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The effect of coherent coupling nonlinearity on modulation instability and rogue wave excitation (2022)
Journal Article
and rogue wave excitation. Communications in Nonlinear Science and Numerical Simulation, 106246. https://doi.org/10.1016/j.cnsns.2021.106246

We study modulation instability (MI) in both anomalous and normal dispersion regimes
of a coherently coupled system. It is found that there exist three types of MI spectra with distinct
characteristics termed baseband, passband, and zero-baseband b... Read More about The effect of coherent coupling nonlinearity on modulation instability and rogue wave excitation.

On the Nth roots of -1 and complex basin boundaries : fractals from Newton-Raphson (2020)
Journal Article
Christian, J., & Middleton-Spencer, H. (2020). On the Nth roots of -1 and complex basin boundaries : fractals from Newton-Raphson. College Mathematics Journal, 51(2), 95-114. https://doi.org/10.1080/07468342.2020.1703452

We consider a systematic generalization of the well-known cube roots of -1 problem to include the Nth roots. The associated fractal basin boundaries are computed, and we also explore how sensitive this class of systems is to fluctuations at its input... Read More about On the Nth roots of -1 and complex basin boundaries : fractals from Newton-Raphson.

Ultrashort nonautonomous similariton solutions and the cascade tunneling of interacting similaritons (2019)
Journal Article
Yang, R., Gao, J., Jia, H., Tian, J., & Christian, J. (2020). Ultrashort nonautonomous similariton solutions and the cascade tunneling of interacting similaritons. Optics Communications, 459, 125025. https://doi.org/10.1016/j.optcom.2019.125025

Similarity transformation and Hirota bilinearization are deployed to derive exact bright and dark ultrashort
one- and two-similariton solutions of a nonautonomous cubic-quintic nonlinear Schrödinger equation. Such wave packets
may emerge when group... Read More about Ultrashort nonautonomous similariton solutions and the cascade tunneling of interacting similaritons.

Using eulerlets to model steady uniform flow past a circular cylinder (2018)
Journal Article
Chadwick, E., Christian, J., & Chalasani, K. (2018). Using eulerlets to model steady uniform flow past a circular cylinder. European Journal of Computational Mechanics, 27(5-6), 469-478. https://doi.org/10.1080/17797179.2018.1550960

Consider uniform, steady flow past a circular cylinder at Reynolds numbers 26, 36 and 40 before the flow becomes unsteady. Model the flow by using eulerlets, new Green’s functions for Euler flow. This is the first time this eulerlet model has been us... Read More about Using eulerlets to model steady uniform flow past a circular cylinder.

Relativistic and pseudorelativistic formulation of nonlinear envelope equations with spatiotemporal dispersion. II. Saturable systems (2018)
Journal Article
Christian, J., McDonald, G., Lundie, M., & Kotsampaseris, A. (2018). Relativistic and pseudorelativistic formulation of nonlinear envelope equations with spatiotemporal dispersion. II. Saturable systems. Physical Review A, 98(5), 053843. https://doi.org/10.1103/PhysRevA.98.053843

We consider an envelope equation with space-time symmetry for describing scalar waves in systems with spatiotemporal dispersion and a generic saturable nonlinearity. Exact bright and gray solitons are derived by direct integration methods and coordin... Read More about Relativistic and pseudorelativistic formulation of nonlinear envelope equations with spatiotemporal dispersion. II. Saturable systems.

Relativistic and pseudorelativistic formulation of nonlinear envelope equations with spatiotemporal dispersion. I. Cubic-quintic systems (2018)
Journal Article
Christian, J., McDonald, G., & Kotsampaseris, A. (2018). Relativistic and pseudorelativistic formulation of nonlinear envelope equations with spatiotemporal dispersion. I. Cubic-quintic systems. Physical Review A, 98(5), 053823. https://doi.org/10.1103/PhysRevA.98.053842

A generic envelope equation is proposed for describing the evolution of scalar pulses in systems with spatiotemporal dispersion and cubic-quintic nonlinearity. Our analysis has application, for instance, in waveguide optics where the physical origin... Read More about Relativistic and pseudorelativistic formulation of nonlinear envelope equations with spatiotemporal dispersion. I. Cubic-quintic systems.

