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.

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.

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

The Bargmann Constraint and Integrable System for a Second-Order Spectral Problem

2015 ◽  
Vol 70 (11) ◽  
pp. 913-917
Author(s):  
Wei Liu ◽  
Yafeng Liu ◽  
Shujuan Yuan
Keyword(s):  
Coordinate System ◽  
Spectral Problem ◽  
Integrable System ◽  
Evolution Equations ◽  
Lagrange Equations ◽  
Lax Pairs ◽  
Infinite Dimensional ◽  
Hamilton System ◽  
Finite Dimensional ◽  
Dimensional Soliton

AbstractIn this article, the Bargmann system related to the spectral problem (∂2+q∂+∂q+r)φ=λφ+λφx is discussed. By the Euler–Lagrange equations and the Legendre transformations, a suitable Jacobi–Ostrogradsky coordinate system is obtained. So the Lax pairs of the aforementioned spectral problem are nonlinearised. A new kind of finite-dimensional Hamilton system is generated. Moreover, the involutive solutions of the evolution equations for the infinite-dimensional soliton system are derived.

Bäcklund transformations, Lax pair and solutions of a Sharma-Tasso-Olver-Burgers equation for the nonlinear dispersive waves

2021 ◽  
Author(s):  
Tian-Yu Zhou ◽  
Bo Tian ◽  
Su-Su Chen ◽  
Cheng-Cheng Wei ◽  
Yu-Qi Chen
Keyword(s):  
Burgers Equation ◽  
Bäcklund Transformation ◽  
Lax Pair ◽  
Ocean Dynamics ◽  
Bäcklund Transformations ◽  
Backlund Transformation ◽  
Dispersive Waves ◽  
Backlund Transformations ◽  
Hybrid Solutions ◽  
Kink Waves

Burgers-type equations are considered as the models of certain phenomena in plasma astrophysics, ocean dynamics, atmospheric science and so on. In this paper, a Sharma-Tasso-Olver-Burgers equation for the nonlinear dispersive waves is studied. Based on the Painlevé-Bäcklund equations, one auto-Bäcklund transformation and two hetero-Bäcklund transformations are derived. Motivated by the Burgers hierarchy, a Lax pair is given. Via two hetero-Bäcklund transformations with different constant seed solutions, we find some multiple-kink solutions, complex periodic solutions, hybrid solutions composed of the lump, periodic and multiple kink waves. Then we discuss the influence of the coefficients of the above equation on such solutions. Via the auto-Bäcklund transformation with the nontrivial seed solutions, we obtain certain lump-type solutions, kink-type solutions and recurrence relation of the above equation.

Bäcklund transformation, Lax pair, and solutions for the Caudrey–Dodd–Gibbon equation

2011 ◽  
Vol 52 (1) ◽  
pp. 013511 ◽  
Author(s):  
Qi-Xing Qu ◽  
Bo Tian ◽  
Kun Sun ◽  
Yan Jiang
Keyword(s):  
Bäcklund Transformation ◽  
Lax Pair ◽  
Backlund Transformation

On a Toda lattice hierarchy: Lax pair, integrable symplectic map and algebraic–geometric solution

2018 ◽  
Vol 32 (28) ◽  
pp. 1850344
Author(s):  
Xiao Yang ◽  
Dianlou Du
Keyword(s):  
Spectral Problem ◽  
Theta Function ◽  
Toda Lattice ◽  
Burgers Equation ◽  
Lax Pair ◽  
Discrete Variable ◽  
Riemann Theta Function ◽  
Symplectic Map ◽  
Discrete Counterpart ◽  
Toda Lattice Hierarchy

A Toda lattice hierarchy is studied by introducing a new spectral problem which is a discrete counterpart of the generalized Kaup–Newell spectral problem. Based on the Lenard recursion equation, Lax pair of the hierarchy is given. Further, the discrete spectral problem is nonlinearized into an integrable symplectic map. As a result, an algebraic–geometric solution in Riemann theta function of the hierarchy is obtained. Besides, two equations, the Volterra lattice and a (2[Formula: see text]+[Formula: see text]1)-dimensional Burgers equation with a discrete variable, yielded from the hierarchy are also solved.

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