The Elliptic Lie Triad: KdV and Riccati Equations, Infinigens, and Elliptic Genera

The Elliptic Lie Triad: Riccati and KdV Equations, Infinigens, and Elliptic Genera

(This site was not correctly updating, so the notes were transcribed to this pdf.)

Addendum to The Elliptic Lie Triad

Errata:

The last line on pg. 8 of the Addendum should have T_n rather than T_4.

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.

Additional notes:

(Added Jan. 31, 2016)  With \omega_1 = q and \omega_2 = 1/q and no other restrictions on these variables, the expansion for the inverse function \frac{df^{-1}(\omega)}{d\omega} in these notes  gives an ordinary generating function for the series a(n) = [n+1]_q = \frac{q^{n+1}-q^{-(n+1)}}{q-q^{-1}}, the so-called quantum numbers or integers of quantum group theory. See OEIS-A010892 for a specific example of q 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 :

\frac{d}{dx}R(x,v)-\frac{1}{2}R^2(x,v) = -2\cdot \psi^{1/2}(x,v) \frac{d^2}{dx^2} \psi^{-1/2}(x,v) = Q(x,v)= S_x\left \{ \int_0^x \psi(u,v) du \right \},

which holds in general for R = \frac{d}{dx} ln(\psi(x,v)), independent of the defn. of \psi(x,v) in the text. Here we have an embedded, more general Ricatti eqn. in R, a Sturm-Liouville equation in the reciprocal of the square root of \psi (a result of Lagrange), and, upon differentiation of the eqn., the spatial part of a viscous Burgers-Hopf eqn. op acting on R.

—–

The derivative of the Lorentz curvature of the curve y=f(x)=\int_0^x \psi(u,v) du  in the Lorentz plane with metric g = dxdy is \frac{d}{dx}\rho(x) = S_x\left \{ f(x)\right \} / {(f')^{1/2}(x)}=-2\cdot  \frac{d^2}{dx^2} \psi^{-1/2}(x,v)=Q(x,v)/\psi^{1/2}(x,v).

—–

Also \frac{d^2}{dx^2}R(x,v)-R(x,v)\frac{d}{dx}R(x,v) = \frac{d}{dx}Q(x,v), so, if R satisfies the inviscid Burgers-Hopf eqn.

\frac{d}{dv}U(x,v) = U(x,v)\frac{d}{dx}U(x,v), then

\frac{d^2}{dx^2}R(x,v)-\frac{d}{dv}R(x,v) = \frac{d}{dx} Q(x,v) and

\frac{d^2}{dx^2}R(x,v)-\frac{1}{2}R(x,v)\frac{d}{dx}R(x,v) - \frac{1}{2} \frac{d}{dv}R(x,v) = \frac{d}{dx} Q(x,v).

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):

R = \frac{1+x-\sqrt{(1-x)^2-4 x t}}{2 (1+t)t}-\frac{x}{t}

=x^2 + (1+2t)x^3+(1+5t+5t^2)x^4+(1+9t+21t^2+14t^3)x^5 + \cdots .

—–

The general relation on page 13 of the text between the viscous Burgers eqn. and the diffusion/heat eqn.

\partial_uR (u,v) + R (u,v) \partial_u R (u,v) + \frac {1}{2} \partial^2_u R (u,v) = \partial_u \frac {\partial_v \psi (u,v) + \frac {1}{2} \partial^2_u \psi (u,v) }{\psi (u,v)}

basically follows, as indicated in the text, from different reps of the Schwarzian derivative,

S_u\left\{q(u,v) \right \} = \partial_u^2 ln(\psi (u,v)) - \frac {1}{2}(\partial_u ln(\psi(u,v)))^2 = \partial_u R (u,v) - \frac {1}{2} (R (u,v))^2

