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Register a new userOn solutions for some class of integrable difference equations
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Added by
Andrei Kirillovich Svinin
Publication date
2021
fields
Physics
and research's language is
English
Authors
Andrei K. Svinin
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In this paper we show that an arbitrary solution of one ordinary difference equation is also a solution for infinite class of difference equations. We also provide an example of such a solution that is related to sequence generated by second-order linear recurrent relations.
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We develop a new approach to the classification of integrable equations of the form $$ u_{xy}=f(u, u_x, u_y, triangle_z u triangle_{bar z}u, triangle_{zbar z}u), $$ where $triangle_{ z}$ and $triangle_{bar z}$ are the forward/backward discrete derivatives. The following 2-step classification procedure is proposed: (1) First we require that the dispersionless limit of the equation is integrable, that is, its characteristic variety defines a conformal structure which is Einstein-Weyl on every solution. (2) Secondly, to the candidate equations selected at the previous step we apply the test of Darboux integrability of reductions obtained by imposing suitable cut-off conditions.
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