A COMPLETENESS STUDY ON CERTAIN 2 2 LAX PAIRS Mike Hay. University of Sydney.

Transcription

1 Lax COMPLETENESS STUDY ON CERTIN 2 2 LX PIRS University of Sydney michaelh@maths.usyd.edu.au

2 Lax pair is a pair of linear problems who s compatibility is associated with a nonlinear equation. φ = φ(l + 1, m) = L(l, m)φ(l, m) ˆφ = φ(l, m + 1) = M(l, m)φ(l, m) Lax ˆ φ = L ˆφ = LMφ ˆ φ = M φ = MLφ : LM = ML nonlinear equation is integrable if it has a Lax pair.

3 LSG 2 The recently discovered, higher order version of the lattice sine-gordon equation, LSG 2 : Lax LSG 2 : ρ ˆx σ x + λ 1µ 1ˆ xŷ = σ ˆ x ρ x + λ 2µ 2 xȳ σ ˆȳ ρ ŷ + λ 2µ 2 = ρ ȳ ˆxy σ y + λ 1µ 1 xˆȳ So-called because using x = y retrieves the well-known lattice sine-gordon equation: LSG : ˆ xx( σ ρ λ 1µ 1 xˆx) = ρ σ xˆx λ 2µ 2

4 Lax The elliptic Lax is L = M = ( /ρ ) λ 1 x λ 2 /x ρȳ/y ( ) ˆx/(σx) /y µ 1 ŷ σ where = (n) is the Weierstrass p elliptic function, and is its derivative wrt. n. We call terms that depend on the spectral variable, n, spectral terms. Those that depend on the lattice variables, l, m, are lattice terms.

5 The (1, 2) entry of the compatibility Lax Explicitly, the compatibility, LM = ML ( ρ ) ( ) λ 1ˆ x ˆx/(σx) /y λ 2 /ˆx ˆȳ/(ŷρ) µ 1 ŷ = σ ( ) ( σˆ x/ x /ȳ /ρ ) λ 1 x µ 1ˆȳ /σ λ 2 /x ρȳ/y The (1, 2) entry of which, is clearly an identity. 2 ρ/y + 2 σλ 1ˆ x = 2 σλ 1ˆ x + 2 ρ/y

6 Consider the starting point for the elliptic LSG 2 Lax pair: ( a L = ) b c d ( M = ) α β γ δ Lax The important feature of the spectral dependence is that it groups the terms into that lead to LSG 2. The (1, 2) entry of the compatibility is 2 âβ + 2ˆbδ = 2 d β + 2 bᾱ âβ = d β ˆbδ = bᾱ

7 Generalized Lax Lax Now consider a general 2 2 Lax pair with one separable term in each entry ( ) a bb L = cc dd ( ) αλ βξ M = γγ δ Lower cases: a=a(l,m), upper cases =(n) We are free to choose a spectral dependence so that the sought after arise. The (1, 2) entry is âβξ + ˆbδB = d βdξ + bᾱbλ = D = Λ Ξ B

8 Lax By analyzing all entries, we find that LSG 2 comes from any Lax pair with the following spectral dependence: ( ) ( ) a bb αλ βξ L =, M = cc dd γγ δ where = F 1 /Ξ Λ = F 2 /B C = BΓ/Ξ D = F 1 /Ξ = F 2 /B âα + ˆbγ = aᾱ + c β ˆdδ + ĉβ = d δ + b γ âβ = d β ˆbα = b δ ĉα = c δ âγ = d γ F 1 kf 2, k a constant

9 General solution of the compatibility Lax Need to solve the compatibility in full generality. ll Lax pairs studied so far have compatibility s that are solvable by using the parametrization: ā k 1 a = ˆb k 2 b a = ˆv k 2 v + k 3 k l 1 b = v k 1 v + k 4 k m 2 Using this tool, often in an exponentiated form, there has been no need to appeal to assumptions about the lattice terms.

10 from this Lax pair Lax We have constructed a family of Lax pairs, all for LSG 2, by: - Grouping the lattice terms into the appropriate set of - Constructing a spectral dependence that gives rise to those How else can the lattice terms be grouped? (1,2) âβξ ˆbδB bᾱbλ d βdξ What other integrable can we find Lax pairs for?

11 Lax rbitrarily, start with the spectral terms in the (1,2) entry of the compatibility (1,2) Ξ B BΛ DΞ Consider the different combinations of different sized groups in this entry. ny combination chosen has implications for the other entries of the compatibility.

12 How the choice affects the other entries Lax Let s use the following groups in the (1,2) entry: (1,2) Ξ B BΛ DΞ Turning to the (2,1) entry: = F 1 /Ξ = F 1 /B Λ = (F 1 + F 2 )/B D = F 2 /Ξ (2,1) CΛ DΓ Γ C (F 1 + F 2 ) C B F 2 Γ Ξ F 1 Γ Ξ F 1 C B

13 The resulting lattice term Lax These groupings lead to the following lattice term : âα = aᾱ ˆbγ = c β ˆdδ + ĉβ = d δ + b γ âβ + ˆbδ = bᾱ kbᾱ = d β ĉα = a γ + c δ kĉα = ˆdγ which are associated with the trivial pair of : λ 2 (ˆx x) = k(λ 2 λ λλ 2 ) ˆx x + k(λ λ 2 ) = k xˆ x (µλ λ 2)

14 survey Lax Only two Lax pairs of the form ( ) ( ) a bb αλ βξ L =, M = cc dd γγ δ Lead to unconstrained, nontrivial lattice. The are: - LSG 2, higher order lattice sine-gordon - LMKdV 2, higher order lattice modified KdV and no others

15 One can, in principle, find all Lax pairs of a general form with a set number of terms. Lax We do this by considering all possible groupings of lattice terms that arise in each entry of the compatibility. It is also possible to conduct this analysis with other types of Lax pairs including differential, differential-difference and isomonodromy Lax pairs.