Compositional Inverse Pairs, the Inviscid Burgers-Hopf Equation, and the Stasheff Associahedra: A brief note (pdf) on some relations among these topics, including the Catalan and Fuss-Catalan numbers. References are provided linking the analysis with the distribution of eigenvalues of random matrices, the Wigner semicircle law, and moduli spaces for marked Riemann surfaces.

**Related stuff**:

For some notes on the Burgers equation (generalized form), see Lectures on Partial Differential Equations (Chapter 4) by R. Salmon.

See also the note on the inviscid Burgers equation in “Free probability theory: random matrices and von Nuemann algebras” by Voiculescu and page 112 of The Semicircle Law, Free Random Variables, and Entropy by Hiai and Petz.

(Added Sept. 2016):

See page 17 of “Mastering the master field” by Gopakumar and Gross (referenced in 2014 in the post “Appell polynomials, cumulants, …”

“On the large N limit of the Itzykson-Zuber Integral” by Matytsin

“Chapter 22: Nonlinear partial differential equations” by Olver

(Added Dec 25, 2018)

“Quantum deformation theory of the Airy curve and mirror symmetry of a point” by Zhou

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See also “Burgers-like equation for breakdown of the chiral symmetrry in QCD” by Blaizot, Nowak and Warchol ( http://arxiv.org/abs/1303.2357), “Large N_c confinement and turbulence” by Blaizot and Nowak (http://arxiv.org/abs/0801.1859).

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See also the pdf Partial Differential Equations by Faris.

See also “Large N Gauge Theory — Expansions and Transitions” by Michael R. Douglas (ref. In Mastering the Master Field) https://arxiv.org/abs/hep-th/9409098

Related: “An Introduction to Wave Equations and Solitons” by Richard S. Palais https://www.ma.utexas.edu/users/uhlen/solitons/notes.pdf and the later post here on “The Lie Triad … .”

See the inviscid Burgers equation in “Convolution powers in the operator-valued framework” by Michael Anshelevich, Serban T. Belinschi, Maxime Fevrier, Alexandru Nica

https://arxiv.org/abs/1107.2894

“Large Nc Confinement, Universal Shocks and Random Matrices” by Blaizot and Nowak discusses the inviscid and viscid Burgers’ equation

For more on the generalized Catalan numbers and Lagrange-Burman inversion, see “Function Series, Catalan Numbers, and Random Walks on Trees” by Bajunaid, Cohen, Colonna, and Singman.

See Thm. 3 of “On quantum and relativistic mechanical analogues in mean field spin models” by Barra, Lorenzo, Guerra, and Moro

See also “Remarks on intersection numbers and integrable

hierarchies. I. Quasi-triviality” by Dubrovin and Yang https://arxiv.org/abs/1905.08106