Topic
Integrable system
About: Integrable system is a research topic. Over the lifetime, 13193 publications have been published within this topic receiving 282677 citations. The topic is also known as: Frobenius integrability & Liouville integrability.
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Book•
23 Oct 1990
TL;DR: In this article, the three-body problem: Moon-Earth-Sun, Three Methods of Section, Periodic Orbits, and Surface of Solution is considered, as well as the Diamagnetic Kepler Problem.
Abstract: Contents: Introduction- The Mechanics of Lagrange- The Mechanics of Hamilton and Jacobi- Integrable Systems- The Three-Body Problem: Moon-Earth-Sun- Three Methods of Section- Periodic Orbits- The Surface of Solution- Models of the Galaxy and of Small Molecules- Soft Chaos and the KAM Theorem- Entropy and Other Measures of Chaos- The Anisotropic Kepler Problem- The Transition From Classical to Quantum Mechanics- The New World of Quantum Mechanics- The Quantization of Integrable Systems- Wave Functions in Classically Chaotic Systems- The Energy Spectrum of a Classically Chaotic System- The Trace Formula- The Diamagnetic Kepler Problem- Motion on a Surface of Constant Negative Curvature- Scattering Problems, Coding and Multifractal Invariant Measures- References- Index
3,239 citations
Book•
01 Jan 1987
TL;DR: The Nonlinear Schrodinger Equation (NS Model) and Zero Curvature Representation (ZCR) as discussed by the authors have been used for the classification and analysis of Integrable Evolution Equations.
Abstract: The Nonlinear Schrodinger Equation (NS Model)- Zero Curvature Representation- The Riemann Problem- The Hamiltonian Formulation- General Theory of Integrable Evolution Equations- Basic Examples and Their General Properties- Fundamental Continuous Models- Fundamental Models on the Lattice- Lie-Algebraic Approach to the Classification and Analysis of Integrable Models- Conclusion- Conclusion
2,539 citations
TL;DR: In this paper, a new class of boundary conditions for quantum systems integrable by means of the quantum inverse scattering (R-matrix) method is described, which allows the author to treat open quantum chains with appropriate boundary terms in the Hamiltonian.
Abstract: A new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering (R-matrix) method. The method proposed allows the author to treat open quantum chains with appropriate boundary terms in the Hamiltonian. The general considerations are applied to the XXZ and XYZ models, the nonlinear Schrodinger equation and Toda chain.
1,774 citations
TL;DR: In this paper, the authors derived the one-loop mixing matrix for anomalous dimensions in N = 4 super Yang-Mills, which can be identified with the Hamiltonian of an integrable SO(6) spin chain with vector sites.
Abstract: We derive the one loop mixing matrix for anomalous dimensions in N=4 Super Yang-Mills. We show that this matrix can be identified with the Hamiltonian of an integrable SO(6) spin chain with vector sites. We then use the Bethe ansatz to find a recipe for computing anomalous dimensions for a wide range of operators. We give exact results for BMN operators with two impurities and results up to and including first order 1/J corrections for BMN operators with many impurities. We then use a result of Reshetikhin's to find the exact one-loop anomalous dimension for an SO(6) singlet in the limit of large bare dimension. We also show that this last anomalous dimension is proportional to the square root of the string level in the weak coupling limit.
1,676 citations