2009/02/27 by Frank Nijhoff, James Atkinson, Jarmo Hietarinta · 7 citations
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Elliptic partial differential equation #Integrable system #Inverse scattering transform #Korteweg–de Vries equation #Lattice (music) #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Partial differential equation #Scalar (mathematics) #Soliton #nlin.SI
paper · pdf · doi:10.1088/1751-8113/42/40/404005
40 pages, 3 figures, submitted to Journal of Physics A: Mathematical and Theoretical, Special issue on nonlinearity and geometry: connections with integrability
arxiv created 2009/02/27 · openalex publication_date 2009/09/16 · arxiv updated 2015/05/12 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
In recent years there have been new insights into the integrability of quadrilateral lattice equations, i.e. partial difference equations which are the natural discrete analogues of integrable partial differential equations in 1+1 dimensions. In the scalar (i.e. single-field) case, there now exist classification results by Adler, Bobenko and Suris (ABS) leading to some new examples in addition to the lattice equations 'of KdV type' that were known since the late 1970s and early 1980s. In this paper, we review the construction of soliton solutions for the KdV-type lattice equations and use those results to construct N -soliton solutions for all lattice equations in the ABS list except for the elliptic case of Q4, which is left to a separate treatment.