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The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
The ODE/IM correspondence in Toda fieldtheories
Stefano NegroAdvisor: Roberto Tateo
Dipartimento di fisicaUniversita degli Studi di Torino
19 february 2013
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Table of Contents
1 Introduction
2 Integrability: some basic facts
3 The ODE/IM Correspondence for the Bullough-Dodd Model
4 Conclusions and perspectives
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Table of Contents
1 Introduction
2 Integrability: some basic facts
3 The ODE/IM Correspondence for the Bullough-Dodd Model
4 Conclusions and perspectives
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Introduction
The ODE/IM Correspondence
• The ODE/IM correspondence is a newly found linkbetween two apparently separated areas of mathematicalphysics
• It is based on an equivalence between functional relationsappearing in the study of eigenvalue problems of ODEs onone side and emerging from the analysis of the partitionfunction in 2D quantum integrable theories on the otherside
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Introduction
The ODE/IM Correspondence
• The ODE/IM correspondence is a newly found linkbetween two apparently separated areas of mathematicalphysics
• It is based on an equivalence between functional relationsappearing in the study of eigenvalue problems of ODEs onone side and emerging from the analysis of the partitionfunction in 2D quantum integrable theories on the otherside
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Introduction
The ODE/IM Correspondence
• The ODE/IM correspondence is a newly found linkbetween two apparently separated areas of mathematicalphysics
• It is based on an equivalence between functional relationsappearing in the study of eigenvalue problems of ODEs onone side and emerging from the analysis of the partitionfunction in 2D quantum integrable theories on the otherside
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Introduction
The first instance of this equivalence arose in a work ofP.Dorey and R.Tateo in 1998
• ODE: d2
dx2ψ(x) + x2Mψ(x) = −Eψ(x)
• IM: The six-vertex model
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Introduction
The first instance of this equivalence arose in a work ofP.Dorey and R.Tateo in 1998
• ODE: d2
dx2ψ(x) + x2Mψ(x) = −Eψ(x)
• IM: The six-vertex model
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Introduction
The Six-Vertex Model
• To each local configuration is associated a Boltzmannweight:
W
↑→ →
↑
= W
↓← ←
↓
= a(ν, η)
W
↓→ →
↓
= W
↑← ←
↑
= b(ν, η)
W
↑→ ←
↓
= W
↓← →
↑
= c(ν, η)
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Introduction
The Six-Vertex Model
• To each local configuration is associated a Boltzmannweight:
W
↑→ →
↑
= W
↓← ←
↓
= a(ν, η)
W
↓→ →
↓
= W
↑← ←
↑
= b(ν, η)
W
↑→ ←
↓
= W
↓← →
↑
= c(ν, η)
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Introduction
The Six-Vertex Model
• The physics of the model is encoded in the PartitionFunction:
Z =∑C
∏sites
W
·· ··
• Which can be written in terms of the Transfer Matrix:Z = TrTN
• The free energy density assume a simple form in thethermodynamic limit
f = − 1
NN ′lnZ = − 1
NN ′lnTrTN ∼ − 1
N ′ln t0
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Introduction
The Six-Vertex Model
• The physics of the model is encoded in the PartitionFunction:
Z =∑C
∏sites
W
·· ··
• Which can be written in terms of the Transfer Matrix:Z = TrTN
• The free energy density assume a simple form in thethermodynamic limit
f = − 1
NN ′lnZ = − 1
NN ′lnTrTN ∼ − 1
N ′ln t0
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Introduction
The Six-Vertex Model
• The physics of the model is encoded in the PartitionFunction:
Z =∑C
∏sites
W
·· ··
• Which can be written in terms of the Transfer Matrix:Z = TrTN
• The free energy density assume a simple form in thethermodynamic limit
f = − 1
NN ′lnZ = − 1
NN ′lnTrTN ∼ − 1
N ′ln t0
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Introduction
The Six-Vertex Model
• The physics of the model is encoded in the PartitionFunction:
Z =∑C
∏sites
W
·· ··
• Which can be written in terms of the Transfer Matrix:Z = TrTN
• The free energy density assume a simple form in thethermodynamic limit
f = − 1
NN ′lnZ = − 1
NN ′lnTrTN ∼ − 1
