Integrable Systems(D)

INTEGRABLE SYSTEMS (D)
Part IB Methods, and Complex Methods or Complex Analysis are essential; Part II Classical Dynamics is desirable.

Integrability of ordinary diﬀerential equations: Hamiltonian systems and the Arnol’d–Liouville Theorem (sketch of proof). Examples. [3]
Integrability of partial diﬀerential equations: The rich mathematical structure and the universality of the integrable nonlinear partial diﬀerential equations (Korteweg-de Vries, sine-Gordon). B¨acklund transformations and soliton solutions. [2] The inverse scattering method: Lax pairs. The inverse scattering method for the KdV equation, and other integrable PDEs. Multi-soliton solutions. Zero curvature representation. [6]
Hamiltonian formulation of soliton equations. [2]
Painleve equations and Lie symmetries: Symmetries of diﬀerential equations, the ODE reductions of certain integrable nonlinear PDEs, Painlev´e equations. [3]

Appropriate books

† Dunajski, M Solitons, Instantons and Twistors. (Ch 1–4) Oxford Graduate Texts in Mathematics, ISBN 9780198570639, OUP, Oxford 2009
S. Novikov, S.V. Manakov, L.P. Pitaevskii, V. Zaharov Theory of Solitons. for KdF and Inverse Scattering P.G. Drazin and R.S. Johnson Solitons: an introduction. (Ch 3, 4 and 5) Cambridge University Press 1989
V.I. Arnol’d Mathematical Methods of Classical Mechanics. (Ch 10) Springer, 1997
P.R. Hydon Symmetry Methods for Diﬀerential Equations:A Beginner’s Guide. Cambridge University Press 2000
P.J. Olver Applications of Lie groups to diﬀerential equations. Springeri 2000
MJ Ablowitz and P Clarkson Solitons, Nonlinear Evolution Equations and Inverse Scattering. CUP 1991
MJ Ablowitz and AS Fokas Complex Variables. CUP, Second Edition 200