Bianchi (hyper-)cubes and a geometric unification of the Hirota and Miwa equations. W.K. Schief. The University of New South Wales, Sydney

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1 Bianchi (hyper-)cubes and a geometric unification of the Hirota and Miwa equations by W.K. Schief The University of New South Wales, Sydney ARC Centre of Excellence for Mathematics and Statistics of Complex Systems [with Alastair King, University of Bath]

2 KP Lax hierarchy of linear equations: 1. The (B)KP hierarchies tn ϕ = L (n) ( x )ϕ, L (n) = x n + u (n) n 2 n 2 x + + u (n) 0 Base equation: ϕ y = ϕ xx + uϕ BKP Lax hierarchy: Impose the condition L (n) x = x L (n) on all odd flows and set aside the even flows. Base equation: ϕ t = ϕ xxx + uϕ x Fact: τ KP = τ 2 BKP

3 2. An open problem Question 1: Are the Hirota (dkp) lattice equation τ 1 τ 23 + τ 2 τ 13 + τ 3 τ 12 = 0 and the Miwa (dbkp) lattice equation ττ τ 1 τ 23 + τ 2 τ 13 + τ 3 τ 12 = 0 related in any sense? [τ = τ(n 1, n 2, n 3 ), τ 1 = τ(n 1 + 1, n 2, n 3 ), τ 2 = τ(n 1, n 2 1, n 3 )] Question 2: Is there a (geometric/algebraic setting) which unifies these two equations?

4 3. Result (King & WKS) Geometric answer: Projective coordinate geometry of Cox lattices Algebraic answer: Consider a function V : Z 3 R 4 subject to the orthogonality condition V(n), V(m) = 0 for any vertices n and m linked by an edge. Result: The components of V give rise to either 8 and 12 or 16 Bäcklund-related solutions of either a novel 14-point equation and the Miwa equation or the Hirota equation.

5 4. A 14-point equation Consider the decomposition Z 3 = D e 3 Do 3 ν 23 σ 3 σ 123 ν 13 and define σ 2 τ = { ν if n D e 3 ν : D e 3 R σ if n D o 3 σ : D o 3 R ν σ 1 ν 12 so that the Miwa equation becomes the pair νσ σ 1 ν 23 + σ 2 ν 13 + σ 3 ν 12 = 0 σ ν + ν 2 3 σ 1 + ν 1 3 σ 2 + ν 1 2 σ 3 = 0 Even shifts and elimination then leads to the integrable 14-point equation on D o 3 σ 123 σ 1 σ 2 σ 3 + σ σ 1 σ 2 σ 3 + σ 12 3 σ 1 σ 2σ 3 + σ 1 23 σ 1 σ 2 σ 3 = σ σ 1 σ 2 σ 3 + σ 123 σ 1 σ 2σ 3 + σ 1 23 σ 1 σ 2 σ 3 + σ 12 3 σ 1 σ 2 σ 3

6 5. Integrability Alternating sum ( 1) χ(a,b,c) a,b,c σ abc σ a σ b σ c = 0 [Universal!] Theorem [Lax pair, Bäcklund transformation]. Given a solution σ of the 14-point equation and suitable Cauchy data for N on an A 2 slice of D3 e, the Lax pair νσ σ 1 ν 23 + σ 2 ν 13 + σ 3 ν 12 = 0 σ ν + ν 2 3 σ 1 + ν 1 3 σ 2 + ν 1 2 σ 3 = 0 determines ν uniquely and, by symmetry, ν obeys the 14-point equation on D e 3. Moreover, {τ} = {ν, σ} constitutes a solution of the Miwa equation on Z 3.

7 We now consider the compatible vector-valued Lax pair Nσ σ 1 N 23 + σ 2 N 13 + σ 3 N 12 = 0 σ N + N 2 3 σ 1 + N 1 3 σ 2 + N 1 2 σ 3 = 0 for N : D3 e R3

8 6. Algebra Theorem. A function r : D o 3 R3 is uniquely defined by the compatible system r 1 r 2 = 1 N N 12, r 1 r 2 σ 1 σ = 1 N N 2 σ σ 2 r 2 r 3 = 1 N N 23, r 2 r 3 σ 2 σ = 1 N N 3 σ σ 3 r 3 r 1 = 1 N N 13, r 3 r 1 σ 3 σ = 1 N N 1 σ 13 3 σ 1 N 23 r 3 r 123 N 13 r 2 N r 1 N 12

9 In terms of so that we obtain R 1 = σ 1 ( r (Projective) Geometry ), N = ( R i, N = Rī, N = 0, N r 1 N ) R 1 R 2 = (N N 12 ), R 1 R 2 = (N N 1 2 ) (and cycl.) Result: The components of R i and N may be interpreted as particular homogeneous coordinates of points p i and planes π respectively in a three-dimensional projective space P 3 which are incident along the edges of Z 3. N 23 R 3 R 123 N 13 R 2 More precisely... N R 1 N 12

10 8. Cox lattices (cf. Bobenko & Suris 2009) Definition. A lattice of points and planes in P 3, that is p : D o 3 {points in P3 }, π : D e 3 {planes in P3 }, is termed a Cox lattice if p(n) π(m) for nearest neighbours. In homogeneous coordinates: π 23 p 3 p 123 π 13 p 2 R i, N = Rī, N = 0 or, equivalently, R i R k = α ik (N N ik ), R i R k = αi k (N N i k ) π p 1 π 12 Theorem. In the generic case, there exist lifts such that α = 1 so that generic Cox lattices are generated by the Bäcklund chain {R 4 } {N 1 }, {N 2 }, {N 3 } {R 1 }, {R 2 }, {R 3 } {N 4 } of solutions of the 14-point equation.

11 9. A Bianchi cube Corollary. Cox lattices encapsulate 8 Bäcklund-related solutions {R 1 }, {R 2 }, {R 3 }, {R 4 } and {N 1 }, {N 2 }, {N 3 }, {N 4 } of the 14-point equation and 12 associated solutions {τ mn } = {N m, R n }, n m of the Miwa equation. Bianchi cube: D o 3 {N 3 } {R 1 } {τ 32 } {τ 41 } {R 2 } {N 4 } D e 3 {N 2 } {τ 23 } {R 4 } {τ 14 } {N 1 } {R 3 } Remark: The above Bianchi cube encodes three Bianchi diagrams for the Miwa equation, e.g. {τ 14 } {τ 23 } BTs {τ 32 } {τ 41 }

12 10. Degenerate Cox lattices Theorem. If the tetrahedra [R 1, R 2, R 3, R 123 ] are planar then the points R 1, R 2, R 3, R 123 are, in fact, collinear and the associated degenerate Cox lattice encapsulates a standard Bianchi hypercube for 16 Bäcklund-related solutions τ,..., τ 1234 of the Hirota equation with the identification τ i = N i, τ 1234\i = R i, τ 1234 = ψ R 1 ψ R 2 τ 34 R2 R 1 R 3 τ 23 N 3 R 4 τ 13 R 3 R 1 R 2 τ N 2 τ 12 N 1 R 3 [complete quadrilateral] τ 24 N 4 τ 14 Conundrum: τ Hirota = τ 2X Miwa

13 11. A Bäcklund transformation for Cox lattices (cf. Bobenko & Suris 2009) Idea: Construct four-dimensional Cox lattices, that is, a lattice version of Cox classical configuration (2 d 1 ) d of points and planes for d = 4. The latter is known as Cox theorem or Möbius theorem.? Remark: This does not work in the degenerate case! Instead, one needs to consider lattices of A 4 type and employ Desargues theorem. [cf. Doliwa s talk]