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Abstract
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 ρ, all the kinetic constants are encoded in a matrix H; (2) find a Darboux–Bäcklund dressing transformation for the Lax representation iρ˙=[H,f(ρ)], where f models a time-dependent environment; (3) find a class of seed solutions ρ=ρ[0] that lead, via a nontrivial chain of dressings ρ[0]→ρ[1]→ρ[2]→… to new solutions, difficult to find by other methods. The latter step is not a trivial one since a non-soliton method has to be employed to find an appropriate initial ρ[0]. Procedures that lead to a correct ρ[0] have been discussed in the literature only for a limited class of H and f. Here, we develop a formalism that works for practically any H, and any explicitly time-dependent f. As a result, we are able to find exact solutions to a system of equations describing an arbitrary number of species interacting through (auto)catalytic feedbacks, with general time dependent parameters characterizing the nonlinearity. Explicit examples involve up to 42 interacting species.
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- Accepted or Published Version
- DOI:
- Digital Object Identifier (open in new tab) 10.1007/s10699-018-9568-9
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- Category:
- Articles
- Type:
- artykuły w czasopismach
- Published in:
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Foundations of Science
no. 24,
pages 95 - 132,
ISSN: 1233-1821 - Language:
- English
- Publication year:
- 2019
- Bibliographic description:
- Kuna M.: Systems, Environments, and Soliton Rate Equations: Toward Realistic Modeling// Foundations of Science -Vol. 24,iss. 1 (2019), s.95-132
- DOI:
- Digital Object Identifier (open in new tab) 10.1007/s10699-018-9568-9
- Verified by:
- Gdańsk University of Technology
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