Elliptic solutions of the lattice CKP equation and its elliptic direct linearisation scheme

TL;DR

The paper develops an elliptic direct linearisation (DL) framework for the lattice CKP equation, leveraging a Weierstrass $\sigma$-function parametrisation and an elliptic Cauchy kernel to obtain a Lax triplet and a continuum-limit path to elliptic solitons. By formulating linear integral equations with a symmetric measure and introducing infinite spectral vectors, it derives a compact nonlinear structure with a determinant-based $\tau$-function, $\tau=\det(I+{\boldsymbol \Omega}_\xi{\boldsymbol C})$, from which elliptic $N$-soliton solutions are constructed. The CKP-specific reductions yield a lattice CKP equation together with a Lax triplet and a coupled OΔEs system for $u$, $v_\alpha$, and $s_{\alpha,\beta}$, all expressible in finite-dimensional data for soliton solutions. Furthermore, straight and skew continuum limits produce semi-discrete CKP equations, illustrating a unifying framework that connects discrete, semi-discrete, and continuous KP-type hierarchies; the approach also opens avenues for algebro-geometric and reduction studies to DKP.

Abstract

A direct linearisation scheme, based on an elliptic Cauchy kernel, is set up for the lattice CKP equation. This leads to an elliptic parametrisation of the lattice CKP equation, together with its Lax triplet, which allows us to perform appropriate continuum limits and construct elliptic solutions. By selecting appropriate integration measures and domains for the singular linear integral equation in the scheme, elliptic multi-soliton solutions of the lattice CKP equation are found.