= \frac {\partial_u^2 \psi (u,v)}{\psi (u,v)} - \frac {3}{2} (\frac{\partial_u\psi (u,v)}{\psi (u,v)})^2 = \frac {\partial_u^3 q (u,v)}{\partial_u q (u,v)}- \frac {3}{2}(\frac{\partial_u^2 q (u,v)}{\partial_u q (u,v)})^2,

implying

 \partial_u R (u,v) + (R (u,v))^2  =  \frac{\partial_u^2 \psi (u,v)}{\psi (u,v)}=\partial_u^2 ln(\psi(u,v)) + (\partial_u ln(\psi(u,v)))^2

for  R(u,v) = \partial_u ln(\psi(u,v)) and \psi (u,v)=\partial_u q (u,v) ,

and holds for general \psi (u,v) , not just for the Appell generating function, since

\partial_v R(u,v) = \partial_v \partial_u ln (\psi(u,v)) = \partial_u \partial_v ln (\psi (u,v)) = \partial_u \frac {\partial_v \psi (u,v)}{\psi (u,v)}.

This clearly generalizes to

\partial_v R (u,v) + \alpha \cdot R (u,v) \partial_u R (u,v)  + \alpha \cdot \frac {1}{2} \partial^2_u R (u,v) = \partial_u \frac {\partial_v \psi (u,v) \; + \; \alpha \cdot \frac {1}{2} \partial^2_u \psi (u,v) }{\psi (u,v)}

for  \alpha independent of u.

With the additional transformation \beta \; ln(G(u,v)) = ln(\psi(u,v)),

S_u\left\{q(u,v) \right \}/ \beta = \partial_u^2 ln(G(u,v)) - \frac {\beta}{2}[\partial_u ln(G(u,v))]^2

and

 \partial_u^2 ln(G(u,v)) + \beta \; [ \partial_u ln(G(u,v))]^2  = \frac{1}{\beta}  \frac{\partial_u^2 \psi (u,v)}{\psi (u,v)}.

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

u_t(x,t) = u_x(x,t) \cdot S_x\left\{u(x,t) \right \}

and concludes that any function satisfying this eqn. also satifies the Schwarzian KdV eqn.

\partial_t S_x\left\{u(x,t) \right \} - 3 S_x\left\{u(x,t) \right \} \partial_x S_x\left\{u(x,t) \right \} - \partial^3_x S_x\left\{u(x,t) \right \}=0.

—–

(Added 9/1/2016) The Legendre polynomials satisfy the differential equation

[(1-x^2)\frac{d}{dx}]^2 L_n(x) + n(n+1)(1-x^2)L_n(x)= 0.

Letting x = tanh(y) and L_n(tanh(y)) = T_n(y), this becomes

\frac{d^2}{dy^2}T_n(y) + n(n+1) sech^2(y)T_n(y) = 0,

and,using the relations above for the Sturm-Liouville equation for \psi(x)=1/T_n^2(x),  we obtain

Q_n(x)/2 = n(n+1) sech^2(x) = -2 [D_x^2 ln(T_n(x))+[D_x ln(T_n(x))]^2]

= \frac{1}{2} S_x\left \{ \int_0^x \psi(u) du \right \}

and

\frac{d^2}{dx^2}T_n(x) + \frac{1}{2} Q_n(x) T_n(x) = 0.

Though \int_0^x \psi(u) du here and ln(T_n(x)) 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:

\frac{\frac{d^2}{dx^2}T_n(x)}{T_n(x)} = - n(n+1)sech^2(x) < 0

for all real x implies that the curve T_n(x) 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 sech^2(x+t/3) = Q_n(x+t/3)/[2n(n+1)] 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,

\nu^{'}(\zeta) = \frac{1}{2 \gamma} (\nu-a)(\nu-b),

we can identify it with the Ricatti eqn. of the elliptic Lie triad in my pdf,

h^{'}(x) = \epsilon_2 (h-\omega_1)(h-\omega_2).

Travelling wave solutions for the Burgers’ Eqn.  22.35  of Olver

u_t+uu_x = \gamma u_{xx}

are u(x,t) = \nu(x-ct) with c = (a+b)/2

\nu(\zeta) = \frac{a \;e^{(b-a)(\zeta - \delta)/(2\gamma)} + \; b}{e^{(b-a)(\zeta - \delta)/(2\gamma)} + \; 1} = \frac{a \; e^{(b-a)(\zeta - \delta)/(4\gamma)} \; + \; b \; e^{-(b-a)(\zeta - \delta)/(4\gamma)}}{e^{(b-a)(\zeta - \delta)/(4\gamma)} \; + \; e^{-(b-a)(\zeta - \delta)/(4\gamma)}} .

