Integrable Hierarchies, Solitons and Infinite Dimensional Algebras Jose Francisco Gomes Instituto de Física Teórica - IFT-Unesp 7th International Workshop on New challenges in Quantum Mechanics: Integrability and Supersymmetry Benasque, September/2019 Integrable Hierarchies, Solitons and Infinite Dimensional Algebras
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Integrable Hierarchies, Solitons and Infinite DimensionalAlgebras
Jose Francisco Gomes
Instituto de Física Teórica - IFT-Unesp
7th International Workshop on New challenges in Quantum Mechanics:Integrability and Supersymmetry
Benasque, September/2019
Integrable Hierarchies, Solitons and Infinite Dimensional Algebras
Aim
Discuss the general structure of Integrable non Linearevolution equations associated to graded Lie algebraicstructure.
Integrable Hierarchies1 Systematic construction Soliton Equations
2 Systematic construction of Soliton solutions.
3 Systematic construction of Backlund transformation and Defects.
Integrable Hierarchies, Solitons and Infinite Dimensional Algebras
Integrable Hierarchies, Solitons and Infinite Dimensional Algebras
Backlund transformation and Integrable Defects
Integrable Defects interpolate two solutions of same eqn.of motion by Backlund Transformation (Bowcock, Corriganand Zambon, 2003),i.e.,
x
1 2
1K( , )2
Eqns. of motion at the Defect position, x , is given byBacklund Transformation.Energy, Momentum, Topological Charges, etc are modifiedin order to include the Defect contribution (border effects).
Integrable Hierarchies, Solitons and Infinite Dimensional Algebras
Backlund transformation and Integrable Defects
SL(3) Toda Model
Eqns. Motion
∂x∂t−1φ1 = e2φ1−φ2 − e−φ1−φ2 ,
∂x∂t−1φ2 = e2φ2−φ1 − e−φ1−φ2 .
Lax
L = ∂x + Eα1 + Eα2 + λE−α1−α2 + ∂xφ1h1 + ∂xφ2h2
Backlund-gauge Transformation
K (φ1, φ2, ψ1, ψ2) =
1 0 λ−1e−φ2−φ1
eφ1+ψ1−ψ2 1 00 e−φ1+φ2+ψ2 1
Integrable Hierarchies, Solitons and Infinite Dimensional Algebras
Backlund transformation and Integrable Defects
Leading to
∂x (φ1 − ψ1) = λ(eφ1+ψ1−ψ2 − e−φ2−ψ1)
∂x (φ2 − ψ2) = λ(e−φ1+φ2+ψ2 − e−φ2−ψ1)
and
∂t (φ1 + ψ1 − ψ2) = λ−1(eφ1−φ2−ψ1+ψ2 − e−φ1+ψ1)
∂t (−φ1 + φ2 + ψ2) = λ−1(eφ2−ψ2 − eφ1−φ2−ψ1+ψ2)
Easely generalized to sl(n + 1).
Integrable Hierarchies, Solitons and Infinite Dimensional Algebras
Backlund transformation and Integrable Defects
More General Backlund Transformations (Araujo,JFG,Zimerman 2011)
e.g. Tzitzeica Model (φ1 = φ2 = φ)
∂x∂t−1φ = eφ − e−2φ.
Type II Backlund-gauge Transformation (Corrigan,Zambon,09)Involves non local auxiliary field, Λ
diagonal Ki,i 6= 1,
∂xq = − i2ζ
eΛ − ζe−2Λ(eq + e−q)2,
∂x (Λ− p) = −ζe−2Λ(e2q − e−2q),
∂tq = −iζep−Λ(eq + e−q)− 14ζ
e2Λ−2p,
∂t Λ = iζe−Λ+p(e−q − eq) (4)
for q = φ− ψ, p = φ+ ψ and Λ is the auxiliary field.Integrable Hierarchies, Solitons and Infinite Dimensional Algebras
Conclusions
Introduced Integrable Hierarchy associated to Affine LieAlgebra.
Proposed systematic construction of Soliton solutions interms of vertex operators and representations of Affine LieAlgebras.
Generalized to other integrable hierarchies allowingconstant vacuum solutions, e.g., negative evensub-hierarchy
Integrable Hierarchies, Solitons and Infinite Dimensional Algebras