N ′ln t0
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Introduction
The Six-Vertex Model
• The six-vertex model is integrable, this means that in thethermodynamic limit, the eigenvalues of T satisfy theBethe Ansatz equations:
t0(E )q0(E ) = q0(ω2E ) + q0(ω−2E )
q0(E ) =∞∏`=1
(1− E
E`
)∞∏`=1
(E` − ω2Ei
E` − ω−2Ei
)= −1
• Where E = e2ν and ω = −e−2iη
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Introduction
The Six-Vertex Model
• The six-vertex model is integrable, this means that in thethermodynamic limit, the eigenvalues of T satisfy theBethe Ansatz equations:
t0(E )q0(E ) = q0(ω2E ) + q0(ω−2E )
q0(E ) =∞∏`=1
(1− E
E`
)∞∏`=1
(E` − ω2Ei
E` − ω−2Ei
)= −1
• Where E = e2ν and ω = −e−2iη
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Introduction
The Six-Vertex Model
• The six-vertex model is integrable, this means that in thethermodynamic limit, the eigenvalues of T satisfy theBethe Ansatz equations:
t0(E )q0(E ) = q0(ω2E ) + q0(ω−2E )
q0(E ) =∞∏`=1
(1− E
E`
)∞∏`=1
(E` − ω2Ei
E` − ω−2Ei
)= −1
• Where E = e2ν and ω = −e−2iη
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Table of Contents
1 Introduction
2 Integrability: some basic facts
3 The ODE/IM Correspondence for the Bullough-Dodd Model
4 Conclusions and perspectives
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Integrable Hamiltonian Systems
Definition (Liouville 1855)
A competely integrable Hamiltonian system is a dynamicalsystem admitting a Hamiltonian description and possessing amaximal set of independent conserved quantities, so that it canbe solved by quadratures.
n-Dimensional Systems
• the phase space M is a 2n-dimensional symplecticmanifold equipped with some Poisson structure {·, ·}
• the conserved quantities are functions Fi on M (F1 = H)Poisson commuting with the Hamiltonian: {Fi ,H} = 0
• the system is completely integrable if there exist nconserved quantities {Fi}ni=1 such that{Fi ,Fj} = 0 ; ∀i , j = 1, . . . , n
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Integrable Hamiltonian Systems
Definition (Liouville 1855)
A competely integrable Hamiltonian system is a dynamicalsystem admitting a Hamiltonian description and possessing amaximal set of independent conserved quantities, so that it canbe solved by quadratures.
n-Dimensional Systems
• the phase space M is a 2n-dimensional symplecticmanifold equipped with some Poisson structure {·, ·}
• the conserved quantities are functions Fi on M (F1 = H)Poisson commuting with the Hamiltonian: {Fi ,H} = 0
• the system is completely integrable if there exist nconserved quantities {Fi}ni=1 such that{Fi ,Fj} = 0 ; ∀i , j = 1, . . . , n
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Integrable Hamiltonian Systems
Definition (Liouville 1855)
A competely integrable Hamiltonian system is a dynamicalsystem admitting a Hamiltonian description and possessing amaximal set of independent conserved quantities, so that it canbe solved by quadratures.
n-Dimensional Systems
• the phase space M is a 2n-dimensional symplecticmanifold equipped with some Poisson structure {·, ·}
• the conserved quantities are functions Fi on M (F1 = H)Poisson commuting with the Hamiltonian: {Fi ,H} = 0
• the system is completely integrable if there exist nconserved quantities {Fi}ni=1 such that{Fi ,Fj} = 0 ; ∀i , j = 1, . . . , n
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Integrable Hamiltonian Systems
Definition (Liouville 1855)
A competely integrable Hamiltonian system is a dynamicalsystem admitting a Hamiltonian description and possessing amaximal set of independent conserved quantities, so that it canbe solved by quadratures.
n-Dimensional Systems
• the phase space M is a 2n-dimensional symplecticmanifold equipped with some Poisson structure {·, ·}
• the conserved quantities are functions Fi on M (F1 = H)Poisson commuting with the Hamiltonian: {Fi ,H} = 0
• the system is completely integrable if there exist nconserved quantities {Fi}ni=1 such that{Fi ,Fj} = 0 ; ∀i , j = 1, . . . , n
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Integrable Hamiltonian Systems
Definition (Liouville 1855)
A competely integrable Hamiltonian system is a dynamicalsystem admitting a Hamiltonian description and possessing amaximal set of independent conserved quantities, so that it canbe solved by quadratures.
