group: A new connection

Schwarzian integrable systems and the Möbius group: A new connection March 4, 2009

History The Schwarzian KdV equation, SKdV (u) := u t u x [ uxxx u x 3 2 uxx 2 ] ux 2 = 0.

History The Schwarzian KdV equation, SKdV (u) := u t u x [ uxxx u x 3 2 uxx 2 ] ux 2 = 0. Involves the Schwarzian derivative (Named after Schwarz by Cayley, discovered by Lagrange in about 1779).

History The Schwarzian KdV equation, SKdV (u) := u t u x [ uxxx u x 3 2 uxx 2 ] ux 2 = 0. Involves the Schwarzian derivative (Named after Schwarz by Cayley, discovered by Lagrange in about 1779). Discovered by John Weiss in 1982 (The Painlevé property for partial differential equations. II: Bäcklund transformation, Lax pairs and the Schwarzian derivative) as the Singularity manifold equation for KdV and mkdv, connected to these equations by a Bäcklund transformation.

History The Schwarzian KdV equation, SKdV (u) := u t u x [ uxxx u x 3 2 uxx 2 ] ux 2 = 0. Involves the Schwarzian derivative (Named after Schwarz by Cayley, discovered by Lagrange in about 1779). Discovered by John Weiss in 1982 (The Painlevé property for partial differential equations. II: Bäcklund transformation, Lax pairs and the Schwarzian derivative) as the Singularity manifold equation for KdV and mkdv, connected to these equations by a Bäcklund transformation. Möbius invariant (so solutions considered naturally as functions R R C { })

History The Schwarzian KdV equation, SKdV (u) := u t u x [ uxxx u x 3 2 uxx 2 ] ux 2 = 0. Involves the Schwarzian derivative (Named after Schwarz by Cayley, discovered by Lagrange in about 1779). Discovered by John Weiss in 1982 (The Painlevé property for partial differential equations. II: Bäcklund transformation, Lax pairs and the Schwarzian derivative) as the Singularity manifold equation for KdV and mkdv, connected to these equations by a Bäcklund transformation. Möbius invariant (so solutions considered naturally as functions R R C { }) Actually a parameter sub case of an equation found by Krichever and Novikov in 1979.

(Auto) Bäcklund transformation and superposition principle u x ũ x = 1 2(1 p) (u ũ)2, SKdV (u) + SKdV (ũ) = 0,

(Auto) Bäcklund transformation and superposition principle u x ũ x = 1 2(1 p) (u ũ)2, SKdV (u) + SKdV (ũ) = 0, (u ũ)(û ũ) (u û)(ũ ũ) = p 1 q 1.

(Auto) Bäcklund transformation and superposition principle u x ũ x = 1 2(1 p) (u ũ)2, SKdV (u) + SKdV (ũ) = 0, (u ũ)(û ũ) (u û)(ũ ũ) = p 1 q 1. First written explicitly (and termed Lattice Schwarzian KdV) in 1995 by Nijhoff and Capel.

(Auto) Bäcklund transformation and superposition principle u x ũ x = 1 2(1 p) (u ũ)2, SKdV (u) + SKdV (ũ) = 0, (u ũ)(û ũ) (u û)(ũ ũ) = p 1 q 1. First written explicitly (and termed Lattice Schwarzian KdV) in 1995 by Nijhoff and Capel. Its actually a parameter sub case of the NQC equation of 1983: (Nijhoff, Quispel, Capel: Direct Linearization of nonlinear difference-difference equations).

(Auto) Bäcklund transformation and superposition principle u x ũ x = 1 2(1 p) (u ũ)2, SKdV (u) + SKdV (ũ) = 0, (u ũ)(û ũ) (u û)(ũ ũ) = p 1 q 1. First written explicitly (and termed Lattice Schwarzian KdV) in 1995 by Nijhoff and Capel. Its actually a parameter sub case of the NQC equation of 1983: (Nijhoff, Quispel, Capel: Direct Linearization of nonlinear difference-difference equations). Interestingly this equation was proposed by Wynn as an efficient convergence accelerator algorithm in numerical analysis in 1961 (The ɛ-algorithm and operational formulas of numerical analysis). (This connection observed by Vassilios Papageorgiou.)

