nwhm/lattice solitons

Finite difference discretizations of wave equations can also generate soliton like solutions. The discovery of this phenomenon is credited to Fermi-Pasta and Ulam. A review of this can be found here Thierry Dauxois and Stefano Ruffo (2008) Fermi-Pasta-Ulam nonlinear lattice oscillations. Scholarpedia, 3(8):5538.

Most of the work on these lattices has been in one dimension. The simplest relation between these lattices and partial differential equations is through a finite difference discretization. It is unclear whether one can repeat the analysis for a finite volume discretization. It may also be of interest to see if one could generalize this to higher dimensions.

Some relevant papers are:

Friesecke, G; Matthies, K. Geometric solitary waves in a 2D mass-spring lattice. Discrete Contin. Dyn. Syst. Ser. B 3 (2003), no. 1, 105–114. Mathscinet review

Herrmann, M. Oscillatory waves in discrete scalar conservation laws. Math. Models Methods Appl. Sci. 22 (2012), no. 1, 1150002, 21 pp. Mathscinet review, arXiv preprint.