Classical and Quantum Nonlinear Integrable Systems: Theory and Application

by Kundu; A

• ISBN13: 9780750309592
• ISBN10: 0750309598
• Format: Hardcover
• Copyright: 2003-09-01
• Publisher: CRC Press

Summary

Covering both classical and quantum models, nonlinear integrable systems are of considerable theoretical and practical interest, with applications over a wide range of topics, including water waves, pin models, nonlinear optics, correlated electron systems, plasma physics, and reaction-diffusion processes. Comprising one part on classical theories and applications and another on quantum aspects, Classical and Quantum Nonlinear Integrable Systems: Theory and Application reviews the advances made in nonlinear integrable systems, with emphasis on the underlying concepts rather than technical details. It forms an outstanding introductory textbook as well as a useful reference for specialists.

Table of Contents

Preface	xi
PART I Classical Systems	1	(146)
A journey through the Korteweg--de Vries equation	3	(36)
M Lakshmanan
Introduction	3	(1)
Nonlinear dispersive waves: Scott Russell phenomenon and solitary waves	4	(3)
KdV equation and cnoidal waves and the solitary waves	6	(1)
The Fermi--Pasta--Ulam (FPU) numerical experiments on anharmonic lattices	7	(3)
The FPU lattice and recurrence phenomenon	8	(2)
The KdV equation again!	10	(2)
Asymptotic analysis and the KdV equation	10	(2)
Numerical experiments of Zabusky and Kruskal: the birth of solitons	12	(3)
Periodic boundary conditions	12	(1)
Initial condition with just two solitary waves	13	(2)
Hirota's bilinearization method: explicit soliton solutions	15	(3)
One-soliton solution	16	(1)
Two-soliton solution	16	(1)
N-soliton solutions	17	(1)
Asymptotic analysis	17	(1)
The Miura transformation and linearization of KdV: the Lax pair	18	(3)
The Miura transformation	18	(1)
Galilean invariance and the Schrodinger eigenvalue problem	19	(1)
Linearization of the KdV equation	20	(1)
Lax pair	20	(1)
Lax pair and the method of inverse scattering	21	(4)
The IST method for the KdV equation	21	(4)
Explicit soliton solutions	25	(5)
One-soliton solution (N = 1)	25	(1)
Two-soliton solution	26	(1)
N-soliton solution	27	(2)
Soliton interaction	29	(1)
Non-reflectionless potentials	29	(1)
Hamiltonian structure of KdV equation: complete integrability	30	(2)
KdV as a Hamiltonian dynamical system	30	(1)
Complete integrability of the KdV equation	31	(1)
Infinite number of conserved densities	32	(2)
Backlund transformations	34	(1)
The Painleve property for the KdV equations	35	(1)
Lie and Lie--Backlund symmetries	35	(1)
Conclusion	36	(3)
The Painleve methods	39	(25)
R Conte
M Musette
The classical programme of the Painleve school and its achievements	39	(3)
Integrability and Painleve property for partial differential equations	42	(2)
The Painleve test for ODEs and PDEs	44	(7)
The Fuchsian perturbative method	49	(1)
The non-Fuchsian perturbative method	50	(1)
Singularity-based methods towards integrability	51	(7)
Linearizable equations	51	(1)
Auto-Backlund transformation of a PDE: the singular manifold method	52	(2)
Single-valued solutions of the Bianchi IX cosmological model	54	(1)
Polynomial first integrals of a dynamical system	55	(1)
Solitary waves from truncations	56	(1)
First-degree birational transformations of Painleve equations	57	(1)
Liouville integrability and Painleve integrability	58	(1)
Discretization and discrete Painleve equations	59	(1)
Conclusion	60	(4)
Discrete integrability	64	(31)
K M Tamizhmani
A Ramani
B Grammaticos
T Tamizhmani
Introduction: who is afraid of discrete systems?	64	(1)
The detector gallery	65	(10)
Singularity confinement	66	(2)
The perturbative Painleve approach to discrete integrability	68	(2)
Algebraic entropy	70	(2)
The Nevanlinna theory approach	72	(3)
The showcase	75	(11)
The discrete KdV and its de-autonomization	75	(2)
The discrete Painleve equations	77	(6)
Linearizable systems	83	(3)
Beyond the discrete horizon	86	(5)
Differential-difference systems	86	(3)
Ultra-discrete systems	89	(2)
Parting words	91	(4)
The dbar method: a tool for solving two-dimensional integrable evolution PDEs	95	(12)
A S Fokas
Introduction	95	(4)
The dbar method	95	(1)
Coherent structures	96	(2)
Organization of this chapter	98	(1)
