18 research outputs found
A Lax pair of a lattice equation whose entropy vanishes
In this paper, we present multi parametric quadgraph equations which are
consistent around the cube. These equations are obtained by applying a `double
twist' to known integrable equations. Furthermore, we perform a limit to one of
these equations to derive the non-symmetric equation which is given by
Hietarinta and Viallet. As a result, we obtain a novel Lax pair of this
equation
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
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 exchange. (2) When
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 ), using replacement 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 possible side-equations. However, we find that only
equations constructed with an even number of 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. Added references and discussion about decouplin