Classical Integrable N=1 and N=2N= 2 Super Sinh-Gordon Models with Jump Defects

The structure of integrable field theories in the presence of jump defects is discussed in terms of boundary functions under the Lagrangian formalism. Explicit examples of bosonic and fermionic theories are considered. In particular, the boundary functions for the N=1 and N=2 super sinh-Gordon models are constructed and shown to generate the Backlund transformations for its soliton solutions. As a new and interesting example, a solution with an incoming boson and an outgoing fermion for the N=1 case is presented. The resulting integrable models are shown to be invariant under supersymmetric transformation.Comment: talk presented at the V International Symposium on Quantum Theory and Symmetries, Valladolid, Spain, July 22-28,200

Affine Lie Algebraic Origin of Constrained KP Hierarchies

We present an affine $sl (n+1)$ algebraic construction of the basic constrained KP hierarchy. This hierarchy is analyzed using two approaches, namely linear matrix eigenvalue problem on hermitian symmetric space and constrained KP Lax formulation and we show that these approaches are equivalent. The model is recognized to be the generalized non-linear Schr\"{o}dinger (\GNLS) hierarchy and it is used as a building block for a new class of constrained KP hierarchies. These constrained KP hierarchies are connected via similarity-B\"{a}cklund transformations and interpolate between \GNLS and multi-boson KP-Toda hierarchies. Our construction uncovers origin of the Toda lattice structure behind the latter hierarchy.Comment: 25 pgs, LaTeX, IFT-P/029/94 and UICHEP-TH/93-1

Darboux-Backlund Derivation of Rational Solutions of the Painleve IV Equation

Rational solutions of the Painleve IV equation are constructed in the setting of pseudo-differential Lax formalism describing AKNS hierarchy subject to the additional non-isospectral Virasoro symmetry constraint. Convenient Wronskian representations for rational solutions are obtained by successive actions of the Darboux-Backlund transformations.Comment: 21 page

The complex Sine-Gordon equation as a symmetry flow of the AKNS Hierarchy

It is shown how the complex sine-Gordon equation arises as a symmetry flow of the AKNS hierarchy. The AKNS hierarchy is extended by the ``negative'' symmetry flows forming the Borel loop algebra. The complex sine-Gordon and the vector Nonlinear Schrodinger equations appear as lowest negative and second positive flows within the extended hierarchy. This is fully analogous to the well-known connection between the sine-Gordon and mKdV equations within the extended mKdV hierarchy. A general formalism for a Toda-like symmetry occupying the ``negative'' sector of sl(N) constrained KP hierarchy and giving rise to the negative Borel sl(N) loop algebra is indicated.Comment: 8 pages, LaTeX, typos corrected, references update

On a negative flow of the AKNS hierarchy and its relation to a two-component Camassa-Holm equation

Different gauge copies of the Ablowitz-Kaup-Newell-Segur (AKNS) model labeled by an angle $\theta$ are constructed and then reduced to the two-component Camassa--Holm model. Only three different independent classes of reductions are encountered corresponding to the angle $\theta$ being 0, $\pi/2$ or taking any value in the interval $0<\theta<\pi/2$. This construction induces B\"{a}cklund transformations between solutions of the two-component Camassa--Holm model associated with different classes of reduction.Comment: This is a contribution to the Proc. of the O'Raifeartaigh Symposium on Non-Perturbative and Symmetry Methods in Field Theory (June 2006, Budapest, Hungary), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Integrable Origins of Higher Order Painleve Equations

Higher order Painleve equations invariant under extended affine Weyl groups $A^{(1)}_n$ are obtained through self-similarity limit of a class of pseudo-differential Lax hierarchies with symmetry inherited from the underlying generalized Volterra lattice structure.Comment: 18 pages Late

Dressing approach to the nonvanishing boundary value problem for the AKNS hierarchy

We propose an approach to the nonvanishing boundary value problem for integrable hierarchies based on the dressing method. Then we apply the method to the AKNS hierarchy. The solutions are found by introducing appropriate vertex operators that takes into account the boundary conditions.Comment: Published version Proc. Quantum Theory and Symmetries 7 (QTS7)(Prague, Czech Republic, 2011

T-Duality in 2-D Integrable Models

The non-conformal analog of abelian T-duality transformations relating pairs of axial and vector integrable models from the non abelian affine Toda family is constructed and studied in detail.Comment: 14 pages, Latex, v.2 misprints corrected, reference added, to appear in J. Phys.