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 θ 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 θ being 0, π/2 or taking any value in the interval 0 < θ < π/2. This construction induces Bäcklund transformations between solutions of the two-component Camassa-Holm model associated with different classes of reduction

Some comments on the bi(tri)-Hamiltonian structure of Generalized AKNS and DNLS hierarchies

We give the correct prescriptions for the terms involving the inverse of the derivative of the delta function, in the Hamiltonian structures of the AKNS and DNLS systems, in order for the Jacobi identities to hold. We also establish that the sl(2) AKNS and DNLS systems are tri-Hamiltonians and construct two compatible Hamiltonian structures for the sl(3) AKNS system. We also give a derivation of the recursion operator for the sl(n+1) DNLS system.Comment: 10 pages, LaTe

A symmetry reduction technique for higher order Painlev\'e systems

The symmetry reduction of higher order Painlev\'e systems is formulated in terms of Dirac procedure. A set of canonical variables that admit Dirac reduction procedure is proposed for Hamiltonian structures governing the ${A^{(1)}_{2M}}$ and ${A^{(1)}_{2M-1}}$ Painlev\'e systems for $M=2,3,...$.Comment: to appear in Phys. Lett.

On Non-Linear W-Infinity Symmetry of Generalized Liouville and Conformal Toda Models

Invariance under non-linear ${\sf {\hat W}}_{\infty}$ algebra is shown for the two-boson Liouville type of model and its algebraic generalizations, the extended conformal Toda models. The realization of the corresponding generators in terms of two boson currents within KP hierarchy is presented.Comment: 10 pgs, LaTeX, IFT-P.038/9

Hirota's Solitons in the Affine and the Conformal Affine Toda Models

We use Hirota's method formulated as a recursive scheme to construct complete set of soliton solutions for the affine Toda field theory based on an arbitrary Lie algebra. Our solutions include a new class of solitons connected with two different type of degeneracies encountered in the Hirota's perturbation approach. We also derive an universal mass formula for all Hirota's solutions to the Affine Toda model valid for all underlying Lie groups. Embedding of the Affine Toda model in the Conformal Affine Toda model plays a crucial role in this analysis.Comment: 36 pages, LaTe

On Discrete Symmetries of the Multi-Boson KP Hierarchies

We show that the multi-boson KP hierarchies possess a class of discrete symmetries linking them to the discrete Toda systems. These discrete symmetries are generated by the similarity transformation of the corresponding Lax operator. This establishes a canonical nature of the discrete transformations. The spectral equation, which defines both the lattice system and the corresponding Lax operator, plays a key role in determining pertinent symmetry structure. We also introduce a concept of the square-root lattice leading to a family of new pseudo-differential operators with covariance under additional B\"{a}cklund transformations.Comment: 11 pgs, LaTeX, IFT-P/75/93, UICHEP-TH/93-1

Integrable Hierarchy for Multidimensional Toda Equations and Topological-Anti-topological Fusion

The negative symmetry flows are incorporated into the Riemann-Hilbert problem for the homogeneous $A_m$-hierarchy and its $\hat{gl} (m+1, C)$ extension. A loop group automorphism of order two is used to define a sub-hierarchy of $\hat{gl} (m+1, C)$ hierarchy containing only the odd symmetry flows. The positive and negative flows of the $\pm 1$ grade coincide with equations of the multidimensional Toda model and of topological-anti-topological fusion.Comment: 24+1 pages, some minus signs correcte

A New Deformation of W-Infinity and Applications to the Two-loop WZNW and Conformal Affine Toda Models

We construct a centerless W-infinity type of algebra in terms of a generator of a centerless Virasoro algebra and an abelian spin-1 current. This algebra conventionally emerges in the study of pseudo-differential operators on a circle or alternatively within KP hierarchy with Watanabe's bracket. Construction used here is based on a special deformation of the algebra $w_{\infty}$ of area preserving diffeomorphisms of a 2-manifold. We show that this deformation technique applies to the two-loop WZNW and conformal affine Toda models, establishing henceforth $W_{\infty}$ invariance of these models.Comment: 8 page

Super WZNW with Reductions to Supersymmetric and Fermionic Integrable Models

A systematic construction for an action describing a class of supersymmetric integrable models as well as for pure fermionic theories is discussed in terms of the gauged WZNW model associated to twisted affine Kac-Moody algebras. Explicit examples of the $N=1,2$ super sinh(sine)-Gordon models are discussed in detail. Pure fermionic theories arises for cosets $sl(p,1)/sl(p)\otimes u(1)$ when a maximal kernel condition is fulfilled. The integrability condition for such models is discussed and it is shown that the simplest example when $p=2$ leads to the constrained Bukhvostov-Lipatov, Thirring, scalar massive and pseudo-scalar massless Gross-Neveu models.Comment: 28 pages, latex, added referenc