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Dr James Christian's Outputs (152)

Electromagnetic scattering by fractal screens: from diffraction integrals towards a boundary element method (2025)
Presentation / Conference Contribution

The scattering of time-harmonic electromagnetic fields by perfectly-conducting plane
screens is a fundamental problem in physics. A standard approach to computing the scattered
wave is to deploy Rayleigh-Sommerfeld diffraction integrals in conjunct... Read More about Electromagnetic scattering by fractal screens: from diffraction integrals towards a boundary element method.

Counterpropagating waves in discrete nonlinear equations: instability spectra & spontaneous patterns (2025)
Presentation / Conference Contribution

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 struc... Read More about Counterpropagating waves in discrete nonlinear equations: instability spectra & spontaneous patterns.

Families of exact similaritons with flexible modulations in N-coupled homogeneous and inhomogeneous systems (2024)
Journal Article
Huo, K., Li, B., Jia, H., Christian, J., & Yang, R. (in press). Families of exact similaritons with flexible modulations in N-coupled homogeneous and inhomogeneous systems. Physics Letters A, Article 129997. https://doi.org/10.1016/j.physleta.2024.129997

In this paper, a family of exact solutions to N-coupled nonlinear Schrödinger (N-CNLS) equations including bright and dark solitons, Akhmediev breathers, Kuznetsov-Ma breathers, and rogue waves are derived with the aid of a generic nonlinear wave ass... Read More about Families of exact similaritons with flexible modulations in N-coupled homogeneous and inhomogeneous systems.

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

A mathematical model for the motion of a micro-robot consisting of three spheres linked with axially aligned retractable arms in Stokes flow that gives expressions for mean distance moved, mean drift velocity and energy efficiency of the non-reciprocal cyclic swimming motion (2023)
Journal Article
Elatrash, L., Chadwick, E., El-Mazuzi, R., Christian, J. M., Wang, Y., Adamu, H. A., & Chadwick, E. (in press). A mathematical model for the motion of a micro-robot consisting of three spheres linked with axially aligned retractable arms in Stokes flow that gives expressions for mean distance moved, mean drift velocity and energy efficiency of the non-reciprocal cyclic swimming motion. Computers and Fluids, 266, https://doi.org/10.1016/j.compfluid.2023.106064

The problem studied was the non-reciprocal cyclic swimming motion of three spheres linked with axially aligned retractable arms in Stokes flow. The arms are assumed to be able to retract at a steady speed to half their length, and then at a later tim... Read More about A mathematical model for the motion of a micro-robot consisting of three spheres linked with axially aligned retractable arms in Stokes flow that gives expressions for mean distance moved, mean drift velocity and energy efficiency of the non-reciprocal cyclic swimming motion.

Exact similariton solution families and diverse composite waves in coherently coupled inhomogeneous systems (2023)
Journal Article
Huo, K., Yang, R., Jia, H., He, Y., & Christian, J. (2023). Exact similariton solution families and diverse composite waves in coherently coupled inhomogeneous systems. Nonlinear Dynamics, https://doi.org/10.1007/s11071-023-08574-9

Seeking analytical solutions of nonlinear Schrödinger (NLS)-like equations remains an open topic. In this paper, we revisit the general inhomogeneous nonautonomous NLS (inNLS) equation and report on exact similaritons under generic constraint relati... Read More about Exact similariton solution families and diverse composite waves in coherently coupled inhomogeneous systems.

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.

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.

The theory and application of eulerlets (2019)
Journal Article
Chadwick, E., Christian, J., Kapoulas, A., & Chalasani, K. (2019). The theory and application of eulerlets. Physics of Fluids, 31(4), 047106. https://doi.org/10.1063/1.5088132

Consider a fixed body in a uniform flow field in the limit as the Reynolds number approaches infinity and the flow field remains steady. Instead of using standard techniques and theory for describing the problem, a new method is employed based upon t... Read More about The theory and application of eulerlets.

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

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

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.