MATH 576 Introduction to Soliton Theory

IVP for Burgers’ equation and Cole-Hopf transformation. Shock solitons and their dynamics. Backlund transformation. General solution of the Liouville equation. The Sine-Gordon equation. Topological soliton. Bianchi permutability theorem and nonlinear superposition principle. Multi-soliton solutions. Collisions and bound states of solitons. From Riccati equation to the inverse scattering transform. Zero curvature and Lax representations. Zakharov-Shabat problem. Hirota direct method in soliton theory. KdV equation and the Schrodinger spectral problem. Elements of quantum scattering theory. The inverse problem. Gel’fand-Levitan-Marchenko equation and N-soliton solution. Analytic properties of the scattering amplitude. Integration of the KdV equation. Infinite hierarchy of integrals of motion. Current developments in soliton theory.