Dr James Christian J.Christian@salford.ac.uk
Lecturer
Models such as the discrete nonlinear Schr¨odinger (dNLS) and Ablowitz-Ladik (AL) equations play fundamental roles in theoretical physics and applied mathematics. They often govern the dis- crete diffraction of waves travelling through periodic structures in the presence of cubic nonlinearity. The origin of the dNLS equation is firmly rooted in the physically local response of the host medium. On a more abstract level, its AL generalization acquires desirable properties (e.g., exact integrabil- ity) that appear through imposing nonlocal nonlinear coupling between nearest-neighbour channels. Previous research on the dNLS and AL equations has considered their connections in the context of spontaneous pattern formation with ring-cavity feedback. Our presentation here will address counterpropagating waves. The counterpropagation scenario is a type of problem distinct, both physically and mathematically, from the ring cavity. It is of the (1+2) class [rather than (1+1)] and the boundary conditions are more subtle. The simplest case involves equal-intensity plane-wave pump fields; we are primarily concerned with searching for the condition under which a static spatial modulation of finite amplitude (i.e., a Turing pattern) may develop on top of such weakly-perturbed background waves. A derivation of the threshold instability spectra will be given, which requires the exponentiation of a 4×4 system matrix (in comparison with the simpler 2×2 ring-cavity case). Such spectra, obtained by deploying linear analysis, quantify the dominant transverse length-scale characterizing emergent patterns. The long-wavelength asymptotics will also be explored, restoring the continuum limit of the two discrete models. To conclude, our predictions from linearization are tested against numerics.
Presentation Conference Type | Conference Abstract |
---|---|
Conference Name | British Mathematical Colloquium & British Applied Mathematics Colloquium (BMC-BAMC 2025 joint meeting) |
Start Date | Jun 23, 2025 |
End Date | Jun 26, 2025 |
Acceptance Date | Apr 11, 2025 |
Deposit Date | Jun 20, 2025 |
Peer Reviewed | Peer Reviewed |
External URL | https://sites.exeter.ac.uk/bmc-bamc2025/ |
Accepted Version
(41 Kb)
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