Theory of Nonlinear Lattices

Soliton theory, the theory of nonlinear waves in lattices composed of particles interacting by nonlinear forces, is treated rigorously in this book.

Author: Morikazu Toda

Publisher: Springer Science & Business Media

ISBN: 9783642832192

Category: Science

Page: 225

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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.
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Jacobi Operators and Completely Integrable Nonlinear Lattices

G. Teschl, Trace Formulas and Inverse Spectral Theory for Jacobi Operators, Comm. Math. Phys. 196, 175–202 (1998). ... M. Toda, Theory of Nonlinear Lattices, 2nd enl. ed., Springer, Berlin, 1989. M. Toda, Theory of Nonlinear Waves and ...

Author: Gerald Teschl

Publisher: American Mathematical Soc.

ISBN: 9780821819401

Category: Mathematics

Page: 351

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This volume can serve as an introduction and a reference source on spectral and inverse spectral theory of Jacobi operators (i.e., second order symmetric difference operators) and applications of those theories to the Toda and Kac-van Moerbeke hierarchy. Beginning with second order difference equations, the author develops discrete Weyl-Titchmarsh-Kodaira theory, covering all classical aspects, such as Weyl $m$-functions, spectral functions, the moment problem, inverse spectral theory, and uniqueness results.Teschl then investigates more advanced topics, such as locating the essential, absolutely continuous, and discrete spectrum, subordinacy, oscillation theory, trace formulas, random operators, almost periodic operators, (quasi-)periodic operators, scattering theory, and spectral deformations. Utilizing the Lax approach, he introduces the Toda hierarchy and its modified counterpart, the Kac-van Moerbeke hierarchy. Uniqueness and existence theorems for solutions, expressions for solutions in terms of Riemann theta functions, the inverse scattering transform, Backlund transformations, and soliton solutions are derived. This text covers all basic topics of Jacobi operators and includes recent advances. It is suitable for use as a text at the advanced graduate level.
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Perspectives of Nonlinear Dynamics Volume 2

4 , 551-8 ; Nonlinear coupled oscillators . II . Comparison of theory with computer solutions . J. Math . Phys . 4 , 686-700 . Jackson , E.A. , Pasta , J.R. & Waters , J.F. ( 1968 ) . Thermal conductivity of one - dimensional lattices ...

Author: E. Atlee Jackson

Publisher: CUP Archive

ISBN: 0521426332

Category: Mathematics

Page: 652

View: 336

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The dynamics of physical, chemical, biological or fluid systems generally must be described by nonlinear models, whose detailed mathematical solutions are not obtainable. To understand some aspects of such dynamics, various complementary methods and viewpoints are of crucial importance. The presentation and style is intended to stimulate the reader's imagination to apply these methods to a host of problems and situations.
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Perspectives of Nonlinear Dynamics Volume 1

4 , 551-8 ; Nonlinear coupled oscillators . II . Comparison of theory with computer solutions . J. Math . Phys . 4 , 686-700 . Jackson , E.A. , Pasta , J.R. & Waters , J.F. ( 1968 ) . Thermal conductivity of one - dimensional lattices ...

Author: E. Atlee Jackson

Publisher: CUP Archive

ISBN: 0521426324

Category: Mathematics

Page: 520

View: 203

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The dynamics of physical, chemical, biological, or fluid systems generally must be described by nonlinear models, whose detailed mathematical solutions are not obtainable. To understand some aspects of such dynamics, various complementary methods and viewpoints are of crucial importance. In this book the perspectives generated by analytical, topological and computational methods, and interplays between them, are developed in a variety of contexts. This book is a comprehensive introduction to this field, suited to a broad readership, and reflecting a wide range of applications. Some of the concepts considered are: topological equivalence; embeddings; dimensions and fractals; Poincaré maps and map-dynamics; empirical computational sciences vis-á-vis mathematics; Ulam's synergetics; Turing's instability and dissipative structures; chaos; dynamic entropies; Lorenz and Rossler models; predator-prey and replicator models; FPU and KAM phenomena; solitons and nonsolitons; coupled maps and pattern dynamics; cellular automata.
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Non Linear Lattice

Chetverikov, A.P.; Ebeling, W.; Velarde, M.G. Non-linear excitations and electric transport in dissipative Morse-Toda lattices. Eur. Phys. J. B 2006, 51,87–99. Toda, M. Theory of Nonlinear Lattices, 2nd ed.; Springer: Berlin, Germany, ...

