Solitons: Inverse Scattering Theory and Its Applications
In: Bifurcation Phenomena in Mathematical Physics and Related Topics ISBN: 9789400990067; (1980)
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Zugriff:
Five lectures on solitons are presented. The first summarises the discovery of the inverse scattering method for solving the initial value problem of the Korteweg de Vries equation by Kruskal and colleagues: the polynomial conserved densities of the KdV are introduced. The second lecture treats Backlund transformations and conserved densities, especially for the sine-Gordon equation, and extends the inverse scattering method to the 2x2 scheme of Zakharov and Shabat and of Ablowitz, Kaup, Newell and Segur. It is shown that all AKNS-ZS systems represent surfaces of constant negative Gaussian curvature. The geometrical analysis is used to derive non-local conserved densities for the sine-Gordon equation. In the third and fourth lectures the canonical structure of these systems is exhibited: all AKNS-ZS systems of the types considered are infinite dimensional completely integrable Hamiltonian systems. The AKNS-ZS scattering transformation is shown to be a canonical transformation. New canonical co-ordinates expressed in terms of scattering data are found. These are used for the semi-classical quantisation of these systems. The sine-Gordon equation is quantised explicitly and its eigen spectrum found. The relation of this to the eigen spectrum of the spin-00BD;’ x-y-z model of statistical mechanics and the massive Thirring model of massive fermions is sketched. In the final lecture the double sine-Gordon equations uxx-u tt=± (sin u + 00BD; sin λ sin 00BD; u) which arise in resonant non-linear optics and in the theory of spin waves in 3He below 2.6 mK are mentioned. The optical problem (+ ve sign) is used as a vehicle for singular perturbation theory about the sine-Gordon equation.
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Solitons: Inverse Scattering Theory and Its Applications
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Autor/in / Beteiligte Person: | Bullough, R. K. |
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Quelle: | Bifurcation Phenomena in Mathematical Physics and Related Topics ISBN: 9789400990067; (1980) |
Veröffentlichung: | Springer Netherlands, 1980 |
Medientyp: | unknown |
ISBN: | 978-94-009-9006-7 (print) |
DOI: | 10.1007/978-94-009-9004-3_17 |
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