Stability and Well-posedness in Integrable Nonlinear Evolution Equations

Abstract

This dissertation is concerned with analysis of orbital stability of solitary waves and well-posedness of the Cauchy problem in the integrable evolution equations. The analysis is developed by using tools from integrable systems, such as higher-order conserved quantities, B\"{a}cklund transformation, and inverse scattering transform. The main results are obtained for the massive Thirring model, which is an integrable nonlinear Dirac equation, and for the derivative NLS equation. Both equations are related with the same Kaup-Newell spectral problem. Our studies rely on the spectral properties of the Kaup-Newell spectral problem, which convey key information about solution behavior of the nonlinear evolution equations.