n-Dimensional Systems
• the phase space M is a 2n-dimensional symplecticmanifold equipped with some Poisson structure {·, ·}
• the conserved quantities are functions Fi on M (F1 = H)Poisson commuting with the Hamiltonian: {Fi ,H} = 0
• the system is completely integrable if there exist nconserved quantities {Fi}ni=1 such that{Fi ,Fj} = 0 ; ∀i , j = 1, . . . , n
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Integrable Hamiltonian Systems
∞-Dimensional Systems
• the phase space M is ∞-dimensional
• thus we need an infinite set of integrals of motion {Fi}∞i=1
in involution
Example: the Korteweg-de Vries (KdV) equation
∂tφ+ ∂3xφ− 6φ∂xφ = 0
L = 12∂xψ∂tψ + (∂xψ)3 − 1
2
(∂2xψ)2
; φ = ∂xψ
• Integrals of motion (Miura, Gardner & Kruskal, 1969):F2n−1 =
∫∞−∞ P2n−1
(φ, ∂φ, ∂
2xφ, . . .
)dx ; F2n = 0
Pn = − ddxPn−1 +
∑n−2i=1 PiPn−1−i ; P1 = φ
• Soliton solutions: φ(x , t) = c2sech
2[√
c x−ct−a2
]
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Integrable Hamiltonian Systems
∞-Dimensional Systems
• the phase space M is ∞-dimensional
• thus we need an infinite set of integrals of motion {Fi}∞i=1
in involution
Example: the Korteweg-de Vries (KdV) equation
∂tφ+ ∂3xφ− 6φ∂xφ = 0
L = 12∂xψ∂tψ + (∂xψ)3 − 1
2
(∂2xψ)2
; φ = ∂xψ
• Integrals of motion (Miura, Gardner & Kruskal, 1969):F2n−1 =
∫∞−∞ P2n−1
(φ, ∂φ, ∂
2xφ, . . .
)dx ; F2n = 0
Pn = − ddxPn−1 +
∑n−2i=1 PiPn−1−i ; P1 = φ
• Soliton solutions: φ(x , t) = c2sech
2[√
c x−ct−a2
]
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Integrable Hamiltonian Systems
∞-Dimensional Systems
• the phase space M is ∞-dimensional
• thus we need an infinite set of integrals of motion {Fi}∞i=1
in involution
Example: the Korteweg-de Vries (KdV) equation
∂tφ+ ∂3xφ− 6φ∂xφ = 0
L = 12∂xψ∂tψ + (∂xψ)3 − 1
2
(∂2xψ)2
; φ = ∂xψ
• Integrals of motion (Miura, Gardner & Kruskal, 1969):F2n−1 =
∫∞−∞ P2n−1
(φ, ∂φ, ∂
2xφ, . . .
)dx ; F2n = 0
Pn = − ddxPn−1 +
∑n−2i=1 PiPn−1−i ; P1 = φ
• Soliton solutions: φ(x , t) = c2sech
2[√
c x−ct−a2
]
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Integrable Hamiltonian Systems
∞-Dimensional Systems
• the phase space M is ∞-dimensional
• thus we need an infinite set of integrals of motion {Fi}∞i=1
in involution
Example: the Korteweg-de Vries (KdV) equation
∂tφ+ ∂3xφ− 6φ∂xφ = 0
L = 12∂xψ∂tψ + (∂xψ)3 − 1
2
(∂2xψ)2
; φ = ∂xψ
• Integrals of motion (Miura, Gardner & Kruskal, 1969):F2n−1 =
∫∞−∞ P2n−1
(φ, ∂φ, ∂
2xφ, . . .
)dx ; F2n = 0
Pn = − ddxPn−1 +
∑n−2i=1 PiPn−1−i ; P1 = φ
• Soliton solutions: φ(x , t) = c2sech
2[√
c x−ct−a2
]
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Integrable Hamiltonian Systems
∞-Dimensional Systems
• the phase space M is ∞-dimensional
• thus we need an infinite set of integrals of motion {Fi}∞i=1
in involution
Example: the Korteweg-de Vries (KdV) equation
∂tφ+ ∂3xφ− 6φ∂xφ = 0
L = 12∂xψ∂tψ + (∂xψ)3 − 1
2
(∂2xψ)2
; φ = ∂xψ
• Integrals of motion (Miura, Gardner & Kruskal, 1969):F2n−1 =
∫∞−∞ P2n−1
(φ, ∂φ, ∂
2xφ, . . .