Compare this with the expression for the forward fct. of the elliptic Lie triad

h(x) = \frac{ \; e^{-\epsilon_2 \omega_2x} - \; e^{-\epsilon_2 \omega_1x}}{e^{-\epsilon_2 \omega_2x} / \omega_2 - \; e^{-\epsilon_2 \omega_1x}/ \omega_1},

where h^{'}(x-\hat{c}t) = u(x-\hat{c}t),  with \hat{c} = \frac{\epsilon_2}{12 } (\omega_2-\omega_1)^2,  is a soliton solution of the KdV eqn.

u_t -uu_x + \frac{1}{12 \epsilon_2} u_{xxx}=0.

If we identify \omega_2 =b, \; \; \omega_1 = a, \; \;  \epsilon_2 = 1/(2\gamma), and define the integration constant or coordinate shift to be

\delta = \frac{-2 \gamma}{b-a} \; (ln(b/a)+ i \pi) = \frac{-1}{\epsilon_2 (\omega_2-\omega_1)} (ln(\omega_2 / \omega_1)+ i \pi),

then \nu(x) = h(x).

From Olver’s Burgers’ eqn. with \gamma =1,

c = -\frac{D_x^2 \nu(x) - \frac{1}{2}D_x (\nu(x))^2}{D_x \nu(x)} = -\frac{ D_xS_x\left\{ \int_0^x \psi(u,v) du \right \}}{D_x \nu(x)}=-D_x ln[D_x \nu(x)] + \nu(x)

with \nu(x) = \frac{d}{dx} ln(\psi(x,v)).

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.

Hirota’s direct method and Sato’s formalism in soliton theory” by Druitt

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.

Jacobi elliptic functions from a dynamical systems point of view” by Meyer
(https://math.uc.edu/~meyer/amm2001.pdf)

Continued fractions and integrable systems” by Beals, Sattinger, and Szmigielski

Elliptic cohomology and modular forms” by Landweber

An introduction to elliptic cohomology and topological modular forms” by Ravenel

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

See also nLab on elliptic cohomology and elliptic genus

The Symmetries of Solitons” by D. Palais

Elliptic solutions of nonlinear integrable equations and related topics” by Krichever

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)

The ubiquitous ‘c’: from the Stefan-Boltzmann law to quantum information” by Cardy

Conformal field theory and statistical mechanics” by Cardy

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)

Geodesic equations on diffeomorphism groups” by Vizman and slides “2-Cocycles and Geodesic Equations

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

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5 Responses to The Elliptic Lie Triad: KdV and Riccati Equations, Infinigens, and Elliptic Genera

  1. Tom Copeland says:

    Related: “Feynman motives and deletion-contraction relations” by Aluffi and Marcolli, particularly p. 42, http://arxiv.org/abs/0907.3225

  2. Tom Copeland says:

    Relater: “Combinatorial aspects of elliptic curves” by Musiker (https://arxiv.org/abs/0707.3179)

  3. Tom Copeland says:

    Related stuff:

    “Ground-state isolation and discrete flows in a rationally extended quantum harmonic oscillator” by José F. Cariñena, Mikhail S. Plyushchay http://arxiv.org/abs/1611.08051

    “Schwarzian derivative treatment of the quantum second-order supersymmetry anomaly, and coupling-constant metamorphosis” by Mikhail S. Plyushchay, http://arxiv.org/abs/1602.02179

  4. Tom Copeland says:

    More on the Schwarzian: pg. 21 of “Cartan for beginners: Differential geometry via moving frames and exterior differential systems” by Ivey and Landsberg

  5. Tom Copeland says:

    Related: “Some Analytical and Numerical Consequences of Sturm Theorems” by Javier Segura

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