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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.

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.

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.

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. 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.

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.

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.

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.

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.