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Integrable nonlinear PDEs and their combinatorial algebra structure.

One interpretation of integrability for nonlinear PDEs is that they are linearisable. By this we mean that solutions can be found by solving the linearised form of the PDE and a linear Fredholm integral equation. The latter characterises the nonlinear PDE hierarchies that can be linearised. We will show this procedure, to determine such hierarchies, can be abstracted to a combinatorial algebra equipped with a quasi-Leibniz product (for Hankel operators). Integrability boils down to establishing polynomial expansions in the combinatorial algebra which in turn boils down to a system of linear algebraic equations for the polynomial coefficients. There are many open associated problems which we will discuss. For example, the nonlinear Schrodinger hierarchy requires a combinatorial triple system, and all such flows are examples of Fredholm Grassmannian flows.