(Auto) Bäcklund transformation and superposition principle u x ũ x = 1 2(1 p) (u ũ)2, SKdV (u) + SKdV (ũ) = 0, (u ũ)(û ũ) (u û)(ũ ũ) = p 1 q 1. First written explicitly (and termed Lattice Schwarzian KdV) in 1995 by Nijhoff and Capel. Its actually a parameter sub case of the NQC equation of 1983: (Nijhoff, Quispel, Capel: Direct Linearization of nonlinear difference-difference equations). Interestingly this equation was proposed by Wynn as an efficient convergence accelerator algorithm in numerical analysis in 1961 (The ɛ-algorithm and operational formulas of numerical analysis). (This connection observed by Vassilios Papageorgiou.) However, the parameters are important...

History The Schwarzian KP equation ( u t u xxx + 2u2 y 3uxx 2 x u x 2ux 2 ) + ( ) uy = 0. y u x

History The Schwarzian KP equation ( u t u xxx + 2u2 y 3uxx 2 x u x 2ux 2 Again Weiss in 1982. ) + ( ) uy = 0. y u x

History The Schwarzian KP equation ( u t u xxx + 2u2 y 3uxx 2 x u x 2ux 2 ) + ( ) uy = 0. y u x Again Weiss in 1982. Found as the singularity manifold equation for the KP equation, Bäcklund transformation to KP, Möbius invariant etc.

History The Schwarzian KP equation ( u t u xxx + 2u2 y 3uxx 2 x u x 2ux 2 ) + ( ) uy = 0. y u x Again Weiss in 1982. Found as the singularity manifold equation for the KP equation, Bäcklund transformation to KP, Möbius invariant etc. Lattice Schwarzian KP (ũ ũ)(û û)(u ũ) = (ũ ũ)(u û)(û ũ).

History The Schwarzian KP equation ( u t u xxx + 2u2 y 3uxx 2 x u x 2ux 2 ) + ( ) uy = 0. y u x Again Weiss in 1982. Found as the singularity manifold equation for the KP equation, Bäcklund transformation to KP, Möbius invariant etc. Lattice Schwarzian KP (ũ ũ)(û û)(u ũ) = (ũ ũ)(u û)(û ũ). Written explicitly by Nijhoff and Dorfman in 1991 and by Bogdanov and Konopelchenko in 1998 both in connection with the Schwarzian KP equation.

History The Schwarzian KP equation ( u t u xxx + 2u2 y 3uxx 2 x u x 2ux 2 ) + ( ) uy = 0. y u x Again Weiss in 1982. Found as the singularity manifold equation for the KP equation, Bäcklund transformation to KP, Möbius invariant etc. Lattice Schwarzian KP (ũ ũ)(û û)(u ũ) = (ũ ũ)(u û)(û ũ). Written explicitly by Nijhoff and Dorfman in 1991 and by Bogdanov and Konopelchenko in 1998 both in connection with the Schwarzian KP equation. Actually gauge related to a three dimensional lattice equation proposed in 1984 by Nijhoff, Capel, Wiersma and Quispel (Bäcklund transformations and three-dimensional lattice equations).

History It is important to mention work of Konopelchenko and Schief who connected these Schwarzian integrable lattice equations with constructions of classical and ancient plane geometry:

History It is important to mention work of Konopelchenko and Schief who connected these Schwarzian integrable lattice equations with constructions of classical and ancient plane geometry: Menelaus theorem, Clifford configurations and inverse geometry of the Schwarzian KP hierarchy. (2002)

History It is important to mention work of Konopelchenko and Schief who connected these Schwarzian integrable lattice equations with constructions of classical and ancient plane geometry: Menelaus theorem, Clifford configurations and inverse geometry of the Schwarzian KP hierarchy. (2002) These are point-circle configurations.

Definitions

Definitions The group of Möbius transformations { M = u au + b a, b, c, d C, ad bc cu + d } permute the extended complex plane C { }.

