Connection Preserving Deformation Approach to the Q-Difference - - PowerPoint PPT Presentation
Connection Preserving Deformation Approach to the Q-Difference Equations Chuan-Tsung Chan Department of Applied Physics, Tunghai University 24th Annual Workshop on Differential Equations National Sun Yat-sen University, Kaohsiung, Taiwan
24th Annual Workshop on Differential Equations National Sun Yat-sen University, Kaohsiung, Taiwan
– Notion of universality classes from the physics of phase transitions and critical phenomena. – Landscape/multiverse picture as inferred from the study of quan- tum gravity. – M theory (or even higher structure) as a unified arena of string theories.
– Classifications of algebraic and geometrical structures. – Classifications of differential equations. – Langlands Program as a GUT in mathematics.
d2y dt2 = 6y2 + t (1) d2y dt2 = 2y3 + ty + α (2) tyd2y dt2 = t
dy
dt
2
− ydy dt + δt + βy + αy3 + γty4 (3) yd2y dt2 = 1 2
dy
dt
2
+ β + 2(t2 − α)y2 + 4ty3 + 3 2y4 (4) d2y dt2 =
1
2y + 1 y − 1
dy
dt
2
− 1 t dy dt + (y − 1)2 t2
y
t + δy(y + 1) y − 1 (5) d2y dt2 = 1 2
1
y + 1 y − 1 + 1 y − t
dy
dt
2
−
1
t + 1 t − 1 + 1 y − t
dy
dt + y(y − 1)(y − t) t2(t − 1)2
y2 + γ t − 1 (y − 1)2 + δt(t − 1) (y − t)2
the 2nd order differential equations which admit ”Painlev´ e property”, and the general solutions can not be expressed in terms of ”standard functions”.
The only movable singularities of the solutions to these eqs. are poles!
tions of systems of linear ordinary differential equations. The most general form of PVI (6th Painlev´ e eq.) was discovered by Richard Fuchs in 1905 through this approach.
e.g., P-Q systems of the string equations.
e.g., correlators in the statistical model (e.g., Ising model), susceptibility of free-energy of 2-dim gravity/matrix model.
xn+1 + xn−1 + xn = an + b xn + c (7) xn+1 + xn−1 = xn(an + b) + c 1 − x2
n
(8) xn−1xn+1 = γx2
n + ζ0λnxn + µ0λ2n
x2
n + βxn + γ
(9) xn+1xn−1 + xn(xn+1 + xn−1) = −(an + b)x3
n + [ǫ0 − 1 4(an + b)2]x2 n + µ
x2
n + (an + b)xn + [γ0 + 1 4(an + b)2]
(10) (2xn − 1)xn+1xn−1 − xn(xn+1 + xn−1) =
1 2(σ − α0λ2n)x3 n + {θ + 1 4(σ + α0λ2n − 2ρ0λn)}x2 n − 2µxn + µ
α0λ2nx2
n + (ρ0λn − α0λ2n)xn + 1 4(σ + α0λ2n − 2ρ0λn)
(11)
Starting from the QRT (Quispel, Roberts, and Thompson) class of the second order difference equations, f3(xn) · Π − f2(xn) · Σ + f1(xn) = 0, with Π := xn−1·xn+1, Σ := xn−1+xn+1, and fi are quartic polynomials.
parallel to the sprit of Painlev´ e property, that if there exists an instance where the solution of a difference equation diverges, it will only last for a finite iterations and beyond this finite interval, the solution will become regular and restores the memory of previous data. Hence the name singularity confinement.
e equations (continuous and discrete) follow the same conflu- ence/degeneration pattern as that of the hypergeometric equations:
VI Gauss V Kummer II Airy I None IV Hermite-Weber III Bessel
How much can we trust calculus as a useful language for a description of the microscopic world? e.g., quantum theories are inherently discrete?
the numerical computations of the integrable differential equations?
qx − x , q := 1 − ǫ.
Y (qx, q) = A(x)Y (x, q), A(x) = A0 + A1 + ... + ANxN. Assumptions: A0, AN are both semi-simple and invertible, and their eigenvalues are θj and κj, j = 1, 2.
