Symmetries and Integrability of Difference Equations.
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| Author / Creator: | Levi, D. (Decio) |
|---|---|
| Imprint: | Cambridge : Cambridge University Press, 2011. |
| Description: | 1 online resource (362 pages) |
| Language: | English |
| Series: | London Mathematical Society Lecture Note Series, 381 ; v. 381 London Mathematical Society Lecture Note Series, 381. |
| Subject: | |
| Format: | E-Resource Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/11830388 |
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| 100 | 1 | |a Levi, D. |q (Decio) |0 http://id.loc.gov/authorities/names/n80151915 | |
| 245 | 1 | 0 | |a Symmetries and Integrability of Difference Equations. |
| 260 | |a Cambridge : |b Cambridge University Press, |c 2011. | ||
| 300 | |a 1 online resource (362 pages) | ||
| 336 | |a text |b txt |2 rdacontent | ||
| 337 | |a computer |b c |2 rdamedia | ||
| 338 | |a online resource |b cr |2 rdacarrier | ||
| 490 | 1 | |a London Mathematical Society Lecture Note Series, 381 ; |v v. 381 | |
| 505 | 0 | |a Cover; Title; Copyright; Contents; List of figures; List of contributors; Preface; Introduction; 1 Lagrangian and Hamiltonian Formalism for Discrete Equations: Symmetries and First Integrals V. orodnitsyn and R. Kozlov; Abstract; 1.1 Introduction; 1.2 Invariance of Euler-Lagrange equations; 1.3 Lagrangian formalism for second-order difference equations; 1.4 Hamiltonian formalism for differential equations; 1.4.1 Canonical Hamiltonian equations; 1.4.2 The Legendre transformation; 1.4.3 Invariance of canonical Hamiltonian equations; 1.5 Discrete Hamiltonian formalism. | |
| 505 | 8 | |a 1.5.1 Discrete Legendre transform1.5.2 Variational formulation of the discrete Hamiltonian equations; 1.5.3 Symplecticity of the discrete Hamiltonian equations; 1.5.4 Invariance of the Hamiltonian action; 1.5.5 Discrete Hamiltonian identity and discrete Noether theorem; 1.5.6 Invariance of the discrete Hamiltonian equations; 1.6 Examples; 1.6.1 Nonlinear motion; 1.6.2 A nonlinear ODE; 1.6.3 Discrete harmonic oscillator; 1.6.4 Modified discrete harmonic oscillator (exact scheme); 1.7 Conclusion; Acknowledgments; References. | |
| 505 | 8 | |a 2 Painlevé Equations: Continuous, Discrete and Ultradiscrete B. Grammaticos and A. RamaniAbstract; 2.1 Introduction; 2.2 A rough sketch of the top-down description of the Painlevé equations; The Hamiltonian formulation of Painlevé equations; 2.3 A succinct presentation of the bottom-up description of the Painlevé equations; Derivation of continuous Painlevé equations; 2.4 Properties of the, continuous and discrete, Painlevé equations: a parallel presentation; 2.4.1 Degeneration cascade; 2.4.2 Lax pairs; 2.4.3 Miura and Bäcklund relations; 2.4.4 Particular solutions; 2.4.5 Contiguity relations. | |
| 505 | 8 | |a 2.5 The ultradiscrete Painlevé equations2.5.1 Degeneration cascade; 2.5.2 Lax pairs; 2.5.3 Miura and Bäcklund relations; 2.5.4 Particular solutions; 2.5.5 Contiguity relations; 2.6 Conclusion; References; 3 Definitions and Predictions of Integrability for Difference Equations J. Hietarinta; Abstract; 3.1 Preliminaries; 3.1.1 Points of view on integrability; 3.1.2 Preliminaries on discreteness and discrete integrability; 3.2 Conserved quantities; 3.2.1 Constants of motion for continuous ODE; 3.2.2 The standard discrete case; 3.2.3 The Hirota-Kimura-Yahagi (HKY) generalization. | |
