RUI: Mathematical Analysis of Several Models in Nonlinear Optics

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The most recent advances in nonlinear optics include the generation and applications of ultrashort optical pulses, whose time duration is typically of the order of femtoseconds (one quadrillionth of a second). Duration of these pulses is approaching the timescales of fundamental atomic and molecular processes. They deliver energy so quickly that it allows them to probe living structures without damage and to make material modifications on the micron scale with minimal heat effects. Femtoscale lasers are used, for example, in microsurgery or materials processing. For ultrashort pulses, the traditional mathematical models used for longer optical pulses, the Nonlinear Schrödinger (NLS) and Coupled Nonlinear Schrödinger (CNLS) equations, lose their predictive capabilities. This research project is concerned with two novel mathematical models: the Complex Short Pulse (CSP) equation and the Coupled Complex Short Pulse (CCSP) equation. The project is devoted to analysis and computation of solutions of these equations for the description of ultrashort optical pulses.The models to be investigated are integrable; the project concerns the investigation of the CSP equation, the CCSP equation and their integrable semi-discrete analogues, where the spatial variable takes values in a lattice, while the integrability of the resulting equations is preserved. In particular, the investigator plans to: (1) construct the semi-discrete CSP equation and derive exact solutions including bright and dark solitons and rogue-wave-like solutions; (2) construct bright-bright, dark-dark and bright-dark (mixed) soliton solutions, as well as rogue wave solutions, of the CCSP equation; (3) utilize the semi-discrete CSP equation as a novel self-adaptive moving mesh numerical method for simulating solutions of the continuous CSP model. Being at the second largest Hispanic-serving institution in the country, the investigator strives to motivate and encourage undergraduate and graduate students to engage in cutting-edge research in applied mathematics.