The course is meets from 1130am to 1230pm usually on Tuesdays, Wednesdays and Fridays at CMI.
Some brief lecture notes are here.
Topics covered in this course: Hamiltonian mechanics, Poisson brackets, Angle-action variables, Liouville integrability,
Lax pairs for mechanical systems, isospectral evolution and conserved quantities, examples of harmonic oscillator, Euler top, Toda chain, fundamental Poisson brackets and r-matrices, conserved quantities in involution.
KdV equation, similarity and solitary wave solutions, Hamiltonian and Gardner-Faddeev-Zakharov Poisson brackets, Lagrangian and Noether symmetries, Lax pair and Schrodinger operator, Miura transform, Riccati and mKdV equations; Inverse scattering transform for KdV, Scattering data and its GGKM evolution: discrete and continuous spectrum, Inverse scattering and GLM equation, example of direct and inverse scattering for Dirac delta potential, reflectionless scattering and evolution of 1- and 2-soliton solutions of KdV, Infinitely many local conserved quantities, bi-Hamiltonian formulation: Gardner and Magri brackets, KdV hierarchy and infinitely many conserved quantities in involution, from Lax pair to zero curvature representation for KdV.
Nonlinear Schrodinger field equation, Hamiltonian and Lagrangian formulations, Noether conserved charges, Zero curvature representation of the nonlinear Schrodinger equation, Fundamental Poisson brackets and classical r-matrix, anti-symmetry and Jacobi identity and classical Yang-Baxter equation, transition and monodromy matrices and conserved quantities in involution. Quantum nonlinear Schrodnger field, transition matrix, fundamental commutation relations, quantum R-matrix, RTT relation, quantum Yang-Baxter equation.
Backlund transformations for Liouville and wave equations, Auto-Backlund transformations for Laplace's equation, Sine-Gordon equation and KdV equation.
Painleve property. Ordinary differential equations with fixed and movable singularities: linear and nonlinear equations, movable poles and movable critical points, Painleve property, First order ODEs: Fuchs and generalized Riccati equations, 2nd order ODEs: Painleve equations; singularity analysis to detect ODEs of Painleve-type, resonances; symmetry reduction of PDEs to ODEs and ARS conjecture on nonlinear PDEs solvable by inverse scattering transform.