CANONICAL BÄCKLUND TRANSFORMATION — A NEW DISCRETE INTEGRABLE SYSTEM AND BETHE ANSATZ

2005 ◽  
Vol 20 (18) ◽  
pp. 4355-4361
Author(s):  
SUPRIYA MUKHERJEE ◽  
A. ROY CHOWDHURY ◽  
A. GHOSE CHOUDHURY
Keyword(s):  
Explicit Form ◽  
Integrable System ◽  
Canonical Variable ◽  
Bäcklund Transformation ◽  
Backlund Transformation ◽  
R Matrix ◽  
Discrete Integrable System ◽  
Lax Operator ◽  
Bethe Equations ◽  
Quantization Problem

A new discrete Lax operator involving discrete canonical variable is introduced which generate new integrable system, and is analyzed in the light of the new concept of canonical Bäcklund transformation and classical r-matrix. The generating function of the transformation is explicitly deduced. The second half of the paper deals with the quantization problem where an explicit form of the Bethe equations are deduced.

CANONICAL BÄCKLUND TRANSFORMATION FOR A DISCRETE INTEGRABLE CHAIN AND ITS ASSOCIATED PROPERTIES

2003 ◽  
Vol 18 (16) ◽  
pp. 1127-1139
Author(s):  
A. GHOSE CHOUDHURY ◽  
BARUN KHANRA ◽  
A. ROY CHOWDHURY
Keyword(s):  
Poisson Structure ◽  
Algebraic Structure ◽  
Bäcklund Transformation ◽  
Inverse Transformation ◽  
Strict Sense ◽  
Backlund Transformation ◽  
Lax Equation ◽  
R Matrix ◽  
Lax Operator ◽  
Time Part

The concept of a canonical Bäcklund transformation as laid down by Sklyanin is extended to a discrete integral chain, with a Poisson structure which is not canonical in the strict sense. The transformation is induced by an auxiliary Lax operator with a classical r-matrix which is similar in its algebraic structure to that of the original Lax operator governing the dynamics of the chain. Moreover, the transformation can be obtained from a suitable generating function. It is also shown how successive transformations can be composed to construct a new transformation. Finally an inverse transformation is also constructed. The compatibility of the transformation with the "time" part of the Lax equation is explicitly demonstrated. It is also shown that the Bianchi theorem of permutability holds good.

Mukherjee–Choudhury–Chowdhury spectral problem and the semi-discrete integrable system

2016 ◽  
Vol 30 (28n29) ◽  
pp. 1640027
Author(s):  
Xi-Xiang Xu ◽  
Ye-Peng Sun
Keyword(s):  
Spectral Problem ◽  
Integrable System ◽  
Bäcklund Transformation ◽  
Proper Time ◽  
Lax Pair ◽  
Hamiltonian Form ◽  
Backlund Transformation ◽  
Discrete Integrable System ◽  
Finite Dimensional ◽  
Symplectic Map

Starting from the Mukherjee–Choudhury–Chowdhury spectral problem, we derive a semi-discrete integrable system by a proper time spectral problem. A Bäcklund transformation of Darboux type of this system is established with the help of gauge transformation of the Lax pairs. By means of the obtained Bäcklund transformation, an exact solution is given. Moreover, Hamiltonian form of this system is constructed. Further, through a constraint of potentials and eigenfunctions, the Lax pair and the adjoint Lax pair of the obtained semi-discrete integrable system are nonlinearized as an integrable symplectic map and a completely integrable finite-dimensional Hamiltonian system in the Liouville sense. Finally, the involutive representation of solution of the obtained semi-discrete integrable system is presented.

Complete Integrability Prolongation Structure and Backlund Transformation for the Konno-Onno Equation

2000 ◽  
Vol 55 (5) ◽  
pp. 545-549
Author(s):  
Chandan Kr. Das ◽  
A. Roy Chowdhury
Keyword(s):  
Positive Result ◽  
Explicit Form ◽  
Bäcklund Transformation ◽  
Lax Pair ◽  
Independent Study ◽  
Complete Integrability ◽  
Painlevé Test ◽  
Backlund Transformation ◽  
Prolongation Structure ◽  
Prolongation Theory

Abstract Painleve analysis is used to study the complete integrability of the recently proposed Konno-Onno equation, which also leads to a general form of solutions of the system. An independent study, using the prolongation theory, gives the explicit form of the Lax pair which is then used to obtain the Backlund transformation connecting two sets of solutions of the system. The existence of the Lax pair and the positive result of the Painleve test indicate the complete integrability of the system

Fermionic covariant prolongation structure theory for a (2+1)-dimensional super nonlinear evolution equation

2018 ◽  
Vol 15 (07) ◽  
pp. 1850114 ◽  
Author(s):  
Zhaowen Yan ◽  
Chuanzhong Li
Keyword(s):  
Evolution Equation ◽  
Integrable System ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Bäcklund Transformation ◽  
Structure Theory ◽  
Backlund Transformation ◽  
Prolongation Structure ◽  
New Formula

In the framework of the fermionic covariant prolongation structure theory (PST), we investigate the integrability of a new [Formula: see text]-dimensional super nonlinear evolution equation (NEE). The prolongation structure of the super integrable system is presented. Moreover, we derive the Bäcklund transformation of the super integrable system.

Multi-soliton solutions and Bäcklund transformation for a two-mode KdV equation in a fluid

2016 ◽  
Vol 27 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Zi-Jian Xiao ◽  
Bo Tian ◽  
Hui-Ling Zhen ◽  
Jun Chai ◽  
Xiao-Yu Wu
Keyword(s):  
Bäcklund Transformation ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Backlund Transformation
Sign in / Sign up

Export Citation Format

Share Document