10.46298/ocnmp.9798
https://ocnmp.episciences.org/9798
Hietarinta, Jarmo
Jarmo
Hietarinta
Search for integrable two-component versions of the lattice equations in the ABS-list
We search and classify two-component 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 two-component 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 side-equations. 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.
Comment: 14 pages, final version
episciences.org
Nonlinear Sciences - Exactly Solvable and Integrable Systems
2022-07-25
2022-08-02
2022-08-02
eng
journal article
arXiv:2009.12208
10.48550/arXiv.2009.12208
2802-9356
https://ocnmp.episciences.org/9798/pdf
VoR
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Open Communications in Nonlinear Mathematical Physics
Volume 2
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