ar X iv : s ol v - in t / 9 70 80 05 v 1 1 4 A ug 1 99 7 On the Integrable Hierarchies

ar X iv : s ol v - in t / 9 70 80 06 v 1 1 4 A ug 1 99 7 Introduction to the Hirota bilinear method

ar X iv : s ol v - in t / 9 80 20 05 v 1 3 F eb 1 99 8 Hierarchy of Higher Dimensional Integrable System

Integrable equations in (1 + 1) dimensions have their own higher order integrable equations, like the KdV, mKdV and NLS hierarchies etc. In this paper we consider whether integrable equations in (2 + 1) dimensions have also the analogous hierarchies to those in (1 + 1) dimensions. Explicitly is discussed the Bogoyavlenskii-Schiff(BS) equation. For the BS hierarchy, there appears an ambiguity in...

ar X iv : s ol v - in t / 9 70 80 01 v 1 4 A ug 1 99 7 Two - dimensional soliton cellular automaton of deautonomized Toda - type

A deautonomized version of the two-dimensional Toda lattice equation is presented. Its ultra-discrete analogue and soliton solutions are also discussed. PACS: 03.20.+i; 03.40.Kf; 04.60.Nc

ar X iv : s ol v - in t / 9 40 70 05 v 1 2 7 Ju l 1 99 4 Integrable dynamics of a discrete curve and

We show that the following elementary geometric properties of the motion of a discrete (i.e. piecewise linear) curve select the integrable dynamics of the Ablowitz-Ladik hierarchy of evolution equations: i) the set of points describing the discrete curve lie in the sphere S; ii) the distance between any two subsequent points does not vary in time; iii) the dynamics does not depend explicitly on...

ar X iv : s ol v - in t / 9 80 80 19 v 1 2 7 A ug 1 99 8 Hypercomplex Integrable Systems

In this paper we study hypercomplex manifolds in four dimensions. Rather than using an approach based on differential forms, we develop a dual approach using vector fields. The condition on these vector fields may then be interpreted as Lax equations, exhibiting the integrability properties of such manifolds. A number of different field equations for such hypercomplex manifolds are derived, one...