| 505 | 8 | |a 3.3 Singularity confinement and algebraic entropy3.3.1 Singularity analysis for difference equations; 3.3.2 Singularity confinement in projective space; 3.3.3 Singularity confinement is not sufficient; 3.4 Integrability in 2D; 3.4.1 Definitions and examples; 3.4.2 Quadrilateral lattices; 3.4.3 Continuum limit; 3.4.4 Conservation laws; 3.5 Singularity confinement in 2D; 3.6 Algebraic entropy for 2D lattices; 3.6.1 Default growth of degree and factorization; 3.6.2 Search based on factorization; 3.7 Consistency around a cube; 3.7.1 Definition; 3.7.2 Lax pair; 3.7.3 CAC as a search method. | |
| 500 | |a 3.8 Soliton solutions. | ||
| 520 | |a A comprehensive introduction to and survey of the state of the art, suitable for graduate students and researchers alike. | ||
| 588 | 0 | |a Print version record. | |
| 504 | |a Includes bibliographical references. | ||
| 650 | 0 | |a Difference equations. |0 http://id.loc.gov/authorities/subjects/sh85037879 | |
| 650 | 0 | |a Symmetry (Mathematics) |0 http://id.loc.gov/authorities/subjects/sh2006001303 | |
| 650 | 0 | |a Integrals. |0 http://id.loc.gov/authorities/subjects/sh85067099 | |
| 650 | 7 | |a MATHEMATICS |x Calculus. |2 bisacsh | |
| 650 | 7 | |a MATHEMATICS |x Mathematical Analysis. |2 bisacsh | |
| 650 | 7 | |a Difference equations. |2 fast |0 (OCoLC)fst00893419 | |
| 650 | 7 | |a Integrals. |2 fast |0 (OCoLC)fst00975518 | |
| 650 | 7 | |a Symmetry (Mathematics) |2 fast |0 (OCoLC)fst01739417 | |
| 655 | 4 | |a Electronic books. | |
| 700 | 1 | |a Olver, Peter. | |
| 700 | 1 | |a Thomova, Zora. | |
| 700 | 1 | |a Winternitz, Pavel. | |
| 776 | 0 | 8 | |i Print version: |a Levi, Decio. |t Symmetries and Integrability of Difference Equations. |d Cambridge : Cambridge University Press, ©2011 |z 9780521136587 |
| 830 | 0 | |a London Mathematical Society Lecture Note Series, 381. | |
| 880 | 0 | 0 | |6 505-00/(S |g Machine generated contents note: |g 1. |t Lagrangian and Hamiltonian Formalism for Discrete Equations: Symmetries and First Integrals / |r R. Kozlov -- |g 1.1. |t Introduction -- |g 1.2. |t Invariance of Euler-Lagrange equations -- |g 1.3. |t Lagrangian formalism for second-order difference equations -- |g 1.4. |t Hamiltonian formalism for differential equations -- |g 1.4.1. |t Canonical Hamiltonian equations -- |g 1.4.2. |t Legendre transformation -- |g 1.4.3. |t Invariance of canonical Hamiltonian equations -- |g 1.5. |t Discrete Hamiltonian formalism -- |g 1.5.1. |t Discrete Legendre transform -- |g 1.5.2. |t Variational formulation of the discrete Hamiltonian equations -- |g 1.5.3. |t Symplecticity of the discrete Hamiltonian equations -- |g 1.5.4. |t Invariance of the Hamiltonian action -- |g 1.5.5. |t Discrete Hamiltonian identity and discrete Noether theorem -- |g 1.5.6. |t Invariance of the discrete Hamiltonian equations -- |g 1.6. |t Examples -- |g 1.6.1. |t Nonlinear motion -- |g 1.6.2. |t nonlinear ODE -- |g 1.6.3. |t Discrete harmonic oscillator -- |g 1.6.4. |t Modified discrete harmonic oscillator (exact scheme) -- |g 1.7. |t Conclusion -- |g 2. |t Painleve Equations: Continuous, Discrete and Ultradiscrete / |r A. Ramani -- |g 2.1. |t Introduction -- |g 2.2. |t rough sketch of the top-down description of the Painleve equations -- |g 2.3. |t succinct presentation of the bottom-up description of the Painleve equations -- |g 2.4. |t Properties of the, continuous and discrete, Painleve equations: a parallel presentation -- |g 2.4.1. |t Degeneration cascade -- |g 2.4.2. |t Lax pairs -- |g 2.4.3. |t Miura and Backlund relations -- |g 2.4.4. |t Particular solutions -- |g 2.4.5. |t Contiguity relations -- |g 2.5. |t