)dx ; F2n = 0
Pn = − ddxPn−1 +
∑n−2i=1 PiPn−1−i ; P1 = φ
• Soliton solutions: φ(x , t) = c2sech
2[√
c x−ct−a2
]
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Integrable Hamiltonian Systems
∞-Dimensional Systems
• the phase space M is ∞-dimensional
• thus we need an infinite set of integrals of motion {Fi}∞i=1
in involution
Example: the Korteweg-de Vries (KdV) equation
∂tφ+ ∂3xφ− 6φ∂xφ = 0
L = 12∂xψ∂tψ + (∂xψ)3 − 1
2
(∂2xψ)2
; φ = ∂xψ
• Integrals of motion (Miura, Gardner & Kruskal, 1969):F2n−1 =
∫∞−∞ P2n−1
(φ, ∂φ, ∂
2xφ, . . .
)dx ; F2n = 0
Pn = − ddxPn−1 +
∑n−2i=1 PiPn−1−i ; P1 = φ
• Soliton solutions: φ(x , t) = c2sech
2[√
c x−ct−a2
]
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Inverse Scattering Transform
Lax Representation
• in 1968 P.D.Lax observed that the KdV equation can berealized as the commutation condition[L, ∂t − A] = 0 ⇒ ∂tL = [A, L]for the auxiliary linear differential operatorsL = −∂2
x +φ(x , t) ; A = 4∂3x −3 [2φ(x , t)∂x + φx(x , t)]
• he further showed that it’s possible to recover the solitonsolution of KdV equation through the Inverse ScatteringTransform (IST); schematically:
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Inverse Scattering Transform
Lax Representation
• in 1968 P.D.Lax observed that the KdV equation can berealized as the commutation condition[L, ∂t − A] = 0 ⇒ ∂tL = [A, L]for the auxiliary linear differential operatorsL = −∂2
x +φ(x , t) ; A = 4∂3x −3 [2φ(x , t)∂x + φx(x , t)]
• he further showed that it’s possible to recover the solitonsolution of KdV equation through the Inverse ScatteringTransform (IST); schematically:
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Inverse Scattering Transform
Lax Representation
• in 1968 P.D.Lax observed that the KdV equation can berealized as the commutation condition[L, ∂t − A] = 0 ⇒ ∂tL = [A, L]for the auxiliary linear differential operatorsL = −∂2
x +φ(x , t) ; A = 4∂3x −3 [2φ(x , t)∂x + φx(x , t)]
• he further showed that it’s possible to recover the solitonsolution of KdV equation through the Inverse ScatteringTransform (IST); schematically:
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Inverse Scattering Transform
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Zero Curvature Representation
More generally the IST method is based on the observationthat some 2D PDEs appear as consistency conditions for linearsystems
∂xΨ = U(x , t;λ)Ψ ; ∂tΨ = V (x , t;λ)Ψ
which takes the form of a Zero Curvature Condition (ZCC)
∂tU(x , t;λ)− ∂xV (x , t;λ) + [U(x , t;λ),V (x , t;λ)] = 0
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Table of Contents
1 Introduction
2 Integrability: some basic facts
3 The ODE/IM Correspondence for the Bullough-Dodd Model
4 Conclusions and perspectives
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
The Bullough-Dodd Model
The Bullough-Dodd Model
• The modified EOMs are
∂∂η(z , z) + e−η(z,z) − p(z)p(z)e2η(z,z) = 0
where p(x) = x3M − E .
• We search for periodic solutions η(ρ, φ+ 2π3M ) = η(ρ, φ),
real-valued and finite for real ρ 6= 0 and φ, satisfying theasymptotics:
η(ρ, φ) ∼ρ→∞
−2M ln ρ ; η(ρ, φ) ∼ρ→0−2g ln ρ
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
The Bullough-Dodd Model
The Bullough-Dodd Model
• The modified EOMs are
∂∂η(z , z) + e−η(z,z) − p(z)p(z)e2η(z,z) = 0
where p(x) = x3M − E .
