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

The structure of affine Kac-Moody algebras has provided considerable insight underlining the construction of integrable hierarchies in 1+1 spacetime dimensions. An integrable hierarchy consists of a series of equations of motion with a common algebraic structure. They are constructed out of a zero curvature condition which involves a pair of gauge potentials, lying within the affine Lie algebra. A crucial ingredient is the decomposition of the affine Lie algebra G into graded subspaces, determined by a grading operator Q. The integrable hierarchy is further specified by a constant, grade one semi-simple element E (1) . The choice of G, Q and E (1) are the basic ingredients that fully determine and classify the entire hierarchy (see for instance ref. [1] for a review).

Representation theory of Kac-Moody algebras play also an important role in constructing systematically soliton solutions of integrable hierarchies. The main idea of the dressing method consists in, by gauge transformation, to map a simple vacuum configuration into a non trivial one-or multi-soliton solution. The pure gauge solution of the zero curvature representation leads to explicit spacetime dependence for the vacuum configuration which in turn, generates by gauge transformation, the non-trivial soliton solutions of the hierarchy [2,3]. It becomes clear that the solutions are classified into conjugacy classes according to the choice of vacuum configuration. In general, the vacuum is taken as the zero field configuration and the soliton solutions are constructed and classified in terms of vertex operators [4].

A dressing method approach to nontrivial constant vacuum configuration was proposed in [5] when considering the negative even flows of the modified Korteweg-de Vries (mKdV) hierarchy, which do not admit trivial vacuum configuration, i.e. vanishing boundary condition. A deformed vertex operator was introduced and the method was further applied to the whole mKdV hierarchy with nonvanishing boundary condition and to a hierarchy containing the Gardner equation [6]. Here in these notes, we shall provide further clarifications discussing nonvanishing boundary conditions for integrable hierarchies in general, constructed with the structure proposed in [7] (and references therein). We point out that not all the models within the hierarchy admit solutions with nonvanishing (constant) boundary condition. We then consider as an example the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy with nontrivial constant vacuum configuration. Results for the focusing nonlinear Schroedinger (NLS) equation were recently obtained in ref. [8]. A hybrid of the dressing and Hirota method for a multi-component generalization of the AKNS hierarchy was also considered recently in [9].

We first describe the general aspects of the dressing method in section 3, together with a discussion on the possible different boundary conditions. We point out the requirements for the individual models of the hierarchy to admit a nonvanishing constant boundary value solutions. Later in section 4, we discuss the construction of the AKNS hierarchy, followed by the construction of its solutions, section 5. In section 6 we present the deformed vertex operators used to construct solitonic solutions with nonvanishing boundary condition and illustrate with explicit examples.

In this section we introduce the main algebraic concepts used for the construction of integrable models and to obtain its solutions. A more detailed exposition can be found in [10,11].

Let G be a semi-simple finite-dimensional Lie algebra. The infinite-dimensional loop algebra L(G) is defined as the tensor product of G with integer powers of the so called complex spectral parameter λ, L(G)

The commutator of the loop algebra is given by

where T a , T b is the G commutator.

The central extension is performed by the introduction of the operator ĉ, which commutes with all the others ĉ, T n a = 0. Furthermore, consider the spectral derivative operator d ≡ λ d dλ such that d, T n a = nT n a , so it measures the power of the spectral parameter. The affine Kac-Moody Lie algebra is defined by G ≡ L(G) ⊕ Cĉ ⊕ C d and for T n a , T m b ∈ G the commutator is provided by

where T a |T b is the Killing form on G.

The algebra G can be decomposed into Z-graded subspaces by the introduction of a grading operator

where

The integer j is the grade of the operators defined with respect to Q. It follows from the Jacobi identity that if i+j) . Let E be a semi-simple element of G. Its kernel is defined by K ≡ T n a ∈ G | E, T n a = 0 . Its complement is called the image subspace M and G = K ⊕ M. It can be readily verified from this definition and the Jacobi identity that K, K ⊂ K and K, M ⊂ M. We also assume the symmetric space structure M, M ⊂ K.

Consider the linear system in 1+1 spacetime dimension

where U, V ∈ G and Ψ is an element of the Lie group of G. The compatibility of this system ensures integrability, yielding the zero c

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