J. F. Gomes

Type-II Backlund transformations via gauge transformations

A bstractThe construction of type II Bäcklund transformation for the sine-Gordon and the Tzitzéica-Bullough-Dodd models are obtained from gauge transformation. An infinite number of conserved quantities are constructed from the defect matrices. This guarantees that the introduction of type II defects for these models does not spoil their integrability. In particular, modified energy and momentum are derived and compared with those presented in recent literature.

Classical integrable super sinh-Gordon equation with defects

The dressing and vertex operator formalism is emploied to study the soliton solutions of the N = 1 super mKdV and sinh-Gordon models. Explicit four vertex soliton solution is presented. The introduction of defects is discussed under the Lagrangian formalism and Backlund transformations for the super sinh-Gordon model. Modified conserved momentum and energy are constructed for this case. Some explicit examples of different Backlund solitons solutions are discussed.The introduction of defects is discussed under the Lagrangian formalism and Backlund transformations for the N = 1 super sinh-Gordon model. Modified conserved momentum and energy are constructed for this case. Some explicit examples of different Backlund soliton solutions are discussed. The Lax formulation within the space split by the defect leads to the integrability of the model and henceforth to the existence of an infinite number of constants of motion.

Affine Lie Algebraic Origin of Constrained KP Hierarchies

An affine sl(n+1) algebraic construction of the basic constrained KP hierarchy is presented. This hierarchy is analyzed using two approaches, namely linear matrix eigenvalue problem on hermitian symmetric space and constrained KP Lax formulation and it is shown that these approaches are equivalent. The model is recognized to be the generalized non‐linear Schrodinger (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‐Backlund transformations and interpolate between GNLS and multi‐boson KP‐Toda hierarchies. Our construction uncovers the origin of the Toda lattice structure behind the latter hierarchy.

Constrained KP models as integrable matrix hierarchies

We formulate the constrained KP hierarchy (denoted by cKP K+1,M) as an affine sl(M+K+1) matrix integrable hierarchy generalizing the Drinfeld–Sokolov hierarchy. Using an algebraic approach, including the graded structure of the generalized Drinfeld–Sokolov hierarchy, we are able to find several new universal results valid for the cKP hierarchy. In particular, our method yields a closed expression for the second bracket obtained through Dirac reduction of any untwisted affine Kac–Moody current algebra. An explicit example is given for the case sl(M+K+1), for which a closed expression for the general recursion operator is also obtained. We show how isospectral flows are characterized and grouped according to the semisimple non-regular element E of sl(M+K+1) and the content of the center of the kernel of E.

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 la- beled by an angle 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 being 0, / 2 or taking any value in the interval 0 < < / 2. This construction induces Backlund transformations between solutions of the two-component Camassa-Holm model associated with different classes of reduction.

Integrablility of a classical N=2 super sinh-Gordon model with jump defects

The Lagrangian formalism for the N = 2 supersymmetric sinh-Gordon model with a jump defect is considered. The modified conserved momentum and energy are constructed in terms of border functions. The supersymmetric Backlund transformation is given and an one-soliton solution is obtained. The Lax formulation based on the affine super Lie algebra sl(2, 2) within the space split by the defect leads to the integrability of the model and henceforth to the existence of an infinite number of constants of motion.

Nonautonomous mixed mKdV–sinh–Gordon hierarchy

The construction of a nonautonomous mixed mKdV/sine–Gordon model is proposed by employing an infinite-dimensional affine Lie algebraic structure within the zero-curvature representation. A systematic construction of soliton solutions is provided by an adaptation of the dressing method which takes into account arbitrary time-dependent functions. A particular choice of those arbitrary functions provides an interesting solution describing the transition of a pure mKdV system into a pure sine–Gordon soliton.

Backlund transformation for integrable hierarchies: example - mKdV hierarchy

In this note we present explicitly the construction of the mKdV hierarchy and show that it decomposes into positive and negative graded sub-hierarchies. We extend the construction of the Backlund transformation for the sinh-Gordon model to all other positive and negative odd graded equations of motion generated by the same affine algebraic structure. Some simple examples of solutions are explicitly verified to satisfy, in a universal manner, the Backlund transformations for the first few odd (positive and negative) sub-hierarchies.

Type-II Super-Bäcklund Transformation and Integrable Defects for the

A bstractA new super-Bäcklund transformation for the N = 1 supersymmetric sinhGordon equation is constructed. Based on this construction we propose a type-II integrable defect for the supersymmetric sinh-Gordon model consistent with this new transformation through the Lagrangian formalism. Explicit expressions for the modified conserved energy, momentum and supercharges are also computed. In addition, we show for the model that the type-II defect can also been regarded as a pair of fused defects of a previously introduced type. The explicit derivation of the associated defect matrices is also presented as a necessary condition for the integrability of the model.

N =

The permutability of two Backlund transformations is employed to construct a nonlinear superposition formula and to generate a class of solutions for the N=2 super sine-Gordon model. We present explicitly the one and two soliton solutions.