245 research outputs found
Darboux integrability of trapezoidal and families of lattice equations I: First integrals
In this paper we prove that the trapezoidal and the families
of quad-equations are Darboux integrable systems. This result sheds light on
the fact that such equations are linearizable as it was proved using the
Algebraic Entropy test [G. Gubbiotti, C. Scimiterna and D. Levi, Algebraic
entropy, symmetries and linearization for quad equations consistent on the
cube, \emph{J. Nonlinear Math. Phys.}, 23(4):507543, 2016]. We conclude with
some suggestions on how first integrals can be used to obtain general
solutions.Comment: 34 page
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
Algebraic entropy, symmetries and linearization of quad equations consistent on the cube
We discuss the non autonomous nonlinear partial difference equations
belonging to Boll classification of quad graph equations consistent around the
cube. We show how starting from the compatible equations on a cell we can
construct the lattice equations, its B\"acklund transformations and Lax pairs.
By carrying out the algebraic entropy calculations we show that the
trapezoidal and the families are linearizable and in a few examples we
show how we can effectively linearize them
- …