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From another perspective, since we have been looking at travelling wave (soliton) solutions we can simply remove the time dependence and note that such a solution of the KdV must satisfy $u'''= \alpha \; uu'$ which then implies $(u')^2=4\; u^3-g_2u-g_3$ and so the solution of the KdV equation is a travelling Weierstrass elliptic function. This also implies that the KdV equation reduces to the inviscid Burgers equation for such solutions, so we come full circle back to the original example. These third order differential relations are discussed in "Category of vector bundles on algebraic curves and infinite dimensional Grassmannians" by Mulase.

From another perspective, since we have been looking at travelling wave (soliton) solutions we can simply remove the time dependence and note that such a solution of the KdV must satisfy $u'''= \alpha \; uu'$ which then implies $(u')^2=4\; u^3-g_2u-g_3$ and so the solution of the KdV equation is a travelling Weierstrass elliptic function. This also implies that the KdV equation reduces to the inviscid Burgers equation for such solutions, so we come full circle back to the original example. These third order differential relations are discussed in "Category of vector bundles on algebraic curves and infinite dimensional Grassmannians" by Mulase.

A Ricatti equation related to quadratic infinigens in Sl(2) is linked to a soliton solution $1-tanh^2(x-ct)=d[tanh(x-ct)]/dx$ of a Kdv equation in The Elliptic Lie Triad (following up on my comment below on Rzadowski's paper). The hyperbolic tangent can be regarded as an exponential generating function for the number of connected components in the space of M-polynomials in hyperbolic functions (ref. in OEIS A000111) or for a proportionality factor in the Kervaire-Milnor formula in homotopy theory for hyper-spheres involving normalized Bernoulli numbers.

A Ricatti equation related to quadratic infinigens in sl(2) is linked to a soliton solution $1-tanh^2(x-ct)=d[tanh(x-ct)]/dx$ of a Kdv equation in The Elliptic Lie Triad (following up on my comment below on Rzadowski's paper). The hyperbolic tangent can be regarded as an exponential generating function for the number of connected components in the space of M-polynomials in hyperbolic functions (ref. in OEIS A000111) or for a proportionality factor in the Kervaire-Milnor formula in homotopy theory for hyper-spheres involving normalized Bernoulli numbers.

More generally, the bivariate e.g.f. for the Eulerian numbers (A008292/A123125) with its associated quadratic (sl2) infinigen provides a soliton solution of the 1-D KdV equation, and the Eulerians are rife with ($A_n$ and $B_n$) connnections to enumerative algebraic geometry, as discussed by Hirzebruch, Losev and Manin, Batryev and Blume, Cohen, et al.