• We search for periodic solutions η(ρ, φ+ 2π3M ) = η(ρ, φ),
real-valued and finite for real ρ 6= 0 and φ, satisfying theasymptotics:
η(ρ, φ) ∼ρ→∞
−2M ln ρ ; η(ρ, φ) ∼ρ→0−2g ln ρ
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
The Bullough-Dodd Model
The Bullough-Dodd Model
• The modified EOMs are
∂∂η(z , z) + e−η(z,z) − p(z)p(z)e2η(z,z) = 0
where p(x) = x3M − E .
• We search for periodic solutions η(ρ, φ+ 2π3M ) = η(ρ, φ),
real-valued and finite for real ρ 6= 0 and φ, satisfying theasymptotics:
η(ρ, φ) ∼ρ→∞
−2M ln ρ ; η(ρ, φ) ∼ρ→0−2g ln ρ
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
The Associated Linear Problem
The Differential Equations
• By appropriately defining the components of Ψ we canwrite the linear problem LΨ = 0 ; LΨ = 0 as a pair ofdifferential equations:
∂3ψ −[(∂η)2 + 2∂2η
]∂ψ +
[λ3p(z)− ∂η∂2η − ∂3η
]ψ = 0
∂3ψ −[(∂η)2 + 2∂2η
]∂ψ +
[λ−3p(z)− ∂η∂2η − ∂3η
]ψ = 0
• The functions are defined asΨ1 = λ−
12 e
η2 ∂[eη∂(e−ηψ)] = λ−
32 e−
η2 ψ
• The asymptotics for ρ→ 0 is
∂3ψ − g(g + 2)
z2∂ψ +
g(g + 2)
z3ψ = 0
which has solutions {χn ∼ zµn}1n=−1 with
µn = 1 + n(g + 1)
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
The Associated Linear Problem
The Differential Equations
• By appropriately defining the components of Ψ we canwrite the linear problem LΨ = 0 ; LΨ = 0 as a pair ofdifferential equations:
∂3ψ −[(∂η)2 + 2∂2η
]∂ψ +
[λ3p(z)− ∂η∂2η − ∂3η
]ψ = 0
∂3ψ −[(∂η)2 + 2∂2η
]∂ψ +
[λ−3p(z)− ∂η∂2η − ∂3η
]ψ = 0
• The functions are defined asΨ1 = λ−
12 e
η2 ∂[eη∂(e−ηψ)] = λ−
32 e−
η2 ψ
• The asymptotics for ρ→ 0 is
∂3ψ − g(g + 2)
z2∂ψ +
g(g + 2)
z3ψ = 0
which has solutions {χn ∼ zµn}1n=−1 with
µn = 1 + n(g + 1)
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
The Associated Linear Problem
The Differential Equations
• By appropriately defining the components of Ψ we canwrite the linear problem LΨ = 0 ; LΨ = 0 as a pair ofdifferential equations:
∂3ψ −[(∂η)2 + 2∂2η
]∂ψ +
[λ3p(z)− ∂η∂2η − ∂3η
]ψ = 0
∂3ψ −[(∂η)2 + 2∂2η
]∂ψ +
[λ−3p(z)− ∂η∂2η − ∂3η
]ψ = 0
• The functions are defined asΨ1 = λ−
12 e
η2 ∂[eη∂(e−ηψ)] = λ−
32 e−
η2 ψ
• The asymptotics for ρ→ 0 is
∂3ψ − g(g + 2)
z2∂ψ +
g(g + 2)
z3ψ = 0
which has solutions {χn ∼ zµn}1n=−1 with
µn = 1 + n(g + 1)
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
The Associated Linear Problem
The Differential Equations
• By appropriately defining the components of Ψ we canwrite the linear problem LΨ = 0 ; LΨ = 0 as a pair ofdifferential equations:
∂3ψ −[(∂η)2 + 2∂2η
]∂ψ +
[λ3p(z)− ∂η∂2η − ∂3η
]ψ = 0
∂3ψ −[(∂η)2 + 2∂2η
]∂ψ +
[λ−3p(z)− ∂η∂2η − ∂3η
]ψ = 0
• The functions are defined asΨ1 = λ−
12 e
η2 ∂[eη∂(e−ηψ)] = λ−
32 e−
η2 ψ
• The asymptotics for ρ→ 0 is
∂3ψ − g(g + 2)
z2∂ψ +
g(g + 2)
z3ψ = 0
which has solutions {χn ∼ zµn}1n=−1 with
µn = 1 + n(g + 1)
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
The Associated Linear Problem
The WKB approximation
• In the limit ρ→∞ the equation becomes (λ = eθ)(∂3 + e3θz3M
)ψ = e3θEψ
• From WKB approximation we find that on the real axis
ψ ∼ρ→∞
ρ−Me−2 ρM+1
M+1
is the unique subdominant solution.