The KPI equation	99	(3)
The DSII equation	102	(3)
The defocusing DS equation	105	(1)
Summary	105	(2)
Introduction to solvable lattice models in statistical and mathematical physics	107	(40)
Tetsuo Deguchi
Introduction	107	(2)
Solvable vertex models	109	(16)
The six-vertex model	109	(2)
The partition function and the transfer matrix	111	(1)
Diagonalization of the transfer matrix	112	(3)
The free energy of the six-vertex model	115	(3)
Critical singularity in the antiferroelectric regime near the phase boundary	118	(2)
XXZ spin chain and the transfer matrix	120	(1)
Low-lying excited spectrum of the transfer matrix and conformal field theory	120	(5)
Various integrable models on two-dimensional lattices	125	(5)
Ising model and Potts model	125	(1)
Chiral Potts model	126	(1)
The eight-vertex model	127	(1)
IRF models	128	(2)
Yang--Baxter equation and the algebraic Bethe ansatz	130	(6)
Solutions to the Yang--Baxter equation	130	(2)
Algebraic Bethe ansatz	132	(4)
Mathematical structures of integrable lattice models	136	(11)
Braid group	136	(2)
Quantum groups (Hopf algebras)	138	(1)
Appendix. Commuting transfer matrices and the Yang--Baxter equations	139	(8)
PART II Quantum Systems	147	(141)
Unifying approaches in integrable systems: quantum and statistical, ultralocal and non-ultralocal	149	(33)
Anjan Kundu
Introduction	149	(2)
Integrable structures in ultralocal models	151	(5)
List of well-known ultralocal models	152	(4)
Unifying algebraic approach in ultralocal models	156	(11)
Generation of models	157	(6)
Fundamental and regular models	163	(3)
Fusion method	166	(1)
Construction of classical models	166	(1)
Integrable statistical systems: vertex models	167	(1)
Directions for constructing new classes of ultralocal models	168	(2)
Inhomogeneous models	168	(1)
Hybrid models	169	(1)
Non-fundamental statistical models	169	(1)
Unified Bethe ansatz solution	170	(3)
Quantum integrable non-ultralocal models	173	(5)
Braided extensions of QYBE	173	(2)
List of quantum integrable non-ultralocal models	175	(2)
Algebraic Bethe ansatz	177	(1)
Open directions in non-ultralocal models	177	(1)
Concluding remarks	178	(4)
The physical basis of integrable spin models	182	(26)
Indrani Bose
Introduction	182	(2)
Spin models in one dimension	184	(11)
Ladder models	195	(8)
Concluding remarks	203	(5)
Exact solvability in contemporary physics	208	(26)
Angela Foerster
Jon Links
Huan-Qiang Zhou
Introduction	208	(1)
Quantum inverse scattering method	209	(3)
Realizations of the Yang--Baxter algebra	211	(1)
Algebraic Bethe ansatz method of solution	212	(2)
Scalar products of states	213	(1)
A model for two coupled Bose--Einstein condensates	214	(6)
Asymptotic analysis of the solution	216	(4)
A model for atomic--molecular Bose--Einstein condensation	220	(6)
Asymptotic analysis of the solution	223	(1)
Computing the energy spectrum	224	(2)
The BCS Hamiltonian	226	(8)
A universally integrable system	227	(3)
Asymptotic analysis of the solution	230	(4)
The thermodynamics of the spin-1/2 XXX chain: free energy and low-temperature singularities of correlation lengths	234	(22)
Andreas Klumper
Christian Scheeren
Introduction	234	(2)
Lattice path integral and quantum transfer matrix	236	(4)
Mapping to a classical model	237	(2)
Bethe ansatz equations	239	(1)
Manipulation of the Bethe ansatz equations	240	(3)
Derivation of nonlinear integral equations	240	(2)
Integral expressions for the eigenvalue	242	(1)
Numerical results	243	(2)
Low-temperature asymptotics	245	(7)
Calculation to order O (β) and O(1)	247	(3)
O(1/β) corrections	250	(1)
O(1/β) corrections to the nonlinear integral equations	251	(1)
O(1/β) corrections to the eigenvalue	251	(1)
Summary and discussion	252	(4)
Appendix	253	(3)
Reaction--diffusion processes and their connection with integrable quantum spin chains	256	(32)
Malte Henkel
Reaction--diffusion processes	256	(4)
Quantum Hamiltonian formulation	260	(2)
Hecke algebra and integrability	262	(3)
Single-species models	265	(5)
The seven-vertex model	270	(2)
Further applications	272	(9)
Spectral integrability	273	(1)
Similarity transformations	273	(2)
Free fermions	275	(1)
Partial integrability	276	(2)
Multi-species models	278	(2)
Diffusion algebras	280	(1)
Outlook: local scale-invariance	281	(7)
Index	288