Definitions The group of Möbius transformations { M = u au + b a, b, c, d C, ad bc cu + d } permute the extended complex plane C { }. Definition: The stabilizer of u C { }, S(u) = { m M m(u) = u } which is clearly a subgroup of M.

Definitions Natural homomorphism, example: S( ) = { u au + b a, b C, a 0 },

Definitions Natural homomorphism, example: S( ) = { u au + b a, b C, a 0 }, α : S( ) f a C \ {0}.

Definitions Natural homomorphism, example: S( ) = { u au + b a, b C, a 0 }, α : S( ) f a C \ {0}. For any m M and u C { } m S(u) m 1 = S(m(u)),

Definitions Natural homomorphism, example: S( ) = { u au + b a, b C, a 0 }, α : S( ) f a C \ {0}. For any m M and u C { } m S(u) m 1 = S(m(u)), so choose m so that m(u) = and define α u in terms of α : α u : S(u) C \ {0}.

Observation Suppose f, g S(u),

Observation Suppose f, g S(u), more specifically let f(ũ) = ũ, α u (f) = p 1, g(û) = û, α u (g) = q 1.

Observation Suppose f, g S(u), more specifically let f(ũ) = ũ, α u (f) = p 1, g(û) = û, α u (g) = q 1. Of course g 1 f S(u),

Observation Suppose f, g S(u), more specifically let f(ũ) = ũ, α u (f) = p 1, g(û) = û, α u (g) = q 1. Of course g 1 f S(u), and α u (g 1 f) = p/q.

Observation Suppose f, g S(u), more specifically let f(ũ) = ũ, α u (f) = p 1, g(û) = û, α u (g) = q 1. Of course g 1 f S(u), and α u (g 1 f) = p/q. It is natural to ask what is the other fixed-point, i.e., the solution of the equation for ũ [g 1 f]( ũ) = ũ

Observation Suppose f, g S(u), more specifically let f(ũ) = ũ, α u (f) = p 1, g(û) = û, α u (g) = q 1. Of course g 1 f S(u), and α u (g 1 f) = p/q. It is natural to ask what is the other fixed-point, i.e., the solution of the equation for ũ [g 1 f]( ũ) = ũ f( ũ) = g( ũ).

Observation Suppose f, g S(u), more specifically let f(ũ) = ũ, α u (f) = p 1, g(û) = û, α u (g) = q 1. Of course g 1 f S(u), and α u (g 1 f) = p/q. It is natural to ask what is the other fixed-point, i.e., the solution of the equation for ũ [g 1 f]( ũ) = ũ f( ũ) = g( ũ). It turns out that ũ is determined by the equation (u ũ)(û ũ) (u û)(ũ ũ) = p 1 q 1.

Theorem

Theorem Suppose the points u, ũ, û, u, û, ũ, ũ C { } are all distinct. The following are then equivalent:

Theorem Suppose the points u, ũ, û, u, û, ũ, ũ C { } are all distinct. The following are then equivalent: (i) There exist f, g, h S(u) \ {e} such that f(ũ) = ũ, g(û) = û, h(u) = u, f( ũ) = g( ũ), g(û) = h(û), h(ũ) = f(ũ).

Theorem Suppose the points u, ũ, û, u, û, ũ, ũ C { } are all distinct. The following are then equivalent: (i) There exist f, g, h S(u) \ {e} such that f(ũ) = ũ, g(û) = û, h(u) = u, f( ũ) = g( ũ), g(û) = h(û), h(ũ) = f(ũ). (ii) l, m, n M defined uniquely by the equations commute. l(u) = ũ, m(u) = û, n(u) = u, l(û) = ũ, m(u) = û, n(ũ) = ũ, l(u) = ũ, m(ũ) = ũ, n(û) = û,

Theorem (iii) The Möbius transformation i M defined uniquely by the equations i(ũ) = û, i(û) = ũ, i(u) = ũ, is an involution (i 2 = e, i e).

