Integrable mappings of KdV type and hyperelliptic addition theorems. (English) Zbl 0928.37015
Clarkson, Peter A. (ed.) et al., Symmetries and integrability of difference equations. Proceedings of the 2nd international conference, Canterbury, UK, July 1–5, 1996. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 255, 64-78 (1999).
The paper is devoted to study the following lattice version of the KdV equation:
\[
v'_{2j-1}=v_{2j},\quad v_{2j}'=v_{2j+1}+\frac a{v_{2j}}-\frac a{v_{2j+2}},\qquad j=1,\dots,P \tag{1}
\]
where \(a=p^2-q^2\), \(p,q\in\mathbb R\) are parameters of the lattice and the prime denotes the discrete time-shift corresponding to a translation in the second lattice direction. The periodic boundary conditions \(v_{2P+i}=v_i\) are imposed.
The explicit formulae for the solutions of (1) are obtained in terms of hyperelliptic abelian functions. The approach is based on the employment of special addition theorems for hyperelliptic functions, which can be interpreted, analogously to the elliptic case, as multidimensional rational maps.
For the entire collection see [Zbl 0910.00045].
The explicit formulae for the solutions of (1) are obtained in terms of hyperelliptic abelian functions. The approach is based on the employment of special addition theorems for hyperelliptic functions, which can be interpreted, analogously to the elliptic case, as multidimensional rational maps.
For the entire collection see [Zbl 0910.00045].
Reviewer: Serguei Zelik (Moskva)
MSC:
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |
| 37K60 | Lattice dynamics; integrable lattice equations |
| 39A70 | Difference operators |
