A3 Vertaisarvioitu kirjan tai muun kokoomateoksen osa
Elementary introduction to discrete soliton equations
Tekijät: Jarmo Hietarinta
Toimittaja: Norbert Euler
Julkaisuvuosi: 2018
Kokoomateoksen nimi: Nonlinear Systems and Their Remarkable Mathematical Structures: Volume 1
Aloitussivu: 74
Lopetussivu: 92
ISBN: 978-1-1386-0100-0
Verkko-osoite: https://www.crcpress.com/Nonlinear-Systems-and-Their-Remarkable-Mathematical-Structures-Volume-1/Euler/p/book/9781138601000
Rinnakkaistallenteen osoite: https://arxiv.org/abs/1805.10492
We will give a short introduction to discrete or lattice soliton equations, with the particular example
of the Korteweg-de Vries as illustration. We will discuss briefly how Bäcklund transformations lead
to equations that can be interpreted as discrete equations on a Z2 lattice. Hierarchies of equations
and commuting flows are shown to be related to multidimensionality in the lattice context, and multidimensional
consistency is one of the necessary conditions for integrability. The multidimensional
setting also allows one to construct a Lax pair and a Bäcklund transformation, which in turn leads
to a method of constructing soliton solutions. The relationship between continuous and discrete
equations is discussed from two directions: taking the continuum limit of a discrete equation and
discretizing a continuous equation following the method of Hirota.