Colin Rogers

In 1+1-dimensions, an extension of the canonical solitonic Dym equation has previously been derived both in a geometric torsion evolution context and in the analysis of peakon solitonic phenomena in hydrodynamics. Here, a novel 2+1-dimensional S-integrable extended Dym-type equation is introduced. As Lax pair is constructed and an associated $\bar{\partial}$-dressing scheme detailed. Integrable modulated versions of the 2+1-dimensional extended Dym equation are generated via application of a class of involutory transformations with genesis in classical Ermakov theory.

A novel extension of the canonical solitonic mKdV equation is introduced which admits hybrid Ermakov-Painlevé II symmetry reduction. Application of the latter is made to obtain exact solution of Airy-type to a class of moving boundary problems of Stefan kind for this extended mKdV equation. A reciprocal transformation is then applied to the latter to generate an associated exactly solvable class of moving boundary problems for an extension of a base Casimir member of a compacton hierachy. The extended mKdV equation is shown to be embedded in a range of nonlinear evolution equations with temporal modulation as determined via the action of a class of involutory transformations with origin in Ermakov theory. Associated temporal modulation for the hybrid mKdV and KdV equation as embedded in the classical solitonic Gardner equation is delimited. 15 pages

An overview is presented of quantum and resonant nonlinear Schrödinger equation links to Whitham–Broer–Kaup type systems. A novel [ ] dimensional extension of the Whitham–Broer–Kaup hydrodynamic system is constructed with connection to an equivalent multi-dimensional resonant NLS equation. Hybrid Ermakov–Painlevé II and associated Painlevé XXXIV integrable similarity reductions are derived.

Moving boundary problems of Stefan-type for a novel third order nonlinear evolution equation with temporal modulation are here shown to be amenable to exact Airy-type solution via a classical Ermakov equation with its admitted nonlinear superposition principle. Application of the latter together with a class of involutory transformations sets the original moving boundary problem in a wide class with temporal modulation. As an appendix, reciprocally associated exactly solvable moving boundary problems are derived.11 pages

A class of moving boundary problems of Stefan-type is shown to be amenable to exact solution via Painlevé II similarity reduction both for the canonical solitonic Gardner equation and a reciprocally associated novel third order nonlinear evolution equation incorporating a source term.

Two-point boundary value problems for a discrete Ermakov-Painlevé II equation are analysed by means of topological methods. In addition, an alternative variational approach is detailed. Existence of solutions is established for appropriate choice of parameters.

A seed class of exact solutions of the Euler system of two-dimensional relativistic gasdynamics is derived with an underlying Chaplygin–Kármán–Tsien type constitutive law. The invariance of the nonlinear relativistic system under multi-parameter reciprocal Bäcklund transformations is applied to generate a wide class of novel associated solutions with a barotropic relation. The latter reduces to the standard Chaplygin–Kármán–Tsien gas law in the non-relativistic limit. An additional invariance of the general relativistic gasdynamics system is established under a substitution principle. The latter represents an extension of a classical result in non-relativistic gasdynamics.

Evidence-based policing (EBP) posits that police practices should be based upon evidence from research, and those in policing should be trained to value and utilize sources of knowledge, research, practices, and experiences. Set within the national and international context of EBP and volunteer policing, this paper explores whether volunteer police officers, called special constables (SCs) in England and Wales, are aware of EBP, integrate EBP into their volunteering and the evidence sources utilized. The findings, from a sample of SCs volunteering in England and Wales, demonstrate that over half of respondents have heard of EBP, a significant number suggest regular use, yet some are either not sure or do not use EBP. SCs are expected to take part in similar frontline roles as their regular police colleagues, with the same policing powers and responsibilities. The paper concludes though that EBP is integrated into regular officer training but training of SCs in the adoption of EBP seems more limited. This research is one of the first to examine the adoption and understanding of EBP by volunteers within policing.