(This site was not correctly updating, so the notes were transcribed to this pdf.)
The last line on pg. 8 of the Addendum should have rather than .
The differential equation for the Legendre polynomials on page 9 has an erroneous first derivative in the second term. The correct diff eqn is given below.
(Added Jan. 31, 2016) With and and no other restrictions on these variables, the expansion for the inverse function in these notes gives an ordinary generating function for the series , the so-called quantum numbers or integers of quantum group theory. See OEIS-A010892 for a specific example of and the definition of quantum numbers/integers in Knot polynomial identities and quantum group coincidences by Morison, Peters, and Snyder, in A minus sign …. (Two constructions of Jones polynomials) by Tingley, and in Symmetric polynomials and Uq(sl2) by Jing.
(Added 8/2016) From pg. 281 of Osgood and also Ovsienko and Tabachnikov :
which holds in general for , independent of the defn. of in the text. Here we have an embedded, more general Ricatti eqn. in , a Sturm-Liouville equation in the reciprocal of the square root of (a result of Lagrange), and, upon differentiation of the eqn., the spatial part of a viscous Burgers-Hopf eqn. op acting on .
The derivative of the Lorentz curvature of the curve in the Lorentz plane with metric is .
Also , so, if satisfies the inviscid Burgers-Hopf eqn.
An example of a function that satisfies the inviscid Burgers eqn. is related to OEIS A086810, a generating function for the face numbers of the Stasheff associahedra (or their simplicial duals):
The general relation on page 13 of the text between the viscous Burgers eqn. and the diffusion/heat eqn.
basically follows, as indicated in the text, from different reps of the Schwarzian derivative,
for and ,
and holds for general , not just for the Appell generating function, since
This clearly generalizes to
for independent of .
With the additional transformation ,
See pages 24 and 25 in “Lectures on 2-D gravity and 2-D string theory” by Ginsparg and Moore for connections to a stress-energy tensor and note the transformation law has the form of that for the Schwarzian derivative under composition of functions.
See also Burgers equation presented in “An integrable hierarchy from vector models” by Damgaard and Shigemoto; the excellent discussions in “Chapter 22: Nonlinear partial differential equations” by Olver on the viscous Burgers, the heat, and the KdV eqns.; and some relations of the viscous Burgers’ equation to twist rates in scroll (spiral) waves in “Evidence for Burgers’ equation describing the untwisting of scroll rings” by Marts, Bansagi, and Steinbock. Google also references to Winfree and the forced Burgers’ equation with applications to excitable media for beautiful illustrations of the B-Z reaction.
G. Beffa introduces the Schwarzian evolution equation
and concludes that any function satisfying this eqn. also satifies the Schwarzian KdV eqn.
(Added 9/1/2016) The Legendre polynomials satisfy the differential equation
Letting and , this becomes
and,using the relations above for the Sturm-Liouville equation for , we obtain
Though here and are generally ill-defined, we are free to choose the lower limit of integration and a region of evaluation where they are well-defined, and the results analytically continue over the complete real line. In fact, the Sturm-Liouville equation has an easy geometric interpretation in this case:
for all real implies that the curve is continually curved towards the real axis with an inflection point at any zero of the curve. The differential relations (the S-L eqn. and the expanded Schwarzian giving identical results) are valid over the real axis for every point on the curve.
Note that is a soln. for the KdV eqn. given in the pdf notes.
See also “Quantum reflection from the Casimer-Polder potential” by Dufour.
(Added 9/21/2016) Inspecting Eqn. 22.37 on page 1190 of the Olver ref above,
we can identify it with the Ricatti eqn. of the elliptic Lie triad in my pdf,
Travelling wave solutions for the Burgers’ Eqn. 22.35 of Olver
Compare this with the expression for the forward fct. of the elliptic Lie triad
where , with , is a soliton solution of the KdV eqn.
If we identify and define the integration constant or coordinate shift to be
From Olver’s Burgers’ eqn. with ,
Related stuff (additional bibliography):
Google Fibonacci oscillators (Borzov, Marinho, etc.)
See also references in the previous entry on the Kervaire-Milnor formula related to elliptic cohomology/genera.
Blog piece on the Schwarzian derivative by Lamington
For a resume on research on the KdV equation, see “The history of q-calculus and a new method” by Ernst.
“Hyperbolic expressions of polynomial sequences …” by He, Shiue, and Weng.
“Continued fractions and integrable systems” by Beals, Sattinger, and Szmigielski
“Elliptic cohomology and modular forms” by Landweber
“Periodic cohomology theories defined by elliptic curves” by Landweber, Ravenel, and Stong
“Elliptic curves–basics” course notes, Univ. of Oslo, by Ellingsrud (?)
“Legendre polynomials and the elliptic genus” by Brylinski
“Legendre polynomials and applications” by Meziani
“Symbolic computaton of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs” by Baldwin, Goktas, Hereman, Hong, Martino, and Miller
“What is … an elliptic genus?” by Ochanine
“The Symmetries of Solitons” by D. Palais
“Sketches of KdV” by Arbarello
“Integer Sequences and Periodic Points” by Everest, Poorten, Puri, and Ward (pg. 5, characteristic polynomial)
“Moving frames for pseudo-groups. I The Maurer-Cartan forms” by Olver and Pohjanpelto (Example 6.3, pg. 18)
“Field theory models with infinite-dimensional symmeries, integrability, and supersymmetry” by Nissimov (page 21)
Statistical Field Theory: Vol. II by Itzykson and Drouffe (on the Schwarzian and Virasoro connections)
“Aspects of quantum groups and integrable systems” by Carroll
“Slice and Dice” Chapter 13 of the book Chaos: Classical and Quantum by Cvitanovic, Artuso, Mainieri, Tanner, and Vattay (pg. 235 on specific applications to Burgers’ equation and general nonlinear parabolic PDEs)
Integrable systems: An overview” by Ruijsenaars
“An introduction to conformal field theory” by Zuber (on the Schwarzian and Virasoro connections)
“Schwarzian derivatives and flows of surfaces” by Burstall, Pedit, and Pinkall
“Monstrous moonshine and the classification of CFT” by Gannon, p. 22