• We define ψk(z , z ; θ) = ψ(z , z ; θ − 2πi k3 )
• It can be shown that W [ψk , ψk+1, ψk+2] = −3i√
3.
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
The Associated Linear Problem
The WKB approximation
• In the limit ρ→∞ the equation becomes (λ = eθ)(∂3 + e3θz3M
)ψ = e3θEψ
• From WKB approximation we find that on the real axis
ψ ∼ρ→∞
ρ−Me−2 ρM+1
M+1
is the unique subdominant solution.
• We define ψk(z , z ; θ) = ψ(z , z ; θ − 2πi k3 )
• It can be shown that W [ψk , ψk+1, ψk+2] = −3i√
3.
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
The Associated Linear Problem
The WKB approximation
• In the limit ρ→∞ the equation becomes (λ = eθ)(∂3 + e3θz3M
)ψ = e3θEψ
• From WKB approximation we find that on the real axis
ψ ∼ρ→∞
ρ−Me−2 ρM+1
M+1
is the unique subdominant solution.
• We define ψk(z , z ; θ) = ψ(z , z ; θ − 2πi k3 )
• It can be shown that W [ψk , ψk+1, ψk+2] = −3i√
3.
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
The Associated Linear Problem
The WKB approximation
• In the limit ρ→∞ the equation becomes (λ = eθ)(∂3 + e3θz3M
)ψ = e3θEψ
• From WKB approximation we find that on the real axis
ψ ∼ρ→∞
ρ−Me−2 ρM+1
M+1
is the unique subdominant solution.
• We define ψk(z , z ; θ) = ψ(z , z ; θ − 2πi k3 )
• It can be shown that W [ψk , ψk+1, ψk+2] = −3i√
3.
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
The Associated Linear Problem
The WKB approximation
• In the limit ρ→∞ the equation becomes (λ = eθ)(∂3 + e3θz3M
)ψ = e3θEψ
• From WKB approximation we find that on the real axis
ψ ∼ρ→∞
ρ−Me−2 ρM+1
M+1
is the unique subdominant solution.
• We define ψk(z , z ; θ) = ψ(z , z ; θ − 2πi k3 )
• It can be shown that W [ψk , ψk+1, ψk+2] = −3i√
3.
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
From the Linear Problem to the Bethe Ansatz
The ψ-system
• Direct calculation shows that u−k,k = W [ψ−k , ψk ] satisfythe equation
∂3u−k,k −[(∂η)2 + 2∂2η
]∂u−k,k+
−(e−2πikp(z) + ∂η∂2η + ∂3η
)u−k,k = 0
which coincides with the original one for k ∈ 12Z
• Evaluating the WKB expansion for one can show thatu− 1
2, 1
2= i√
3ψ0, obtaining our ψ-system:
W [ψ− 12, ψ 1
2] = i√
3ψ0
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
From the Linear Problem to the Bethe Ansatz
The ψ-system
• Direct calculation shows that u−k,k = W [ψ−k , ψk ] satisfythe equation
∂3u−k,k −[(∂η)2 + 2∂2η
]∂u−k,k+
−(e−2πikp(z) + ∂η∂2η + ∂3η
)u−k,k = 0
which coincides with the original one for k ∈ 12Z
• Evaluating the WKB expansion for one can show thatu− 1
2, 1
2= i√
3ψ0, obtaining our ψ-system:
W [ψ− 12, ψ 1
2] = i√
3ψ0
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
From the Linear Problem to the Bethe Ansatz
The ψ-system
• Direct calculation shows that u−k,k = W [ψ−k , ψk ] satisfythe equation
∂3u−k,k −[(∂η)2 + 2∂2η
]∂u−k,k+
−(e−2πikp(z) + ∂η∂2η + ∂3η
)u−k,k = 0
which coincides with the original one for k ∈ 12Z
• Evaluating the WKB expansion for one can show thatu− 1
2, 1
2= i√
3ψ0, obtaining our ψ-system:
W [ψ− 12, ψ 1
2] = i√
3ψ0
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Towards the BAEWe now set χn;k(z , z ; θ) = ωn(g+1)χn(z , z ; θ) where
ω = e2πi
3(M+1) and expand the ψk functions in terms of the χbasis:
ψk(z , z ; θ) =1∑
n=−1
Qn(θ − 2πik
3)χn;k(z , z ; θ)
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
From the Linear Problem to the Bethe Ansatz
The BAENow, using the last equation and the ψ-system and consideringterms proportional to z−g , we obtain
i√
3Q1(θ) =(g + 1)ω−g+1
2 Q1(θ + iπ
3)Q0(θ − i
π
3)+
−(g + 1)ωg+1
2 Q1(θ − iπ
3)Q0(θ + i
π
3)
which, considered at θ = θn ± i π3 with Q1(θn) = 0 yelds
Q1(θn + i 2π3 )Q1(θn − i π3 )
