68 research outputs found
Elliptic Solutions of ABS Lattice Equations
Elliptic N-soliton-type solutions, i.e. solutions emerging from the
application of N consecutive B\"acklund transformations to an elliptic seed
solution, are constructed for all equations in the ABS list of quadrilateral
lattice equations, except for the case of the Q4 equation which is treated
elsewhere. The main construction, which is based on an elliptic Cauchy matrix,
is performed for the equation Q3, and by coalescence on certain auxiliary
parameters, the corresponding solutions of the remaining equations in the list
are obtained. Furthermore, the underlying linear structure of the equations is
exhibited, leading, in particular, to a novel Lax representation of the Q3
equation.Comment: 42 pages, 3 diagram
On the discrete and continuous Miura Chain associated with the Sixth PainlevƩ Equation
A Miura chain is a (closed) sequence of differential (or difference) equations that are related by Miura or B\"acklund transformations. We describe such a chain for the sixth Painlev\'e equation (\pvi), containing, apart from \pvi itself, a Schwarzian version as well as a second-order second-degree ordinary differential equation (ODE). As a byproduct we derive an auto-B\"acklund transformation, relating two copies of \pvi with different parameters. We also establish the analogous ordinary difference equations in the discrete counterpart of the chain. Such difference equations govern iterations of solutions of \pvi under B\"acklund transformations. Both discrete and continuous equations constitute a larger system which include partial difference equations, differential-difference equations and partial differential equations, all associated with the lattice Korteweg-de Vries equation subject to similarity constraints
Hyperspherical Trigonometry and Corresponding Elliptic Functions
We develop the basic formulae of hyperspherical trigonometry in
multidimensional Euclidean space, using multidimensional vector products, and
their conversion to identities for elliptic functions. We show that the basic
addition formulae for functions on the 3-sphere embedded in 4-dimensional space
lead to addition formulae for elliptic functions, associated with algebraic
curves, which have two distinct moduli. We give an application of these
formulae to the cases of a multidimensional Euler top, using them to provide a
link to the Double Elliptic model
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