Bistable Helmholtz dark spatial optical solitons in materials with self-defocusing saturable nonlinearity (2017)
Journal Article
Christian, J., & Lundie, M. (2017). Bistable Helmholtz dark spatial optical solitons in materials with self-defocusing saturable nonlinearity. Journal of Nonlinear Optical Physics and Materials, 26(02), https://doi.org/10.1142/S0218863517500242

We present, to the best of our knowledge, the first exact dark spatial solitons of a nonlinear Helmholtz equation with a self-defocusing saturable refractive-index model. These solutions capture oblique (arbitrary-angle) propagation in both the forwa... Read More about Bistable Helmholtz dark spatial optical solitons in materials with self-defocusing saturable nonlinearity.

Spontaneous spatial fractal pattern formation in dispersive systems (2017)
Journal Article
Huang, J., Christian, J., & McDonald, G. (2017). Spontaneous spatial fractal pattern formation in dispersive systems. Journal of Nonlinear Optical Physics and Materials, 26(01), https://doi.org/10.1142/S0218863517500096

We report spontaneous spatial optical fractal patterns in a ring cavity containing a thin slice of diffusive Kerr-type material. The Turing threshold instability condition is derived through linear analysis, and static patterns are found to be descri... Read More about Spontaneous spatial fractal pattern formation in dispersive systems.

Exact dipole solitary wave solution in metamaterials with higher-order dispersion (2016)
Journal Article
Min, X., Yang, R., Tian, J., Xue, W., & Christian, J. (2016). Exact dipole solitary wave solution in metamaterials with higher-order dispersion. Journal of Modern Optics, 63(Sup.3), 544-550. https://doi.org/10.1080/09500340.2016.1185178

We present an exact dipole solitary wave solution in a mutual modulation form of bright and dark
solitons for a higher-order nonlinear Schrödinger equation with third- and fourth-order dispersion
in metamaterials (MMs) using an ansatz method. Based... Read More about Exact dipole solitary wave solution in metamaterials with higher-order dispersion.

Vector dark solitons in systems with spatiotemporal dispersion and cubic nonlinearity : solutions and stability, transformations and relativity (2015)
Presentation / Conference
Carter, L., Christian, J., McDonald, G., & Chamorro-Posada, P. (2015, July). Vector dark solitons in systems with spatiotemporal dispersion and cubic nonlinearity : solutions and stability, transformations and relativity. Presented at 12th International Conference on the Mathematical and Numerical Aspects of Waves, Karlsruhe, Germany

The origin of conventional models for nonlinear optical pulse propagation lies in the ubiquitous slowly-varying envelope approximation (SVEA) accompanied by a Galilean boost to a local-time frame. While such a near-universal procedure typically resul... Read More about Vector dark solitons in systems with spatiotemporal dispersion and cubic nonlinearity : solutions and stability, transformations and relativity.

Diffraction of Weierstrass scalar fractal waves by circular apertures : symmetry and patterns, complexity and dimension (2015)
Presentation / Conference
Christian, J., Woodroofe, E., & McDonald, G. (2015, July). Diffraction of Weierstrass scalar fractal waves by circular apertures : symmetry and patterns, complexity and dimension. Presented at 12th International Conference on the Mathematical and Numerical Aspects of Wave Propagation, Karlsruhe, Germany

The diffraction of plane waves from simple hard-edged apertures constitutes a class of boundary value problem that is well understood in optics, at least within the scalar approximation. Similarly, the diffraction of such waves from fractal apertures... Read More about Diffraction of Weierstrass scalar fractal waves by circular apertures : symmetry and patterns, complexity and dimension.

Unstable resonators with polygon and von Koch-type boundary conditions : virtual source modelling of fractal eigenmodes (2015)
Presentation / Conference
Christian, J., Begleris, I., Wickham, S., McDonald, G., & Huang, J. (2015, July). Unstable resonators with polygon and von Koch-type boundary conditions : virtual source modelling of fractal eigenmodes. Presented at 12th International Conference on the Mathematical and Numerical Aspects of Wave Propagation, Karlsruhe, Germany

We will report on our latest research into modelling fractal lasers (linear systems that involve geometrically-unstable resonators with inherent magnification), and propose two new classes of cavity configuration. These devices are of fundamental the... Read More about Unstable resonators with polygon and von Koch-type boundary conditions : virtual source modelling of fractal eigenmodes.