Q1(θn − i 2π3 )Q1(θn + i π3 )
= −ωg+1
This equation, along with the corresponding for Q−1(θ),coincides with the BAE for the ground state of theIzergin-Korepin model.
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Table of Contents
1 Introduction
2 Integrability: some basic facts
3 The ODE/IM Correspondence for the Bullough-Dodd Model
4 Conclusions and perspectives
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Conclusions
• A first work published: Phil. Trans A371 (2013) 23,[arXiv:1209.5517]
• The work of these years focused on the general TodaModels, of which the BD model is a representative
• The result are being collected and ordered in view of apublication
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Conclusions
• A first work published: Phil. Trans A371 (2013) 23,[arXiv:1209.5517]
• The work of these years focused on the general TodaModels, of which the BD model is a representative
• The result are being collected and ordered in view of apublication
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Conclusions
• A first work published: Phil. Trans A371 (2013) 23,[arXiv:1209.5517]
• The work of these years focused on the general TodaModels, of which the BD model is a representative
• The result are being collected and ordered in view of apublication
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Conclusions
• A first work published: Phil. Trans A371 (2013) 23,[arXiv:1209.5517]
• The work of these years focused on the general TodaModels, of which the BD model is a representative
• The result are being collected and ordered in view of apublication
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Perspectives
• The ODE/IM is a surprising and interesting mathematicalstructure which is not yet well understood; it might beworthy to investigate more deeply.
• It has also proven to be a powerful tool in variousbranches of the physics, such as condensed matter physics,boundary CFT and PT-symmetric QM
• It can be applied also to the amplidude calculations inN = 4 Super Yang-Mills theory, via the AdS/CFTcorrespondence
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Perspectives
• The ODE/IM is a surprising and interesting mathematicalstructure which is not yet well understood; it might beworthy to investigate more deeply.
• It has also proven to be a powerful tool in variousbranches of the physics, such as condensed matter physics,boundary CFT and PT-symmetric QM
• It can be applied also to the amplidude calculations inN = 4 Super Yang-Mills theory, via the AdS/CFTcorrespondence
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Perspectives
• The ODE/IM is a surprising and interesting mathematicalstructure which is not yet well understood; it might beworthy to investigate more deeply.
• It has also proven to be a powerful tool in variousbranches of the physics, such as condensed matter physics,boundary CFT and PT-symmetric QM
• It can be applied also to the amplidude calculations inN = 4 Super Yang-Mills theory, via the AdS/CFTcorrespondence
The ODE/IMcorrespon-
dence
S. Negro
Introduction
Integrability:some basicfacts
The ODE/IMCorrespon-dence for theBullough-DoddModel
Conclusionsandperspectives
Perspectives
• The ODE/IM is a surprising and interesting mathematicalstructure which is not yet well understood; it might beworthy to investigate more deeply.
• It has also proven to be a powerful tool in variousbranches of the physics, such as condensed matter physics,boundary CFT and PT-symmetric QM
• It can be applied also to the amplidude calculations inN = 4 Super Yang-Mills theory, via the AdS/CFTcorrespondence