episciences.org_9798_1665089723
1665089723
episciences.org
raphael.tournoy+crossrefapi@ccsd.cnrs.fr
episciences.org
Open Communications in Nonlinear Mathematical Physics
28029356
10.46298/journals/ocnmp
https://ocnmp.episciences.org
08
02
2022
Volume 2
Search for integrable twocomponent versions of the lattice equations in the ABSlist
Jarmo
Hietarinta
We search and classify twocomponent versions of the quad equations in the
ABS list, under certain assumptions. The independent variables will be called
$y,z$ and in addition to multilinearity and irreducibility the equation pair is
required to have the following specific properties: (1) The two equations
forming the pair are related by $y\leftrightarrow z$ exchange. (2) When $z=y$
both equations reduce to one of the equations in the ABS list. (3) Evolution in
any corner direction is by a multilinear equation pair. One straightforward way
to construct such twocomponent pairs is by taking some particular equation in
the ABS list (in terms of $y$), using replacement $y \leftrightarrow z$ for
some particular shifts, after which the other equation of the pair is obtained
by property (1). This way we can get 8 pairs for each starting equation. One of
our main results is that due to condition (3) this is in fact complete for H1,
H3, Q1, Q3. (For H2 we have a further case, Q2, Q4 we did not check.) As for
the CAC integrability test, for each choice of the bottom equations we could in
principle have $8^2$ possible sideequations. However, we find that only
equations constructed with an even number of $y \leftrightarrow z$ replacements
are possible, and for each such equation there are two sets of "side" equation
pairs that produce (the same) genuine B\"acklund transformation and Lax pair.
08
02
2022
9798
arXiv:2009.12208
10.48550/arXiv.2009.12208
https://arxiv.org/abs/2009.12208v2
10.46298/ocnmp.9798
https://ocnmp.episciences.org/9798

https://ocnmp.episciences.org/9881/pdf