Anindya Ghose Choudhury

Quantum integrable systems

NONLINEAR SYSTEMS AND CLASSICAL IST Introduction Definition of Integrability Lax Pair Technique Inverse Scattering Transform Hamiltonian Structure COORDINATE BETHE ANSATZ Introduction Nonlinear Systems and the CBA Fermionic System Boundary Condition in Bethe Ansatz Heisenberg Spin Chain Spin of the Bethe Ansatz State Other Integrable Models YANG-BAXTER EQUATION Introduction General Description Factorized Scattering Baxters Star Triangle Relation Vertex Models Reflection Equation Algebra CONTINUOUS INTEGRABLE SYSTEMS Introduction Quantum Continuous Integrable Systems Conserved Quantities Nonultralocal systems and the YBE Operator Product Expansion and YBE Finite Boundary Conditions Modified Classical Yang-Baxter Equation ALGEBRAIC BETHE ANSATZ Introduction Discrete Self Trapping Model Asymmetric XXZ Model in a Magnetic Field Analytical Bethe Ansatz Off-Shell Bethe Ansatz Nested Bethe Ansatz Fusion Procedure Fusion Procedure for Open chains Fusion Procedure for Transfer Matrices Application of Fusion Procedure INTEGRABLE LONG-RANGE MODELS Introduction Long-Range Models from the ABA Symmetry Transformation Calogero-Moser Models SEPARATION OF VARIABLES Introduction Hamilton-Jacobi Equation Sklyanins Method for SoV Goryachev-Chaplygin Top Quantum Case and the Role of Lie Algebra Bi-Hamiltonian Structure and SoV SoV for GCM Model SoV and Boundary Conditions BACKLUND TRANSFORMATIONS Introduction Permutability Theorem Backlund Transformations and Classical Inverse Scattering Backlund Transformations from Riccati Equation Darboux Backlund Transformations The Exponential Lattice Canonical Transformations Group Property of Backlund Transformations Recent Developments in Backlund Transformation Theory Sklyanins Formalism for Canonical Backlund Transformations Extended Phase Space Method Quantization of Backlund Transformations Method of Projection Operators QUANTUM GLM EQUATION Introduction Quantum GLM Equation Quantum Floquet Function Exact Quantization Quantum GLM Equation in a Continuous System Bound States and an Alternative Approach APPENDICES Direct Product Calculus Grassman Algebra Bethe Ansatz Equation AKNS Problem BIBLIOGRAPHY INDEX

Bi-Hamiltonian Structures of 3D Chaotic Dynamical Systems

We study Hamiltonian structures of dynamical systems with three degrees of freedom which are known for their chaotic properties, namely Lu, modified Lu, Chen, T and Qi systems. We show that all these flows admit bi-Hamiltonian structures depending on the values of their parameters.

Dynamical Studies of Equations from the Gambier Family

We consider the hierarchy of higher-order Riccati equations and establish their connection with the Gambier equation. Moreover we investigate the relation of equations of the Gambier family to other nonlinear differential systems. In particular we explore their connection to the generalized Ermakov-Pinney and Milne-Pinney equations. In addi- tion we investigate the consequence of introducing Okamotos folding transformation which maps the reduced Gambier equation to a Lienard type equation. Finally the conjugate Hamiltonian aspects of certain equations belonging to this family and their connection with superintegrability are explored.

Superintegrable cases of four-dimensional dynamical systems

Degenerate tri-Hamiltonian structures of the Shivamoggi and generalized Raychaudhuri equations are exhibited. For certain specific values of the parameters, it is shown that hyperchaotic Lü and Qi systems are superintegrable and admit tri-Hamiltonian structures.

Canonical and noncanonical variables, Baxter’s Q-operator and the XXX model☆

Abstract Baxter’s Q -operator for the XXX model is analysed to study the different roles played by the canonical and noncanonical variables using the formalism of Sklyanin. In this approach to the study of Backlund transformation (BT) and Baxter’s Q -operator one requires two Lax operators obeying the same Poisson algebra (i.e. having the same classical r matrix). Usually the nonlinear variables in the Lax operator are canonically conjugate quantities. In this communication we have shown that even when the variables in one Lax operator are not canonical, it is still possible to construct the BT and Q -operator by a proper representation of the corresponding nonlinear variables. The price for this is that, the BT is not easily interpretable as a canonical transformation in the conventional sense of the term. However the Q -operator for the XXX chain turns out to be similar to that obtained by Derkachov [J Phys A 32 (1999) 316] using a similar representation but in a different manner. It is important to note that although, in contrast to the results obtained by Kuznetsov et al. [J Phys A 33 (2000) 171], the Q -operator depends on two adjacent sites ( x i , x i −1 ), its relevant properties can be explicitly established.

Bäcklund transformation for the first flows of the relativistic Toda hierarchy and associated properties

In this communication we study a class of one parameter dependent auto-Bäcklund transformations for the first flow of the relativistic Toda lattice and also a variant of the usual Toda lattice equation. It is shown that starting from the Hamiltonian formalism such transformations are canonical in nature with a well defined generating function. The notion of spectrality is also analyzed and the separation variables are explicitly constructed.