scholarly journals Bidifferential Calculus, Matrix SIT and Sine-Gordon Equations

10.14311/1352 ◽  
2011 ◽  
Vol 51 (2) ◽  
Author(s):  
A. Dimakis ◽  
N. Kanning ◽  
F. Müller-Hoissen
Keyword(s):  
Linear System ◽  
Exact Solutions ◽  
Gordon Equation ◽  
Infinite Family ◽  
The Self ◽  
Sine Gordon Equation ◽  
Arbitrary Matrix ◽  
Matrix Size ◽  
Induced Transparency ◽  
Side Result

We express a matrix version of the self-induced transparency (SIT) equations in the bidifferential calculus framework. An infinite family of exact solutions is then obtained by application of a general result that generates exact solutions from solutions of a linear system of arbitrary matrix size. A side result is a solution formula for the sine-Gordon equation.

On a 2+1-dimensional integrable Ernst-type equation

1994 ◽  
Vol 446 (1927) ◽  
pp. 381-398 ◽  
Keyword(s):  
General Relativity ◽  
Integrable Systems ◽  
Gordon Equation ◽  
Type Equation ◽  
The Self ◽  
Coordinate Systems ◽  
Sine Gordon Equation ◽  
Type Transformation ◽  
Darboux System ◽  
Induced Transparency

A strong 2+1-dimensional integrable extension of Ernst’s equation of general relativity is proposed. Its richness is demonstrated by means of various canonical dimensional reductions and specializations which lead to formal analogues of well-known 1+1- and 2+1-dimensional integrable systems such as the self-induced transparency equations, the Konopelchenko-Rogers equations, a 2+1- dimensional Darboux system descriptive of conjugate coordinate systems, a single 2+1-dimensional sine-Gordon equation and the equations representing its Bäcklund transformation. A Darboux-Levi-type transformation is given and its compatibility with the above-mentioned reductions is shown.

Integrable and solvable systems, and relations among them

1985 ◽  
Vol 315 (1533) ◽  
pp. 451-457 ◽  
Keyword(s):  
Gordon Equation ◽  
Soliton Solutions ◽  
The Self ◽  
Sine Gordon Equation ◽  
Field Equations ◽  
Four Dimensions ◽  
Painlevé Property ◽  
All Solutions ◽  
Special Cases ◽  
Integrable Dynamical Systems

There are several different classes of differential equations that may be described as ‘integrable’ or ‘solvable’. For example, there are completely integrable dynamical systems; equations such as the sine—Gordon equation, which admit soliton solutions; and the self-dual gauge-field equations in four dimensions (with generalizations in arbitrarily large dimension). This lecture discusses two ideas that link all of these together: one is the Painlevé property, which says (roughly speaking) that all solutions to the equations are meromorphic; the other is that many of the equations are special cases (i.e. reductions) of others.

EXPLICIT AND EXACT SOLUTIONS TO THE mKdV-SINE-GORDON EQUATION

2008 ◽  
Vol 22 (15) ◽  
pp. 1471-1485 ◽  
Author(s):  
YUANXI XIE
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Exact Solutions ◽  
Gordon Equation ◽  
Variable Separation ◽  
Sine Gordon Equation ◽  
Simple Manner

By introducing an auxiliary ordinary differential equation and solving it by the method of variable separation, rich types of explicit and exact solutions of the mKdV-sine-Gordon equation are presented in a simple manner.

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