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Dr James Christian's Outputs (118)

Dynamics and chaos in extensible pendulum systems (2024)
Presentation / Conference
Jafari, M., & Christian, J. (2024, March). Dynamics and chaos in extensible pendulum systems. Paper presented at 11th Undergraduate Mathematics Conference "Tomorrow's Mathematicans Today" (TMT 2024), University of Greenwich (hosted online), UK

Dynamics on ₵ with generalized Newton-Raphson maps: Julia sets, uncertainty dimension, and periodic orbits (2023)
Presentation / Conference
Christian, J., & Jensen, G. (2023, April). Dynamics on ₵ with generalized Newton-Raphson maps: Julia sets, uncertainty dimension, and periodic orbits. Poster presented at 64th British Applied Mathematics Colloquium (BAMC 2023), University of the West of England and University of Bristol, UK

The Newton-Raphson (NR) method is a well-known iterative scheme for approximating the roots of
functions. Deployed on the complex plane, ₵, perhaps its most famous application is to finding the
cube roots of –1. One often regards any specific outco... Read More about Dynamics on ₵ with generalized Newton-Raphson maps: Julia sets, uncertainty dimension, and periodic orbits.

Vector cnoidal waves in spatiotemporal propagation: exact solutions beyond slowly-varying envelopes (2023)
Presentation / Conference
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

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.... Read More about Vector cnoidal waves in spatiotemporal propagation: exact solutions beyond slowly-varying envelopes.

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.

Dark & anti-dark spatiotemporal solitons: From cubic to cubic-quintic systems (2013)
Presentation / Conference
Cowey, R., Christian, J. M., & McDonald, G. S. (2013, June). Dark & anti-dark spatiotemporal solitons: From cubic to cubic-quintic systems. Presented at College of Science & Technology Research Showcase, University of Salford, Greater Manchester M5 4WT, U.K

The origin of conventional models for optical pulse propagation lies in the universal slowly-varying envelope approximation (SVEA) accompanied by a Galilean boost to the local time frame. However, Biancalana and Creatore [1] have recently pointed out... Read More about Dark & anti-dark spatiotemporal solitons: From cubic to cubic-quintic systems.

The spatiotemporal ginzburg-landau equation: Dissipative solitons & stability (2013)
Presentation / Conference
Bresnahan, D., Christian, J., & McDonald, G. (2013, June). The spatiotemporal ginzburg-landau equation: Dissipative solitons & stability. Presented at College of Science & Technology Research Showcase, University of Salford

The complex Ginzburg-Landau (GL) equation describes universal wave propagation in dispersive systems that also exhibit competition between amplification and dissipation [1,2]. The balance between dispersive effects (group-velocity dispersion and self... Read More about The spatiotemporal ginzburg-landau equation: Dissipative solitons & stability.

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.

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.

Algebraic soliton refraction: New wave contexts for nonlinear Snell’s law (2013)
Presentation / Conference
Yates, R., Christian, J., & McDonald, G. (2013, June). Algebraic soliton refraction: New wave contexts for nonlinear Snell’s law. Presented at College of Science & Technology Research Showcase, University of Salford

The refraction of light waves at planar interfaces is a problem of fundamental interest to the optics community. To date, our Snell’s law-type analyses have considered only families of hyperbolic solitons – robust nonlinear beams that are strongly (e... Read More about Algebraic soliton refraction: New wave contexts for nonlinear Snell’s law.

Coupled-waveguide arrays: Oblique injection & soliton propagation (2013)
Presentation / Conference
Christian, J., McCoy, E., & McDonald, G. (2013, June). Coupled-waveguide arrays: Oblique injection & soliton propagation. Presented at College of Science & Technology Research Showcase, University of Salford

The interaction between light beams and periodically-patterned host materials (such as coupled-waveguide arrays or photonic crystals) is a fundamental class of problem in nonlinear optics [1,2]. While oblique (off-axis) propagation effects play a cen... Read More about Coupled-waveguide arrays: Oblique injection & soliton propagation.

Ultrabroad-band Raman light: Linear detuning & gain suppression (2013)
Presentation / Conference
Kelly, R., Christian, J., & McDonald, G. (2013, June). Ultrabroad-band Raman light: Linear detuning & gain suppression. Presented at College of Science & Techology Research Showcase, University of Salford

Ultrabroad-band multifrequency Raman generation is a (laser-driven) resonant-symmetric two-colour pumping technique for exciting polychromatic light beams that are characterized by potentially very wide “top-hat” temporal bandwidths [1]. The spectra... Read More about Ultrabroad-band Raman light: Linear detuning & gain suppression.

Diffraction of fractal light: New frontiers for the mathematics of edge waves (2013)
Presentation / Conference
Mylova, M., McDonald, G., & Christian, J. (2013, June). Diffraction of fractal light: New frontiers for the mathematics of edge waves. Presented at College of Science & Technology Research Showcase, University of Salford

The diffraction pattern produced by a plane wave (i.e., a perfectly uniform wavefront) scattering from an infinite hard edge is well-described by the Fresnel integral [1]. Such one-dimensional (1D) edge waves [see Fig. 1(a)] turn out to be truly elem... Read More about Diffraction of fractal light: New frontiers for the mathematics of edge waves.