Fourth-order pattern forming PDEs: partial and approximate symmetries

This paper considers pattern forming nonlinear models arising in the study of thermal convection and continuous media. A primary method for the derivation of symmetries and conservation laws is Noether’s theorem. However, in the absence of a Lagrangian for the equations investigated, we propose the use of partial Lagrangians within the framework of calculating conservation laws. Additionally, a nonlinear Kuramoto-Sivashinsky equation is recast into an equation possessing a perturbation term. To achieve this, the knowledge of approximate transformations on the admissible coefficient parameters is required. A perturbation parameter is suitably chosen to allow for the construction of nontrivial approximate symmetries. It is demonstrated that this selection provides approximate solutions

Selected Topics in Gravity, Field Theory and Quantum Mechanics

Quantum field theory has achieved some extraordinary successes over the past sixty years; however, it retains a set of challenging problems. It is not yet able to describe gravity in a mathematically consistent manner. CP violation remains unexplained. Grand unified theories have been eliminated by experiment, and a viable unification model has yet to replace them. Even the highly successful quantum chromodynamics, despite significant computational achievements, struggles to provide theoretical insight into the low-energy regime of quark physics, where the nature and structure of hadrons are determined. The only proposal for resolving the fine-tuning problem, low-energy supersymmetry, has been eliminated by results from the LHC. Since mathematics is the true and proper language for quantitative physical models, we expect new mathematical constructions to provide insight into physical phenomena and fresh approaches for building physical theories

Environmental change and ecosystem functioning drive transitions in social-ecological systems: A stylized modelling approach

Sustainable management of social-ecological systems requires an understanding of how anthropogenic climate- and land use change may disrupt interactions between human societies and the ecosystem processes they depend on. In this study, we expand an existing stylized social-ecological system model by explicitly considering how urbanizing societies may become less dependent on local ecosystem functioning. This expansion is motivated by a previously developed conceptual framework suggesting that societies may reside in either a green loop and be strongly dependent on local ecosystem processes, or in a red loop where this dependency is weaker due to imports of natural resources from elsewhere. Analyzing the feasibility and stability of local social-ecological system states over a wide range of environmental and socio-economic conditions, we observed dynamics consistent with the notion of green loop-dominated and red loop-dominated societies comprising alternate stable social-ecological states. Based on systems' inherent dependencies on local ecosystem processes, responses to environmental change could comprise either transitions between green loop- and red loop-dominated states, or collapse of either of these states. Our quantitative model provides an internally consistent mapping of green loop- and red loop-dominated states, as well as transitions between or collapses of these states, along a gradient of environmental conditions

Noether Symmetries Quantization and Superintegrability of Biological Models

It is shown that quantization and superintegrability are not concepts that are inherent to classical Physics alone. Indeed, one may quantize and also detect superintegrability of biological models by means of Noether symmetries. We exemplify the method by using a mathematical model that was proposed by Basener and Ross (2005), and that describes the dynamics of growth and sudden decrease in the population of Easter Island