Abstract

In order to solve a system of nonlinear rate equations one can try to use some soliton methods. The procedure involves three steps: find a ‘Lax representation’ where all the kinetic variables are combined into a single matrix \, all the kinetic constants are encoded in a matrix H; find a Darboux–Bäcklund dressing transformation for the Lax representation \]\), where f models a time-dependent environment; find a class of seed solutions \ that lead, via a nontrivial chain of dressings \ 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 \. Procedures that lead to a correct \ 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 catalytic feedbacks, with general time dependent parameters characterizing the nonlinearity. Explicit examples involve up to 42 interacting species.