On exact solutions of nonlinear integrable equations via integral linearising transforms and generalised Backlund-Darboux transformations

1990 ◽  
Vol 23 (16) ◽  
pp. 3761-3768 ◽  
Author(s):  
B G Konopelchenko
Keyword(s):  
Exact Solutions ◽  
Darboux Transformations ◽  
Integrable Equations

scholarly journals A Five-Component Generalized mKdV Equation and Its Exact Solutions

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1145
Author(s):  
Bo Xue ◽  
Huiling Du ◽  
Ruomeng Li
Keyword(s):  
Exact Solutions ◽  
Spectral Problem ◽  
Darboux Transformation ◽  
Rational Functions ◽  
Mkdv Equation ◽  
Darboux Transformations ◽  
Lax Pairs ◽  
Gauge Transformations

In this paper, a 3 × 3 spectral problem is proposed and a five-component equation that consists of two different mKdV equations is derived. A Darboux transformation of the five-component equation is presented relating to the gauge transformations between the Lax pairs. As applications of the Darboux transformations, interesting exact solutions, including soliton-like solutions and a solution that consists of rational functions of e x and t, for the five-component equation are obtained.

scholarly journals EXACT SOLUTIONS TO QUANTUM FIELD THEORIES AND INTEGRABLE EQUATIONS

1996 ◽  
Vol 11 (14) ◽  
pp. 1169-1183 ◽  
Author(s):  
A. MARSHAKOV
Keyword(s):  
Exact Solutions ◽  
Gauge Field ◽  
Integrable Systems ◽  
Gauge Field Theories ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Integrable Equations ◽  
First Order

The exact solutions to quantum string and gauge field theories are discussed and their formulation in the framework of integrable systems is presented. In particular we consider in detail several examples of the appearance of solutions to the first-order integrable equations of hydrodynamical type and stress that all known examples can be treated as partial solutions to the same problem in the theory of integrable systems.

A Hierarchy of Lattice Soliton Equations Associated with a New Discrete Eigenvalue Problem and Darboux Transformations

2015 ◽  
Vol 16 (7-8) ◽  
pp. 301-306 ◽  
Author(s):  
Ning Zhang ◽  
Tiecheng Xia
Keyword(s):  
Eigenvalue Problem ◽  
Exact Solutions ◽  
Spectral Parameter ◽  
Darboux Transformation ◽  
Lattice Models ◽  
Darboux Transformations ◽  
Discrete Eigenvalue ◽  
Negative Power ◽  
Soliton Equations ◽  
Gauge Transformations

AbstractBy considering a new discrete isospectral eigenvalue problem, a hierarchy of integrable positive and negative lattice models is derived. It is shown that they correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. And the equation in the resulting hierarchy is integrable in Liouville sense. Further, a Darboux transformation is established for the typical equations by using gauge transformations of Lax pairs, from which the exact solutions are given.

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