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I am dealing with a PDE which can be written in the form

$$\frac{d}{dt} f(t) = \{a, f(t)\} + \{\{b, f(t)\}, f(t)\}$$

A Hamiltonian equation on a Poisson manifold has the following form: $$\frac{d}{dt} f(t) = \{a, f(t)\}$$

My equation seems to be a sort of "quadratic" Hamiltonian evolution. I hope that there exists some theory (existence, uniqueness, etc.) of such equations on Poisson manifolds which leverages the present algebraic structure.

I suppose even more generally one could study equations of the form $$\frac{d}{dt} f(t) = \sum_{k=0}^n \{ \dots \{a_k, \underset{k}{\underbrace{f\}, \dots, f\}}}$$.

If one says that the Hamiltonian evolution is the simplest possible ODE one could study on a Poisson manifold, then clearly these equations come right after that in terms of simplicity and naturality.

Robert Wegner
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  • Maybe only aesthetically related but if $P$ is a Poisson bi-vector field and $X$ is any vector field then $P(t) = P + t[X, P] + \frac{t^2}{2}[X, [X, P]] + \ldots$ (where $[-,-]$ is the Schouten bracket) is a (trivial) formal deformation of $P$. – Ricardo Buring Mar 12 '24 at 13:54

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