Ruling the inverse universe, the inviscid Hopf-Burgers evolution equation: Compositional inversion, free probability, associahedra, diff ids, integrable hierarchies, and translation

A mapping of associations among the iconic inviscid (Poisson-Riemann-Bateman) Hopf-Burgers nonlinear evolution / transport partial differential equation; compositional inversion; Lagrange inversion formulas; differential identities / integrable hierarchies / conservation laws; inversion partition polynomials, in particular the refined Euler characteristic partition polynomials of the associahedra; the formal free moment partition polynomials generating the free cumulants of free probability; and shifts / translations via variables and free moments is presented in the pdf below.

The main initial impetus in exploring the above relationships was to prove the particularly interesting novel empirical observation that the partial derivative w.r.t. the distinguished moment of the free moment partition polynomials (OEIS A350499, still in the draft queue at this time), which give the formal free cumulants in terms of the formal free moments of free probability, gives the refined Euler characteristic partition polynomials of the associahedra (A133437 and A111785) with a simple numerical factor. (See also my MathOverflow question “Combinatorics for the action of Virasoro / Kac–Schwarz operators: partition polynomials of free probability theory“.) This is an analog of a result in classic probability–the partial derivative w.r.t. the moment of the classic moment partition polynomials (A127671) gives the refined Euler characteristic partition polynomials of the permutahedra (A133314, see also A049019) with a simple numerical factor. Soon it became apparent that derivatives w.r.t. all the free moments, or indeterminates of the the refined Euler characteristic partition polynomials of the associahedra, are related to inviscid Hopf-Burgers nonlinear evolution p.d.e.s.

Ruling the inverse universe, the inviscid Hopf-Burgers evolution equation: Compositional inversion, free probability, associahedra, diff ids, integrable hierarchies, and translation

Instances of the integrable hierarchy of diff ids / conservation laws are in the section “5.4 Deformation of flow equations” on p. 57 and in Theorem 5.4 on pp. 66 of “On Emergent Geometry of the Gromov-Witten Theory of Quintic Calabi-Yau Threefold” by Zhou.

Edit (May 26, 2022): Another derivation of a central diff id is presented the MO-Q “A differential equation governing compositional inversion.“

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More Related Stuff (primarily iterated from my earlier notes for emphasis and easier access):

“Enumeration of ladder graphs” by Doob and Barrett, eqn. 7 on p. 4.

“Set partitions and integrable hierarchies” by Adler

“Mastering the Master Field” by Gopakumar and Gross, p. 18.

“Large N Gauge Theory – Expansions and Transitions” by Douglas, eqn. 32 on p. 8 and eqns. 73 and 74 on p. 20.

“An Introduction to Random Matrices” by Anderson, Guionnet, and Zeitouni

“Introduction to Random Matrices from a physicist’s perspective” by Zuber, a presentation

“Introduction to random matrices” by Zuber

“Burgers turbulence, kinetic theory of shock clustering, and complete integrability” by Srinivasan