# Systems, Environments, and Soliton Rate Equations: Toward Realistic Modeling

@article{Kuna2018SystemsEA, title={Systems, Environments, and Soliton Rate Equations: Toward Realistic Modeling}, author={Maciej Kuna}, journal={Foundations of Science}, year={2018}, volume={24}, pages={95-132} }

In order to solve a system of nonlinear rate equations one can try to use some soliton methods. The procedure involves three steps: (1) find a ‘Lax representation’ where all the kinetic variables are combined into a single matrix $$\rho$$ρ, all the kinetic constants are encoded in a matrix H; (2) find a Darboux–Bäcklund dressing transformation for the Lax representation $$i{{\dot{\rho }}}=[H,f(\rho )]$$iρ˙=[H,f(ρ)], where f models a time-dependent environment; (3) find a class of seed solutions…

## References

SHOWING 1-10 OF 74 REFERENCES

Darboux-integrable equations with non-Abelian nonlinearities

- Physics, Mathematics
- 2000

We introduce a new class of nonlinear equations admitting a representation in terms of Darboux-covariant compatibility conditions. Their special cases are, in particular, (i) the "general" von…

Abstract DNA-type systems

- Mathematics, Biology
- 2006

It is explained why non-Kolmogorovian probability models occurring in soliton kinetics are naturally associated with chemical reactions and why fluctuations based on Darboux transformations will not destroy the dynamics but only switch between a finite number of helical structures.

Darboux Transformations and Solitons

- Mathematics
- 1992

In 1882 Darboux proposed a systematic algebraic approach to the solution of the linear Sturm-Liouville problem. In this book, the authors develop Darboux's idea to solve linear and nonlinear partial…

von neumann equations with time-dependent hamiltonians and supersymmetric quantum mechanics

- Medicine, PhysicsPhysical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
- 2000

The method is analogous to supersymmetric quantum mechanics but is based on a different version of a Darboux transformation and represents a scattering of a solitonlike pulse on a three-level system.

Systems, environments, and soliton rate equations: A non-Kolmogorovian framework for population dynamics

- Physics, Biology
- 2013

The general formalism of soliton rate equations is introduced, links of contextuality to non-Kolmogorovity are introduced, and explicit examples of subsystems interacting with environments are worked out.

Construction of exact solutions of Bloch-Maxwell equation based on Darboux transformation

- Physics, Mathematics
- 2004

A new strategy, using Darboux transformations, of finding self-switching solutions of $i\dot{\rho} = [H, f({\rho})]$ is introduced. Unlike the previous ones, working for any f but for Hamiltonians…

Pattern formation outside of equilibrium

- Physics
- 1993

A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples…

Soliton Kinetic Equations with Non‐Kolmogorovian Structure: A New Tool for Biological Modeling?

- Mathematics
- 2006

Non‐commutative diagrams, where X → Y → Z is allowed and X → Z → Y is not, may equally well apply to Malusian experiments with photons traversing polarizers, and to sequences of elementary chemical…

A non-linear instability theory for a wave system in plane Poiseuille flow

- Physics
- 1971

The initial-value problem for linearized perturbations is discussed, and the asymptotic solution for large time is given. For values of the Reynolds number slightly greater than the critical value,…

Nonlinear von Neumann-type equations: Darboux invariance and spectra

- Physics, Mathematics
- 1999

Abstract Generalized Euler-Arnold-von Neumann density matrix equations can be solved by a binary Darboux transformation, given here in a new form: ρ [1]= e P ln( μ / ν ) ρe − P ln( μ / ν ) , where P…