Invariant Solutions of Nonlinear Mathematical Modeling of Natural Phenomena

Abstract: The main objective is to demonstrate the advantages of the invariance method in obtaining new exact analytic solutions expressed in terms of elementary functions for various physical phenomena. As one particular application of the invariance method will be the mathematical modeling of oceanic and atmospheric whirlpools causing weather instabilities and, possibly, linked with climate change. As another particular example, it will be demonstrated that the invariance method allows to obtain the exact solutions of fully nonlinear Navier-Stokes equations within a thin rotating atmospheric shell that serves as a simple mathematical description of an atmospheric circulation caused by the temperature difference between the equator and the poles with included equatorial flows modeling heat waves, known as Kelvin Waves. Special attention will be given to analyzing and visualizing the conserved densities associated with obtained exact solutions. As another modeling scenario, the exact solution of the shallow water equations simulating equatorial atmospheric waves of planetary scales will be analyzed and visualized.

mathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsnumerical analysisexactly solvable and integrable systemsfluid dynamics

Audience: researchers in the topic

Mathematical models and integration methods