Information: Event Type : Conference