Strongly Nonlinear Phenomena and Singularities in Optical, Hydrodynamic and Biological Systems

Singularity formation is an inherent feature of equations in nonlinear physics, in many situations such as in self-focusing of light nonlinearity is essential part of the model and physical events cannot be captured by linearized equations. There are nonlinear systems, such as $1$D NLSE where singularities in analytic continuation would follow a soliton solution at fixed distances and while it is true that soliton determines the position of all the singularities, it is also true that evolution of singularities determines the solution on the real axis. Before we go further to discuss $2$D problems, we want to be more specific about analytic continuation in a $2$D problem: it is well-known that collapse in $2$D NLSE is radially symmetric and introducing radial variable $r = \\sqrt{x^2 + y^2}$ the problem becomes effectively one-dimensional. If we expand the interval spanned by $r$ from $[0,+\\infty)$ to $(-\\infty,+\\infty)$ and continue all the functions evenly across the origin, it starts to make sense to further expand $r$ to complex plane $\\mathbb{C}$ and talk about analytic continuation of functions in $r\\in\\mathbb{C}$. In $2$D nonlinear Schr\\ odinger equation (NLSE) it is common to think that a singularity appears in finite time, but one can also say that a singularity already exists in the analytic continuation of initial data and at critical time $t_c$, the singularity touches the real axis and solution reaches its maximal interval of existence. The latter point of view captures evolution in more detail, in particular it allows to ask many questions that would seem quite meaningless if you follow the philosophy\u27\u27 of a singularity just appearing at a finite time. In particular, one can ask question what is the trajectory of singularity in complex plane and how does the type of singularity change as $t\ o t_c$. For $2$D focusing NLSE and Keller-Segel model (KSE) of chemotactic bacteria, the singularities evolve towards the real axis if sufficient conditions are met by initial distribution of laser intensity (NLSE) and bacteria density (KSE) respectively. Well-established conditions are included in the text and are cited upon in corresponding sections. The central subject of this work is the study of onset of singularity towards the real axis in radially symmetric $2$D NLSE and $2$D reduced KSE models (RKSE) combining two approaches: direct numerical simulations of collapse and asymptotic analysis in the limit $t\ o t_c$. The benefit of this two-sided approach is evident when comparing results of classic theory of critical collapse in $2$D NLSE to numerical simulations: the collapse exhibits dependence on initial data even when intensity reaches enormous magnitudes and as a result is inconsistent with classical theory (e.g loglog law). An intervention of numeric approach allowed us to perform sanity checks of many assumptions and estimate regions of applicability of approximations that were used in asymptotic approach and resulted in a new corrected theory that is able to consistently describe the onset of singularity even for moderate-amplitude, developed collapse while still recovering classic theory in the limit $t\ o t_c$. The problem of $2$D potential flow of ideal fluid in free surface hydrodynamics is another example of nonlinear system containing solutions with singularities. The focus of our investigation lies in fully nonlinear travelling waves on the surface of fluid also known as Stokes waves and in particular we are interested in singularities that are present in the analytic continuation of Stokes waves. These waves computed as a part of this dissertation range from linear waves to the limit of extremely nonlinear waves that were never observed before, in addition a predicted phenomenon of parameter oscillation was confirmed for strongly nonlinear waves

Magnetic Field Effects in Quantum Biology: Beyond the Radical Pair Mechanism

The effects of weak magnetic fields on biological systems have become an area of burgeoning research and interest in recent years. Specifically, the Radical Pair Mechanism (RPM) has been quite successful at beginning to explain phenomena such as avian magnetoreception, the magnetosensitivity of lipid peroxidation reactions and other such biological magnetic field effects (MFEs) - but there are still many questions to answer. This thesis addresses such questions, by proposing a new mechanism (D3M) to offer a new perspective on radical spin dynamics, and methods for amplifying biological MFEs

MUSME 2011 4 th International Symposium on Multibody Systems and Mechatronics

El libro de actas recoge las aportaciones de los autores a través de los correspondientes artículos a la Dinámica de Sistemas Multicuerpo y la Mecatrónica (Musme). Estas disciplinas se han convertido en una importante herramienta para diseñar máquinas, analizar prototipos virtuales y realizar análisis CAD sobre complejos sistemas mecánicos articulados multicuerpo. La dinámica de sistemas multicuerpo comprende un gran número de aspectos que incluyen la mecánica, dinámica estructural, matemáticas aplicadas, métodos de control, ciencia de los ordenadores y mecatrónica. Los artículos recogidos en el libro de actas están relacionados con alguno de los siguientes tópicos del congreso: Análisis y síntesis de mecanismos ; Diseño de algoritmos para sistemas mecatrónicos ; Procedimientos de simulación y resultados ; Prototipos y rendimiento ; Robots y micromáquinas ; Validaciones experimentales ; Teoría de simulación mecatrónica ; Sistemas mecatrónicos ; Control de sistemas mecatrónicosUniversitat Politècnica de València (2011). MUSME 2011 4 th International Symposium on Multibody Systems and Mechatronics. Editorial Universitat Politècnica de València. http://hdl.handle.net/10251/13224Archivo delegad

Preventing premature convergence and proving the optimality in evolutionary algorithms

http://ea2013.inria.fr//proceedings.pdfInternational audienceEvolutionary Algorithms (EA) usually carry out an efficient exploration of the search-space, but get often trapped in local minima and do not prove the optimality of the solution. Interval-based techniques, on the other hand, yield a numerical proof of optimality of the solution. However, they may fail to converge within a reasonable time due to their inability to quickly compute a good approximation of the global minimum and their exponential complexity. The contribution of this paper is a hybrid algorithm called Charibde in which a particular EA, Differential Evolution, cooperates with a Branch and Bound algorithm endowed with interval propagation techniques. It prevents premature convergence toward local optima and outperforms both deterministic and stochastic existing approaches. We demonstrate its efficiency on a benchmark of highly multimodal problems, for which we provide previously unknown global minima and certification of optimality