 
Summary: Abstract DNAtype systems
Diederik Aerts1
and Marek Czachor1,2
1
Centrum Leo Apostel (CLEA) and Foundations of the Exact Sciences (FUND)
Vrije Universiteit Brussel, 1050 Brussels, Belgium
2
Katedra Fizyki Teoretycznej i Metod Matematycznych
Politechnika GdaŽnska, 80952 GdaŽnsk, Poland
Abstract
An abstract DNAtype system is defined by a set of nonlinear kinetic equations with polynomial
nonlinearities that admit soliton solutions associated with helical geometry. The set of equations
allows for two different Lax representations: A von Neumann form and a Darbouxcovariant Lax
pair. We explain why nonKolmogorovian probability models occurring in soliton kinetics are
naturally associated with chemical reactions. The most general known characterization of soliton
kinetic equations is given and a class of explicit soliton solutions is discussed. Switching between
open and closed states is a generic behaviour of the helices. The effect does not crucially depend
on the order of nonlinearity (i.e. types of reactions), a fact that may explain why simplified models
possess properties occuring in realistic systems. We explain also why fluctuations based on Darboux
transformations will not destroy the dynamics but only switch between a finite number of helical
