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In order to solve a system of nonlinear rate equations one can try to usesome soliton methods. The procedure involves three steps: (1) Find a `Laxrepresentation' where all the kinetic variables are combined into a singlematrix $\rho$, all the kinetic constants are encoded in a matrix $H$; (2) finda Darboux-Backund dressing transformation for the Lax representation $i\dot\rho=[H,f(\rho)]$, where $f$ models a time-dependent environment; (3) find aclass of seed solutions $\rho=\rho[0]$ that lead, via a nontrivial chain ofdressings $\rho[0]\to \rho[1]\to \rho[2]\to\dots$ to new solutions, difficultto find by other methods. The latter step is not a trivial one since anon-soliton method has to be employed to find an appropriate initial $\rho[0]$.Procedures that lead to a correct $\rho[0]$ have been discussed in theliterature only for a limited class of $H$ and $f$. Here, we develop aformalism 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 equationsdescribing an arbitrary number of species interacting through (auto)catalyticfeedbacks, with general time dependent parameters characterizing thenonlinearity. Explicit examples involve up to 42 interacting species.

Systems, environments, and soliton rate equations (II): Toward realistic modeling