Applied Nonlinear Dynamical Systems in Mathematical Physics

Dear Colleagues,

Nonlinear phenomena that do not obey the superposition principle appear everywhere in nature. Their description and understanding is therefore of great interest from both theoretical and applicative points of view. Maxwell equations are linear, whereas Einstein equations are highly nonlinear. The nonlinear stability of black hole space-times is a fundamental problem in general relativity whose solution depends on the nonlinear structure of Einstein equations. However, even in the case of Maxwell equations, the behavior of light in nonlinear media gives rise to a host of new nonlinear phenomena which comprise the field of nonlinear optics. Finding solutions to nonlinear equations, be it the nonlinear Schrodinger equation, the equations appearing in nonlinear optics, Einstein equations, Yang–Mills equations, nonlinear wave equations or the nonlinear equations describing the behavior of complex fluids, is difficult. The employment of group theoretic methods facilitates the discovery of solutions to these equations and in generating new solutions from the old. Numerical techniques used to cope with the nonlinear character of the fundamental equations of mathematical physics have provided magnificent results and simulations that offer new insights into the understanding of phenomena. Nonlinear phenomena in gauge and gravity theories are not amenable to perturbation analysis and are essential for their understanding. A class of solutions which explicate such nonlinear phenomena includes solitons, kinks, instantons, and breathers. Such solutions have applications well beyond gauge and gravity theories. They have entered and changed such diverse fields as nonlinear optics, plasma physics, and ocean, atmospheric, and planetary sciences.  If a nonlinear phenomenon can be represented by an integrable system, the existence of sufficiently conserved quantities or first integrals restricts its evolution to a submanifold within its phase space. An interesting example of this kind is the Liouville–Arnold integrability. Integrability structures which emerge in gauge theories and string theory are of fundamental importance for their understanding.

Research areas may include (but are not limited to) the following:

• Nonlinear equations of mathematical physics;
• Nonlinear Schrodinger equation;
• Difference equations and functional equations in mathematical physics;
• (Lie) group theoretic methods for the solution of differential and difference equations;
• Integrability and non-integrability, scattering/inverse scattering, and Painlevé analysis;
• Lie-algebraic characterizations of integrability;
• Tensor Poisson structures;
• Classical and quantum many-body problems;
• Geometry and algebra of nonlinear equations;
• Integrability in gauge and string theory;
• Monopoles, solitons, kinks, breathers, and instantons in gauge theory and in gravity theory;
• Symmetries and invariants in mathematical physics;
• Applications of Lie groups, Lie algebras, and higher structures;
• Quasi-exactly solvable models in quantum mechanics;
• Nonlinear stability of black hole space-times;
• Complex fluids;
• Nonlinear waves;
• Nonlinear optics;
• Numerical simulations in mathematical physics. 

Dr. Evangelos Melas
Guest Editor

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