Author: Ignazio Licata and Sauro Succi

Publisher: MDPI

ISBN: 9783038423065

Category: Science

Page: 290

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This book is a printed edition of the Special Issue "Non-Linear Lattice" that was published in Entropy
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Lattice Dynamical Foundations Of Continuum Theories Elasticity Piezoelectricity Viscoelasticity Plasticity

A.B. D. Woods, W. Cochran, B.N. Brockhouse, Lattice dynamics of alkali halide crystals, Phys. Rev. 119. (1960) 380. ... M. Toda, Theory of nonlinear lattices, Solid State Sci., Wol. 20, Springer-Verlag, Berlin (1981).

Author: Attila Askar

Publisher: World Scientific

ISBN: 9789814518956

Category:

Page: 208

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This book presents a discussion of lattice dynamics for perfect and imperfect lattices and their relation to continuum theories of elasticity, piezoelectricity, viscoelasticity and plasticity. Some of the material is rather classical and close in spirit to solid state physics. A major aim here is to present a coherent theory for the four basic behavior types in the style of continuum mechanics. In each case, emphasis is on an explicit display of the physical mechanisms involved rather than general formalisms. The material is presented in terms of an atomistic picture for the discrete system. The basic ideas are believed to be relevant also at an intermediate scale in the continuum description of media with structure such as granular materials and composites.
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Travelling Waves and Periodic Oscillations in Fermi Pasta Ulam Lattices

Jacobi Operators and Completely Integrable Nonlinear Lattices, Amer. Math. Soc., Providence, R. I. Toda, M. (1989). Theory of Nonlinear Lattices, Springer, Berlin. Treschev, D. (2004). Travelling waves in FPU lattices, Discr. Cont.

Author: Alexander Pankov

Publisher: World Scientific

ISBN: 9781783260362

Category: Science

Page: 212

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This is a unique book that presents rigorous mathematical results on Fermi-Pasta-Ulam lattices, a field of great interest in nonlinear analysis, nonlinear science, mathematical physics, etc. It considers travelling waves and time periodic oscillations in infinite Fermi-Pasta-Ulam lattices, which are not necessarily spatially homogenous. Similar systems, infinite chains of linearly coupled nonlinear oscillators, are also discussed. The book is self-contained and includes a number of open problems, making it suitable for use in a course for graduate students. Contents:Infinite Lattice SystemsTime Periodic OscillationsTravelling Waves: Waves with Prescribed SpeedTravelling Waves: Further Results Readership: Researchers in nonlinear analysis, variational methods, critical point theory, nonlinear science and physics. Key Features:Self-contained and can be used in a course for graduate studentsContains a unified treatment of rigorous results using calculus of variations and critical point theoryKeywords:Fermi-Pasta-Ulam Lattices;Variational Methods;Travelling Waves;Periodic OscillationsReviews:“This well-written book is a reader-friendly and good-organized research monograph in the field of nonlinear science. It can be highly recommended for experts in ODE, PDE, and nonlinear physics.”Zentralblatt MATH
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Encyclopedia of Nonlinear Science

Theory of Nonlinear Lattices, Berlin and New York: Springer Wadati, M. & Toda, M. 1975. Bäcklund transformation for the exponential lattice. Journal of the Physical Society of Japan, 39: 1196–1203 INTEGRABLE TOPS See Rotating rigid ...

Author: Alwyn Scott

Publisher: Routledge

ISBN: 9781135455583

Category: Reference

Page: 1104

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In 438 alphabetically-arranged essays, this work provides a useful overview of the core mathematical background for nonlinear science, as well as its applications to key problems in ecology and biological systems, chemical reaction-diffusion problems, geophysics, economics, electrical and mechanical oscillations in engineering systems, lasers and nonlinear optics, fluid mechanics and turbulence, and condensed matter physics, among others.
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