245 research outputs found

    Darboux integrability of trapezoidal H4H^{4} and H6H^{6} families of lattice equations I: First integrals

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    In this paper we prove that the trapezoidal H4H^{4} and the H6H^{6} 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

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    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,zy,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↔zy\leftrightarrow z exchange. (2) When z=yz=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 yy), using replacement y↔zy \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 828^2 possible side-equations. However, we find that only equations constructed with an even number of y↔zy \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. Added references and discussion about decouplin

    Algebraic entropy, symmetries and linearization of quad equations consistent on the cube

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    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 H4H^4 trapezoidal and the H6H^6 families are linearizable and in a few examples we show how we can effectively linearize them
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