A0 = C0qD0C−1
0 ,
A∞ = C∞qD∞C−1
∞
where D0 := diag(ln θj/ ln q), D∞ :=≡ diag(ln κj/ ln q)
asymptotic behaviours : Y0(x) = ˆ Y0(x)xD0, Y∞(x) = q
N 2 u(u−1)ˆ
Y∞(x)xD∞, u := ln x ln q
Y0(x) is a holomophic and invertible matrix at x = 0, with ˆ Y0(0) = C0, and ˆ Y∞(x) is a holomophic and invertible matrix at x = ∞, with ˆ Y∞(∞) = C∞.
0, j = 1, ..., 2N. may induce poles of the subleading parts of the asymptotic solutions: ˆ Y∞(x)−1 : qαj, q2αj, q3αj, ...... ˆ Y0(x) :, αj q−1αj, q−2αj, ......
P(x) := Y∞(x)[Y0(x)]−1.
Y (qx, t) = A(x, t)Y (x, t), and P(x) → P(x, t)
B(x, t) := Y (x, qt)[Y (x, t)]−1, A(x, t) and B(x, t) are the Lax pair of the linear q-difference system.
to be a rational function! In the case of N = 2, we get P(x, qt) = P(x, t) ↔ B(x, t) = x(xI + B0(x)) (x − qta1)(x − qta2)
From this commuting diagram, we have the following constraint on the Lax pair, A(x, qt) · B(x, t) = B(qx, t) · A(x, t).
A(x, t) = A0(t) + A1(t)x + A2x2 (12) A2 =
κ2
A0(t) has eigenvalues tθ1, tθ2 det A(x, t) = κ1κ2(x − ta1)(x − ta2)(x − a3)(x − a4), (13) with κ1κ2
4
aj = θ1θ2. (14)
e functions: A12(y(t), t) = 0, z1(t) := A11(y(t), t) κ1 , z2(t) := A22(y(t), t) κ2 , and z = (y − ta1)(y − ta2) qκ1z1 (15) A(x, t) =
κ2w(x − y) κ1w−1(γx + δ) κ2((x − y)(x − β) + z2)
z1 · z2 = (y − ta1)(y − ta2)(y − a3)(y − a4) ⇒ Only y(t) and z(t) are independent Painlev´ e functions!
– eigenvalue datum of A2, and A0: κj, θj, j = 1, 2. – degeneration points of A(x): a1, a2, a3, a4. – Painlev´ e functions y(t), and z(t). through the det A(x, t) and trA(x, t) relations.
– Solution of the B0(t) in the dual evolution operator B(x, t). – Coupled first order difference equations in three unknown func- tions, y(t), z(t), w(t). ⇒ d-PVI equation in the first order form!
y¯ y a3a4 = (¯ z − tb1)(¯ z − tb2) (¯ z − b3)(¯ z − b4) , (17) z¯ z b3b4 = (y − ta1)(y − ta2) (y − a3)(y − a4) , (18) ¯ w w = b4 b3 ¯ z − b3 ¯ z − b4 (19) b1 = a1a2 θ1 , b2 = a1a2 θ2 , b3 = 1 qκ1 , b4 = 1 κ2 , b1b2 b3b4 = qa1a2 a3a4 (20)
C
κ2
κ2
C =
1 α
α = 0 (21) C
κ2 κ1
κ2 κ1
C =
b ±1
b is arbitrary. (22)
Jordan type A(x) := A1x + A0(t) (23) where A1 :=
κ2 κ1
A0(t) =
−κ1
γ
ω
α
ω2
1
κ2
−κ1
β
ω
(24)
Deformation : aj → taj, θj → tθj, j = 1, 2 (25) ξ(t) := a1(t) + a2(t) + 1 κ1
α(t) =
ξ
κ1
(27) β(t) = −t
κ1
(28) γ(t) = tξ (29)
B(x, t) = x[xI + B0(t)] (x − qta1)(x − qta2) (30) A(x, qt)B(x, t) = B(qx, t)A(x, t) (31) A(qajt, qt)
(32)
(33) A0(qt)B0(t) = qB0(t)A0(t) (34)
ω¯ ω = ¯ ω − ω, ¯ ω := ω(qt) (35)
ideas among Geometry, Topology and Physics.
Perserving Deformation (CPD) approach.
Jodan type.
tions can be tedious.