ultradiscrete Painleve equations -- |g 2.5.1. |t Degeneration cascade -- |g 2.5.2. |t Lax pairs -- |g 2.5.3. |t Miura and Backlund relations -- |g 2.5.4. |t Particular solutions -- |g 2.5.5. |t Contiguity relations -- |g 2.6. |t Conclusion -- |g 3. |t Definitions and Predictions of Integrability for Difference Equations / |r J. Hietarinta -- |g 3.1. |t Preliminaries -- |g 3.1.1. |t Points of view on integrability -- |g 3.1.2. |t Preliminaries on discreteness and discrete integrability -- |g 3.2. |t Conserved quantities -- |g 3.2.1. |t Constants of motion for continuous ODE -- |g 3.2.2. |t standard discrete case -- |g 3.2.3. |t Hirota-Kimura-Yahagi (HKY) generalization -- |g 3.3. |t Singularity confinement and algebraic entropy -- |g 3.3.1. |t Singularity analysis for difference equations -- |g 3.3.2. |t Singularity confinement in projective space -- |g 3.3.3. |t Singularity confinement is not sufficient -- |g 3.4. |t Integrability in 2D -- |g 3.4.1. |t Definitions and examples -- |g 3.4.2. |t Quadrilateral lattices -- |g 3.4.3. |t Continuum limit -- |g 3.4.4. |t Conservation laws -- |g 3.5. |t Singularity confinement in 2D -- |g 3.6. |t Algebraic entropy for 2D lattices -- |g 3.6.1. |t Default growth of degree and factorization -- |g 3.6.2. |t Search based on factorization -- |g 3.7. |t Consistency around a cube -- |g 3.7.1. |t Definition -- |g 3.7.2. |t Lax pair -- |g 3.7.3. |t CAC as a search method -- |g 3.8. |t Soliton solutions -- |g 3.8.1. |t Background solutions -- |g 3.8.2. |t 1SS -- |g 3.8.3. |t NSS -- |g 3.9. |t Conclusions -- |g 4. |t Orthogonal Polynomials, their Recursions, and Functional Equations / |r M.E.H. Ismail -- |g 4.1. |t Introduction -- |g 4.2. |t Orthogonal polynomials -- |g 4.3. |t spectral theorem -- |g 4.4. |t Freud nonlinear recursions -- |g 4.5. |t Differential equations -- |g 4.6. |t q-difference equations -- |g 4.7. |t inverse problem -- |g 4.8. |t Applications -- |g 4.9. |t Askey-Wilson polynomials -- |g 5. |t Discrete Painleve Equations and Orthogonal Polynomials / |r A. Its -- |g 5.1. |t General setting -- |g 5.1.1. |t Orthogonal polynomials -- |g 5.1.2. |t Connections to integrable systems -- |g 5.1.3. |t Riemann-Hilbert representation of the orthogonal polynomials -- |g 5.1.4. |t Discrete Painleve equations -- |g 5.2. |t Examples -- |g 5.2.1. |t Gaussian weight -- |g 5.2.2. |t d-Painleve I -- |g 5.2.3. |t d-Painleve XXXIV -- |g 6. |t Generalized Lie Symmetries for Difference Equations / |r R.I. Yamilov -- |g 6.1. |t Introduction -- |g 6.1.1. |t Direct construction of generalized symmetries: an example -- |g 6.2. |t Generalized symmetries from the integrability properties -- |g 6.2.1. |t Toda Lattice -- |g 6.2.2. |t symmetry algebra for the Toda Lattice -- |g 6.2.3. |t continuous limit of the Toda symmetry algebras -- |g 6.2.4. |t Backlund transformations for the Toda equation -- |g 6.2.5. |t Backlund transformations vs. generalized symmetries -- |g 6.2.6. |t Generalized symmetries for PδE's -- |g 6.3. |t Formal symmetries and integrable lattice equations -- |g 6.3.1. |t Formal symmetries and further integrability conditions -- |g 6.3.2. |t Why integrable equations on the lattice must be symmetric -- |g 6.3.3. |t Example of classification problem -- |g 7. |t Four Lectures on Discrete Systems / |r S.P. Novikov -- |g 7.1. |t Introduction -- |g 7.2. |t Discrete symmetries and completely integrable systems -- |g 7.3. |t Discretization of linear operators -- |g 7.4. |t Discrete GLn connections and triangle equation -- |g 7.5. |t New discretization of complex