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Wave phenomena appears naturally in biology (gene competition), quantum physics (particles) and theoretical optics (fiber optics), and evolution equations are a great mathematical model for these phenomena. The ideal type of solutions for the latter are traveling wave solutions (they preserve their shape on time). The use of nonlinear ordinary differential equations (ODEs) to find traveling wave solutions has been a standard tool for mathematicians, and they involve mathematically rich areas such as special functions and Fourier analysis. In particular, the Riccati equation (RE) with variable coefficients has been a useful tool in mathematical biology (RE is better known as logistic equation) and in mathematical physics: diffusion and Schrodinger type models. For the last two, a significant number of the explicit solutions that are available in the literature that can be solved in the real line thanks to RE. Further, RE with variable coefficients can be used to find explicit solutions for the celebrated (constant coefficient) nonlinear Schrödinger equation (NLS) that is a standard model of how light propagates inside of a fiber optic with mathematical rich properties such as being integrable.
In this REU students will study explicit solutions for a nonlinear Riccati-Ermakov system with selected variable coefficients, and use the solution of the system to construct explicit solutions for a Nonlinear Schroedinger equation with variable coefficients through the construction of transformations to the standard model with constant coefficients. The techniques that the students will use (based on the PI’s previous research) will provide six parameters. It is expected that students will provide an interpretation of the parameters related to the dynamics of the central axis of symmetry of the traveling wave solution. Also, numerical simulations of more general problems will take place where analytical techniques proposed by the PI’s research can’t work.
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