Abstract
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.
Highlights
The bidifferential calculus approach aims to extract the essence of integrability aspects of integrable partial differential or difference equations (PDDEs) and to express them, and relations between them, in a universal way, i.e. resolved from specific examples
Exchanging d and dleads to what is known in the literature as ‘negative flows’ [3]
Via the Miura transformation (18), Proposition 2 determines a family of sine-Gordon solutions
Summary
The bidifferential calculus approach (see [1] and the references therein) aims to extract the essence of integrability aspects of integrable partial differential or difference equations (PDDEs) and to express them, and relations between them, in a universal way, i.e. resolved from specific examples. A powerful, though simple to prove, result [1, 2, 3] (see section 6) generates families of exact solutions from a matrix linear system. In the following we briefly recall the basic framework and apply the latter result to a matrix generalization of the SIT equations
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