32 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
The Sylvester equation and the elliptic Korteweg-de Vries system
The elliptic potential Korteweg-de Vries lattice system is a multi-component extension of the lattice potential Korteweg-de Vries equation, whose soliton solutions are associated with an elliptic Cauchy kernel (i.e., a Cauchy kernel on the torus). In this paper we generalize the class of solutions by allowing the spectral parameter to be a full matrix obeying a matrix version of the equation of the elliptic curve, and for the Cauchy matrix to be a solution of a Sylvester type matrix equation subject to this matrix elliptic curve equation. The construction involves solving the matrix elliptic curve equation by using Toeplitz matrix techniques, and analysing the solution of the Sylvester equation in terms of Jordan normal forms. Furthermore, we consider the continuum limit system associated with the elliptic potential Korteweg-de Vries system, and analyse the dynamics of the soliton solutions, which reveals some new features of the elliptic system in comparison to the non-elliptic case
Quantum discrete Dubrovin equations
The discrete equations of motion for the quantum mappings of KdV type are
given in terms of the Sklyanin variables (which are also known as quantum
separated variables). Both temporal (discrete-time) evolutions and spatial
(along the lattice at a constant time-level) evolutions are considered. In the
classical limit, the temporal equations reduce to the (classical) discrete
Dubrovin equations as given in a previous publication. The reconstruction of
the original dynamical variables in terms of the Sklyanin variables is also
achieved.Comment: 25 page
Elliptic (N,N^\prime)-Soliton Solutions of the lattice KP Equation
Elliptic soliton solutions, i.e., a hierarchy of functions based on an
elliptic seed solution, are constructed using an elliptic Cauchy kernel, for
integrable lattice equations of Kadomtsev-Petviashvili (KP) type. This
comprises the lattice KP, modified KP (mKP) and Schwarzian KP (SKP) equations
as well as Hirota's bilinear KP equation, and their successive continuum
limits. The reduction to the elliptic soliton solutions of KdV type lattice
equations is also discussed.Comment: 18 page