The Application of Mathematics to Physics and Nonlinear Science

Dear Colleagues,

The importance of understanding nonlinearity has increased over the decades, through the development of newer fields of application: elementary particles, biophysics, wave dynamics, optical fibers, fluids, plasmas, social and ecological systems, astrophysics, and cross-disciplinary fields. The necessary mathematics involves nonlinear evolution equations. Obtaining closed-form solutions for these equations plays an important role in the proper understanding of features of many phenomena, by unraveling the mechanisms of complex phenomena such as pattern formation and selection, the spatial localization of transfer processes, the multiplicity or absence steady states under various conditions, the existence of peaking regimes, etc. Even exact test solutions with no immediate physical meaning are used to verify the consistency of numerical, asymptotic, and approximate analytical methods. In spite of various solving methods like Darboux and Bäcklund transformation, IST, Hirota bilinear, Lie groups, and time reversal, there are still nonlinear phenomena that are not completely understood. Rogue waves, for example, which have aroused scientists’ interest in plasmas, deep ocean, nonlinear optic, Bose–Einstein condensates, biophysics, superfluids, financial markets, and population dynamics, still cannot be predicted, and their generating mechanism is open to debate. Another example is pattern formation and the self-appearance of patterns for spatially extended systems. It is the aim of this Special Issue to gather some of the most debated and interesting topics from nonlinear science for which, in spite of the existence of highly developed mathematical tools of investigation, there are still fundamental open questions. Possible topics are new methods and applications in pattern-formation theory or the relationship between turbulence, solitary waves, breathers and rogue waves in oceanography, optical fibers, superfluids, BEC, and financial markets. Another topic is understanding the self-motion of compact systems like liquid drops, shells, bubbles, super-bubbles, anti-bubbles, vortex filaments, and their relationship to motile cells. The connection between the geometry of the boundaries and the structure of the solutions inside the bounded region is also an invited topic. Two- and four-wave mixing processes in nonlinear fluid dynamics and optics are also an important topic with modern applications in optical imaging, optical communications, real-time holography, and opto-electronic neural networks. Breathers as nonlinear excitations in DNA, self-generating of patterns scale-invariant networks, with applications in social and ecosystems, solitons in nonlinear lattices and stability of matter wave solitons in 2- and 3-dimensions, spontaneous formation of rotational patterns in fluids at different scale and regimes from Leidenfrost drops to oceans and galaxies are examples of topics hosted by this special issue.

Prof. Andrei Ludu
Guest Editor

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