Integrable Models

Contents......Page all_20214_to_00355.cpc0007.djvu
Preface......Page all_20214_to_00355.cpc0005.djvu
1. The Korteweg-de Vries Equation......Page all_20214_to_00355.cpc0013.djvu
Hamiltonian systems......Page all_20214_to_00355.cpc0014.djvu
Integrability of a Hamiltonian system......Page all_20214_to_00355.cpc0019.djvu
KdV equation......Page all_20214_to_00355.cpc0023.djvu
KdV as a Hamiltonian system......Page all_20214_to_00355.cpc0026.djvu
References......Page all_20214_to_00355.cpc0034.djvu
Uniqueness of solutions......Page all_20214_to_00355.cpc0036.djvu
Travelling waves......Page all_20214_to_00355.cpc0040.djvu
Solitons......Page all_20214_to_00355.cpc0042.djvu
Quantum Mechanics of the soliton solutions......Page all_20214_to_00355.cpc0046.djvu
References......Page all_20214_to_00355.cpc0057.djvu
Conserved quantities......Page all_20214_to_00355.cpc0058.djvu
The Miura transformation......Page all_20214_to_00355.cpc0063.djvu
Infinite number of conserved quantities......Page all_20214_to_00355.cpc0067.djvu
Integrability of the KdV system......Page all_20214_to_00355.cpc0077.djvu
Higher order equations of the hierarchy......Page all_20214_to_00355.cpc0083.djvu
References......Page all_20214_to_00355.cpc0085.djvu
The Schrödinger equation......Page all_20214_to_00355.cpc0086.djvu
i) Bound states......Page all_20214_to_00355.cpc0092.djvu
ii) Scattering states......Page all_20214_to_00355.cpc0094.djvu
Gel'fand-Levitan equation......Page all_20214_to_00355.cpc0097.djvu
Example......Page all_20214_to_00355.cpc0098.djvu
References......Page all_20214_to_00355.cpc0103.djvu
Scattering in one dimension......Page all_20214_to_00355.cpc0104.djvu
Analytic behavior of scattering coefficients......Page all_20214_to_00355.cpc0108.djvu
Action-angle variables for KdV......Page all_20214_to_00355.cpc0117.djvu
References......Page all_20214_to_00355.cpc0128.djvu
Origin of the Schrödinger equation......Page all_20214_to_00355.cpc0129.djvu
The Lax pair......Page all_20214_to_00355.cpc0133.djvu
Specialization to KdV......Page all_20214_to_00355.cpc0137.djvu
Alternate construction......Page all_20214_to_00355.cpc0141.djvu
Examples......Page all_20214_to_00355.cpc0143.djvu
Lenard's derivation of the KdV equation......Page all_20214_to_00355.cpc0147.djvu
References......Page all_20214_to_00355.cpc0151.djvu
The spectral parameter......Page all_20214_to_00355.cpc0152.djvu
Involution of the conserved quantities......Page all_20214_to_00355.cpc0162.djvu
KdV equation and the group SL(2,R)......Page all_20214_to_00355.cpc0165.djvu
References......Page all_20214_to_00355.cpc0173.djvu
Bäcklund transformations......Page all_20214_to_00355.cpc0174.djvu
Examples: Liouville equation......Page all_20214_to_00355.cpc0175.djvu
Sine-Gordon equation......Page all_20214_to_00355.cpc0179.djvu
Theorem of permutability......Page all_20214_to_00355.cpc0184.djvu
Bäcklund transformation for the KdV equation......Page all_20214_to_00355.cpc0186.djvu
Soliton solutions......Page all_20214_to_00355.cpc0192.djvu
References......Page all_20214_to_00355.cpc0200.djvu
Symplectic Geometry......Page all_20214_to_00355.cpc0201.djvu
Integrable models......Page all_20214_to_00355.cpc0208.djvu
Example: Geometrical approach to KdV......Page all_20214_to_00355.cpc0219.djvu
References......Page all_20214_to_00355.cpc0223.djvu
The Toda equation......Page all_20214_to_00355.cpc0225.djvu
Dual Poisson bracket structure......Page all_20214_to_00355.cpc0227.djvu
Conserved quantities......Page all_20214_to_00355.cpc0231.djvu
The Nijenhuis tensor......Page all_20214_to_00355.cpc0233.djvu
The Lax equation......Page all_20214_to_00355.cpc0237.djvu
References......Page all_20214_to_00355.cpc0242.djvu
Review of Lie algebra......Page all_20214_to_00355.cpc0244.djvu
Group structure of the Toda equations......Page all_20214_to_00355.cpc0252.djvu
The Lax pair......Page all_20214_to_00355.cpc0256.djvu
Integrability of the Toda system......Page all_20214_to_00355.cpc0259.djvu
References......Page all_20214_to_00355.cpc0264.djvu
The first order formulation......Page all_20214_to_00355.cpc0265.djvu
i) KdV......Page all_20214_to_00355.cpc0271.djvu
ii) MKdV......Page all_20214_to_00355.cpc0272.djvu
iii) Nonlinear Schrödinger equation (NSE)......Page all_20214_to_00355.cpc0274.djvu
iv) Sine-Gordon equation......Page all_20214_to_00355.cpc0275.djvu
v) Sinh-Gordon equation......Page all_20214_to_00355.cpc0276.djvu
NSE as a Hamiltonian system......Page all_20214_to_00355.cpc0277.djvu
Time development of the scattering data......Page all_20214_to_00355.cpc0278.djvu
References......Page all_20214_to_00355.cpc0289.djvu
The zero curvature condition......Page all_20214_to_00355.cpc0290.djvu
The transition matrix......Page all_20214_to_00355.cpc0293.djvu
Time evolution of the transition matrix......Page all_20214_to_00355.cpc0297.djvu
Fundamental Poisson bracket relations......Page all_20214_to_00355.cpc0300.djvu
Action-angle variables......Page all_20214_to_00355.cpc0304.djvu
References......Page all_20214_to_00355.cpc0314.djvu
The Bethe Ansatz......Page all_20214_to_00355.cpc0315.djvu
Quantum Inverse Scattering......Page all_20214_to_00355.cpc0321.djvu
Yang-Baxter Equation and Quantum Groups......Page all_20214_to_00355.cpc0332.djvu
References......Page all_20214_to_00355.cpc0337.djvu
Appendix......Page all_20214_to_00355.cpc0339.djvu
Index......Page all_20214_to_00355.cpc0351.djvu