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Search for integrable two-component versions of the lattice equations in the ABS-list
Full text linkWe 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, final versio
Degree growth of lattice equations defined on a 3x3 stencil
Full text linkWe study complexity in terms of degree growth of one-component lattice equations defined on a stencil. The equations include two in Hirota bilinear form and the Boussinesq equations of regular, modified and Schwarzian type. Initial values are given on a staircase or on a corner configuration and depend linearly or rationally on a special variable, for example , in which case we count the degree in of the iterates. Known integrable cases have linear growth if only one initial values contains , and quadratic growth if all initial values contain . Even a small deformation of an integrable equation changes the degree growth from polynomial to exponential, because the deformation will change factorization properties and thereby prevent cancellations.19 pages, to appear in Open Communications in Nonlinear Mathematical Physics , Special Issue in Memory of Decio Lev
On the parametrization of solutions of the Yang--Baxter equations
Full text linkWe study all five-, six-, and one eight-vertex type two-state solutions of
the Yang-Baxter equations in the form , and analyze the interplay of the `gauge' and `inversion' symmetries of
these solution. Starting with algebraic solutions, whose parameters have no
specific interpretation, and then using these symmetries we can construct a
parametrization where we can identify global, color and spectral parameters. We
show in particular how the distribution of these parameters may be changed by a
change of gauge.Comment: 19 pages in LaTe
Darboux and Binary Darboux Transformations for Discrete Integrable Systems. II. Discrete Potential mKdV Equation
No full textThe paper presents two results. First it is shown how the discrete potential modified KdV equation and its Lax pairs in matrix form arise from the Hirota-Miwa equation by a 2-periodic reduction. Then Darboux transformations and binary Darboux transformations are derived for the discrete potential modified KdV equation and it is shown how these may be used to construct exact solutions.One of the author (YS) would like to acknowledge Professor Jarmo Hietarinta for his useful suggestions and hospitality when the author visited Turku University. The authors (YS and JXZ) also thank for the financial support from NSFC (Grant Numbers 11501510, 11271362, 11271266)
A search for bilinear equations passing Hirota’s three-soliton condition. I. KdV-type bilinear equations
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A search for bilinear equations passing Hirota’s three-soliton condition. III. Sine–Gordon-type bilinear equations
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