Nonlinear Helmholtz wave refraction & Goos-Hänchen shifts in nonparaxial optics : angles and interfaces, solitons and Snell's law (2015)
Presentation / Conference
McCoy, E., Christian, J., Sanchez-Curto, J., & McDonald, G. (2015, July). Nonlinear Helmholtz wave refraction & Goos-Hänchen shifts in nonparaxial optics : angles and interfaces, solitons and Snell's law. Presented at 12th International Conference on the Mathematical and Numerical Aspects of Waves, Karlsruhe, Germany

The interaction of self-localized waves with an abrupt interface is a problem of fundamental importance in many branches of physics, engineering, and applied mathematics. Waveguide optics, for instance, is dominated in an essential way by such consid... Read More about Nonlinear Helmholtz wave refraction & Goos-Hänchen shifts in nonparaxial optics : angles and interfaces, solitons and Snell's law.

Multi-Turing instabilities & spontaneous patterns in discrete nonlinear systems : simplicity and complexity, cavities and counterpropagation (2015)
Presentation / Conference
Bostock, C., Christian, J., Leite, A., McDonald, G., & Huang, J. (2015, June). Multi-Turing instabilities & spontaneous patterns in discrete nonlinear systems : simplicity and complexity, cavities and counterpropagation. Presented at 12th International Conference on the Mathematical and Numerical Aspects of Wave Propagation, Karlsruhe, Germany

Alan Turing's profound insight into morphogenesis, published in 1952, has provided the cornerstone for understanding the origin of pattern and form in Nature. When the uniform states of a nonlinear reaction-diffusion system are sufficiently stressed,... Read More about Multi-Turing instabilities & spontaneous patterns in discrete nonlinear systems : simplicity and complexity, cavities and counterpropagation.

Single interfaces and coupled-waveguide arrays: off-axis nonparaxial analyses (2014)
Presentation / Conference
McCoy, E., Christian, J., McDonald, G., Sanchez-Curto, J., & Chamorro-Posada, P. (2014, September). Single interfaces and coupled-waveguide arrays: off-axis nonparaxial analyses. Presented at 5th European Optical Society Annual Meeting, Berlin, Germany

We report on our most recent results concerning arbitrary-angle spatial soliton
refraction at the interface between dissimilar dielectrics, each of which comprises
both X(3)and X(5) susceptibilities. Attention is also paid to the oblique injectio... Read More about Single interfaces and coupled-waveguide arrays: off-axis nonparaxial analyses.

Refraction at interfaces with X(5) nonlinearity: Snell’s law & Goos-Hänchen shifts (2013)
Presentation / Conference
McCoy, E., Christian, J., & McDonald, G. (2013, June). Refraction at interfaces with X(5) nonlinearity: Snell’s law & Goos-Hänchen shifts. Presented at College of Science & Technology Research Showcase, University of Salford

In this presentation, we give the first detailed overview of spatial soliton refraction at the planar interface between materials whose nonlinear polarization has contributions from both X(3)and X(5)susceptibilities [1]. The governing equation is of... Read More about Refraction at interfaces with X(5) nonlinearity: Snell’s law & Goos-Hänchen shifts.

Coupled spatiotemporal waves: New paradigms in vector soliton physics (2013)
Presentation / Conference
Ashley, J. T., Christian, J., & McDonald, G. (2013, June). Coupled spatiotemporal waves: New paradigms in vector soliton physics. Presented at College of Science & Technology Research Showcase, University of Salford

In this presentation, we propose a novel spatiotemporal generalization of Menyuk’s classic equations [1] describing the propagation of two nonlinearly-coupled waves in a dispersive optical system (such as a fibre or planar waveguide). Our approach is... Read More about Coupled spatiotemporal waves: New paradigms in vector soliton physics.