The inviscid Hopf-Burgers equation is at the heart of combinatorics of convex polytopes surrounding compositional inversion. See, for example, the MO-Q "Why is there a connection between enumerative geometry and nonlinear waves?". For more on the combinatorics of associahedra, the inviscid HB eqn., compositional inversion (and quantum field theory, operads, ...), see links in OEIS A133437.
The KdV equation and its soliton solution is rife with combinatorics associated with the Eulerian numbers A008292. See, for example, the two pdfs presented in my post "The Elliptic Lie Triad: KdV and Riccati Equations, Infinigens, and Elliptic Genera".
See also papers by Yuji Kodama and Lauren Williams relating the combinatorics of the Eulerian numbers and Grassmannians to hydrodynamics (see refs in this MO-A).
See OEIS A036039 for links to connections between the KP hierarchy, which generalizes the KdV hierarchy, and combinatorics.
The entry for the refined Eulerian numbers A145271 illustrates the relations among flow equations and reps for compositional inversion involving several classic number arrays in combinatorics. See the Cf. section of the entry for links to partition polynomials for compositional inversion that have combinatorial models such as noncrossing partitions (central to many expositions on free probability theory and random matrices) and phylogenetic trees (and balls in bins), as well as the associahedra.
Edit (Sept. 2, 2022):
- For vector fields acting on an analytic function,
$$e^{tg(x)\partial_x} W(x) = W[f^{(-1)}(f(x)+t)],$$
with $g(x) = 1/f'(x) = 1/(\partial_x f(x))$. Then
$$\partial_t \; e^{tg(x)\partial} W(x) = g(x)\partial_x \; e^{tg(x)\partial} W(x) = \partial_t W[f^{(-1)}(f(x)+t)],$$
published by Charles Grave around 1850, predating Lie (Abel published related formulas even earlier). This flow equation is at the basis of the the analysis underlying OEIS A145271 and A139605, and the iterated Lie infinitesimal generator $g(x)\partial_x$ has an interpretation in terms of Cayley's analytic trees, among other algebraic-combinatorial-topological constructs for special reps of $g(x)$, such as the associahedra as noted in the ref for item 1 above.
- The exponential generating function of a Sheffer polynomial sequence (see, e.g., "How are Sheffer polynomials related to Lie theory?"), umbrally presented with $(p.(x))^n = p_n(x)$, is
$$e^{tR} p_{0}(x) = e^{tR} \; 1 = e^{tp.(x)} = S(x,t),$$
where $ R$ is a differential raising op defined by $R \; p_n(x) = p_{n+1}(x)$.
Then the associated p.d.e. is
$$\partial_t S(x,t) = \partial_t e^{tR} \; 1 = R e^{tR} \; 1 = R \; S(x,t) .$$
The family of binomial Sheffer sequences include the Stirling polynomials of the first and second kinds, of much importance in operational calculus and combinatorics in particular in normal ordering of products of $x$ and $D_x$. Examples are given in the MSE-Q "Is there a formula similar to $f(x+a) = e^{a\frac{d}{dx}}f(x)$ to express $f(\alpha\cdot x)$?".
For the normal sub-group of Sheffer polynomials, the Appell polynomials, of which the families of Hermite polynomials are examples important in analysis, combinatorics, and physics (see, e.g., "Why is the Gaussian so pervasive in mathematics?", $R$ is a differential raising / creation op of the form $R =x + \sum_{n \geq 0} b_n \frac{D_x^n}{n!}$. The lowering / destruction / annihilation op $L$ is $D_x$; i.e., $D_x p_n(x) = n p_{n-1}(x)$. The action of the exponential map on the identity gives the e.g.f. of the Appell sequence; that is, $e^{tR}1 = f(t)e^{xt} =e^{a.t}e^{xt} = e^{tp.(x)}$, umbrally, with $f(0) = p_0(t) =1$. Another common example of an Appell sequence is the set of Bernoulli polynomials, see "Intuitive explanation why "shadow operator" $\frac D{e^D-1}$ connects logarithms with trigonometric functions?".
For the relation of an Appell Sheffer sequence and the Riemann zeta function to the fractional calculus of Heaviside, see "Lie group heuristics for a raising operator for $(-1)^n \frac{d^n}{d\beta^n}\frac{x^\beta}{\beta!}|_{\beta=0}$" and "What's the matrix of logarithm of derivative operator ($\ln D$)? What is the role of this operator in various math fields?" (and links therein).
The Laguerre and associated Laguerre polynomials are general Sheffer sequences and also examples of confluent hypergeometric functions that are central in many discussions in combinatorics and physics--Laguerre histoires (see. e.g., "An involution on restricted Laguerre histories and its applications" by Chen and Fu), rook polynomials, and the radial part of wave functions for the hydrogen atom. (The Lah polynomials are normalized Laguerre polynomials of order -1, a binomial Sheffer sequence, and a family of Hermite polynomials is related to the Laguerre polynomials of orders 1/2 and -1/2.)
