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Theory of Nonlinear Lattices

Morikazu Toda (Autor)

Buch | Softcover

X, 225 Seiten

1988 | 2. Softcover reprint of the original 2nd ed. 1989
Springer Berlin (Verlag)
978-3-540-18327-3 (ISBN)

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Soliton theory, the theory of nonlinear waves in lattices composed of particles interacting by nonlinear forces, is treated rigorously in this book. The presentation is coherent and self-contained, starting with pioneering work and extending to the most recent advances in the field. Special attention is focused on exact methods of solution of nonlinear problems and on the exact mathematical treatment of nonlinear lattice vibrations. This new edition updates the material to take account of important new advances.

1. Introduction.- 1.1 The Fermi-Pasta-Ulam Problem.- 1.2 Hénon-Heiles Calculation.- 1.3 Discovery of Solitons.- 1.4 Dual Systems.- 2. The Lattice with Exponential Interaction.- 2.1 Finding of an Integrable Lattice.- 2.2 The Lattice with Exponential Interaction.- 2.3 Periodic Solutions.- 2.4 Solitary Waves.- 2.5 Two-Soliton Solutions.- 2.6 Hard-Sphere Limit.- 2.7 Continuum Approximation and Recurrence Time.- 2.8 Applications and Extensions.- 2.9 Poincaré Mapping.- 2.10 Conserved Quantities.- 3. The Spectrum and Construction of Solutions.- 3.1 Matrix Formalism.- 3.2 Infinite Lattice.- 3.3 Scattering and Bound States.- 3.4 The Gel'fand-Levitan Equation.- 3.5 The Initial Value Problem.- 3.6 Soliton Solutions.- 3.7 The Relationship Between the Conserved Quantities and the Transmission Coefficient.- 3.8 Extensions of the Equations of Motion and the Kac-Moerbeke System.- 3.9 The Bäcklund Transformation.- 3.10 A Finite Lattice.- 3.11 Continuum Approximation.- 4. Periodic Systems.- 4.1 Discrete Hill's Equation.- 4.2 Auxiliary Spectrum.- 4.3 Relation Between ?j (k) and ?j (0).- 4.4 Related Integrals on the Riemann Surface.- 4.5 Solution to the Inverse Problem.- 4.6 Time Evolution.- 4.7 A Simple Example (A Cnoidal Wave).- 4.8 Periodic System of Three-Particles.- 5. Application of the Hamilton-Jacobi Theory.- 5.1 Canonically Conjugate Variables.- 5.2 Action Variables.- 6. Recent Advances in the Theory of Nonlinear Lattices.- 6.1 The KdV Equation as a Limit of the TL Equation.- 6.2 Interacting Soliton Equations.- 6.3 Integrability.- 6.4 Generalization of the TL Equation.- 6.5 Two-Dimensional TL.- 6.6 Bethe Ansatz.- 6.7 The Thermodynamic Limit.- 6.8 Hierarchy of Nonlinear Equations.- 6.9 Some Numerical Results.- Appendices.- Simplified Answers to Main Problems.- References.-List of Authors Cited in Text.

Erscheint lt. Verlag	14.11.1988
Reihe/Serie	Springer Series in Solid-State Sciences
Zusatzinfo	X, 225 p.
Verlagsort	Berlin
Sprache	englisch
Maße	155 x 235 mm
Gewicht	375 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
	Naturwissenschaften ► Physik / Astronomie ► Allgemeines / Lexika
	Naturwissenschaften ► Physik / Astronomie ► Festkörperphysik
	Naturwissenschaften ► Physik / Astronomie ► Mechanik
	Naturwissenschaften ► Physik / Astronomie ► Theoretische Physik
Schlagworte	Field • Matrix • Soliton • Transformation
ISBN-10	3-540-18327-2 / 3540183272
ISBN-13	978-3-540-18327-3 / 9783540183273
Zustand	Neuware