analysis -- |g 8. |t Lectures on Moving Frames / |r P.J. Olver -- |g 8.1. |t Introduction -- |g 8.2. |t Equivariant moving frames -- |g 8.3. |t Moving frames on jet space and differential invariants -- |g 8.4. |t Equivalence and signatures -- |g 8.5. |t Joint invariants and joint differential invariants -- |g 8.6. |t Invariant numerical approximations -- |g 8.7. |t invariant bicomplex -- |g 8.8. |t Generating differential invariants -- |g 8.9. |t Invariant variational problems -- |g 8.10. |t Invariant curve flows -- |g 9. |t Lattices of Compact Semisimple Lie Groups / |r J. Patera -- |g 9.1. |t Introduction -- |g 9.2. |t Motivating example -- |g 9.3. |t Simple Lie groups and simple Lie algebras -- |g 9.3.1. |t Simple roots -- |g 9.3.2. |t Standard bases in Rn -- |g 9.3.3. |t Reflections and affine reflections in Rn -- |g 9.3.4. |t Weyl group and Affine Weyl group -- |g 9.4. |t Lattice grids FM [⊂] F [⊂] Rn -- |g 9.4.1. |t Examples of FM -- |g 9.5. |t W-invariant functions orthogonal on FM -- |g 9.6. |t Properties of elements of finite order -- |g 10. |t Lectures on Discrete Differential Geometry / |r Yu. B Suris -- |g 10.1. |t Basic notions -- |g 10.2. |t Backlund transformations -- |g 10.3. |t Q-nets -- |g 10.4. |t Circular nets -- |g 10.5. |t Q-nets in quadrics -- |g 10.6. |t T-nets -- |g 10.7. |t A-nets -- |g 10.8. |t T-nets in quadrics -- |g 10.9. |t K-nets -- |g 10.10. |t Hirota equation for K-nets -- |g 11. |t Symmetry Preserving Discretization of Differential Equations and Lie Point Symmetries of Differential-Difference Equations / |r P. |
| 880 | 0 | 0 | |6 505-00/(S |t Winternitz -- |g 11.1. |t Symmetry preserving discretization of ODEs -- |g 11.1.1. |t Formulation of the problem -- |g 11.1.2. |t Lie point symmetries of ordinary difference schemes -- |g 11.1.3. |t continuous limit -- |g 11.2. |t Examples of symmetry preserving discretizations -- |g 11.2.1. |t Equations invariant under SL1(2, R) -- |g 11.2.2. |t Equations invariant under SL2(2, R) -- |g 11.2.3. |t Equations invariant under the similitude group of the Euclidean plane -- |g 11.3. |t Applications to numerical solutions of ODEs -- |g 11.3.1. |t General procedure for testing the numerical schemes -- |g 11.3.2. |t Numerical experiments for a third-order ODE invariant under SL1(2, R) -- |g 11.3.3. |t Numerical experiments for ODEs invariant under SL2(2, R) -- |g 11.3.4. |t Numerical experiments for third-order ODE invariant under Sim(2) -- |g 11.4. |t Exact solutions of invariant difference schemes -- |g 11.4.1. |t Lagrangian formulation for second-order ODEs -- |g 11.4.2. |t Lagrangian formulation for second order difference equations -- |g 11.4.3. |t Example: Second-order ODE with three-dimensional solvable symmetry algebra -- |g 11.5. |t Lie point symmetries of differential-difference equations -- |g 11.5.1. |t Formulation of the problem -- |g 11.5.2. |t evolutionary formalism and commuting flows for differential equations -- |g 11.5.3. |t evolutionary formalism and commuting flows for differential-difference equations -- |g 11.5.4. |t General algorithm for calculating Lie point symmetries of a differential-difference equation -- |g 11.5.5. |t Theorems simplifying the calculation of symmetries of DδE -- |g 11.5.6. |t Volterra type equations and their generalizations -- |g 11.5.7. |t Toda type equations -- |g 11.5.8. |t Toda field theory type equations -- |g 11.6. |t Examples of symmetries of DδE -- |g 11.6.1. |t YdKN equation -- |g 11.6.2. |t Toda lattice. |
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