The discrete Korteweg-de Vries equation
The lattice version of the KdV equation studied in this paper is \[ (p - q + u_{n, m + 1} - u_{n + 1, m}) (p + q - u_{n + 1, m + 1} + u_{n, m}) = p^2 - q^2, \] where \(p,q \in \mathbb{C}\) are lattice parameters. The discretization has been done both in space and time. This equation was derived and studied in a series of previous papers. In this paper the main properties of this lattice version are shown and taking into account in order to study its solutions. One of the main properties, in contrast with the continuous equation, is the covariance with respect to the interchange of the two discrete variables and the lattice parameters. Nevertheless its relationship with the classical KdV equation is carefully justified by means of continuum limits. The lost of covariance is an artefact of the process of taking continuum limits. Solutions arising from periodic initial value problem in the lattice are considered. This leads to a reduction to integrable finite-dimensional mappings. Other special solutions are also studied. Similar integrable lattice equations are also treated: the modified KdV equation, the lattice analogue of the sine-Gordon equation, etc. In the final part the authors study the algebraic background of these lattice equations, deriving them from the so-called direct linearization approach method, a generalization of the inverse scattering approach.
scientific article; zbMATH DE number 790795
scientific article