Solitons: Inverse Scattering Theory and Its Applications

摘要: Five lectures on solitons are presented. The first summarises the discovery of inverse scattering method for solving initial value problem Korteweg de Vries equation by Kruskal and colleagues: polynomial conserved densities KdV introduced. second lecture treats Backlund transformations densities, especially sine-Gordon equation, extends to 2x2 scheme Zakharov Shabat Ablowitz, Kaup, Newell Segur. It is shown that all AKNS-ZS systems represent surfaces constant negative Gaussian curvature. geometrical analysis used derive non-local equation. In third fourth canonical structure these exhibited: types considered infinite dimensional completely integrable Hamiltonian systems. transformation be a transformation. New co-ordinates expressed in terms data found. These semi-classical quantisation quantised explicitly its eigen spectrum relation this spin-00BD;’ x-y-z model statistical mechanics massive Thirring fermions sketched. final double equations uxx-u tt=± (sin u + 00BD; sin λ u) which arise resonant non-linear optics theory spin waves 3He below 2.6 mK mentioned. optical (+ ve sign) as vehicle singular perturbation about