Theorem (iii) The Möbius transformation i M defined uniquely by the equations i(ũ) = û, i(û) = ũ, i(u) = ũ, is an involution (i 2 = e, i e). (iv) The following condition is satisfied (ũ ũ)(û û)(u ũ) = (ũ ũ)(u û)(û ũ),

Theorem (iii) The Möbius transformation i M defined uniquely by the equations i(ũ) = û, i(û) = ũ, i(u) = ũ, is an involution (i 2 = e, i e). (iv) The following condition is satisfied (ũ ũ)(û û)(u ũ) = (ũ ũ)(u û)(û ũ), ũû ũ + û 1 ûũ û + ũ 1 u ũ u + ũ 1 = 0.

Lemma The core nontrivial property of the Möbius transformations required to prove the equivalence of (i) and (ii) is the following more technical result:

Lemma The core nontrivial property of the Möbius transformations required to prove the equivalence of (i) and (ii) is the following more technical result: For some distinct u, ũ, û, ũ C { } let l, m, f and g satisfy l(u) = ũ, m(u) = û, l(û) = ũ, m(ũ) = ũ, fixed-points(f) = {u, ũ}, fixed-points(g) = {u, û}, f( ũ) = g( ũ).

Lemma The core nontrivial property of the Möbius transformations required to prove the equivalence of (i) and (ii) is the following more technical result: For some distinct u, ũ, û, ũ C { } let l, m, f and g satisfy l(u) = ũ, m(u) = û, l(û) = ũ, m(ũ) = ũ, Then fixed-points(f) = {u, ũ}, fixed-points(g) = {u, û}, f( ũ) = g( ũ). l, m share fixed points l f, m g share fixed points.

LSKP LSKdV Suppose now that any (thus all) of the previous statments hold and define ũ = i(u).

LSKP LSKdV Suppose now that any (thus all) of the previous statments hold and define ũ = i(u). recall (i) There exist f, g, h S(u) \ {e} such that f(ũ) = ũ, g(û) = û, h(u) = u, f( ũ) = g( ũ), g(û) = h(û), h(ũ) = f(ũ).

LSKP LSKdV Suppose now that any (thus all) of the previous statments hold and define ũ = i(u). recall (i) There exist f, g, h S(u) \ {e} such that f(ũ) = ũ, g(û) = û, h(u) = u, f( ũ) = g( ũ), g(û) = h(û), h(ũ) = f(ũ). suppose further that α u (f) = p, α u (g) = q, α u (h) = r.

LSKP LSKdV Suppose now that any (thus all) of the previous statments hold and define ũ = i(u). recall (i) There exist f, g, h S(u) \ {e} such that f(ũ) = ũ, g(û) = û, h(u) = u, f( ũ) = g( ũ), g(û) = h(û), h(ũ) = f(ũ). suppose further that Then the following hold α u (f) = p, α u (g) = q, α u (h) = r. (u ũ)(û ũ) (u û)(ũ ũ) = p 1 q 1, (u ũ)(û ũ) (u û)(ũ ũ) = p 1 q 1.

LSKP LSKdV Recall also (ii) l, m, n M defined uniquely by the equations commute. l(u) = ũ, m(u) = û, n(u) = u, l(û) = ũ, m(u) = û, n(ũ) = ũ, l(u) = ũ, m(ũ) = ũ, n(û) = û,

LSKP LSKdV Recall also (ii) l, m, n M defined uniquely by the equations l(u) = ũ, m(u) = û, n(u) = u, l(û) = ũ, m(u) = û, n(ũ) = ũ, l(u) = ũ, m(ũ) = ũ, n(û) = û, commute. In fact l(û) = ũ, m(ũ) = ũ, n( ũ) = ũ.

LSKP LSKdV Recall also (ii) l, m, n M defined uniquely by the equations commute. In fact l(u) = ũ, m(u) = û, n(u) = u, l(û) = ũ, m(u) = û, n(ũ) = ũ, l(u) = ũ, m(ũ) = ũ, n(û) = û, l(û) = ũ, m(ũ) = ũ, n( ũ) = ũ. And if l, m and n have fixed-points u 0 and u 1, then the following equations hold: u 0 u 1 u 0 + u 1 1 ũû ũ + û 1 u ũ u + ũ 1 = 0, u 0 u 1 u 0 + u 1 1 ũû ũ + û 1 u ũ u + ũ 1 = 0.

Finally

Finally The End Thank you for your attention!