We provide an overview of elliptic algebro-geometric solutions of the KdV and AKNS hierarchies, with special emphasis on Floquet theoretic and spectral theoretic methods. Our treatment includes an effective characterization of all stationary elliptic KdV and AKNS solutions based on a theory developed by Hermite and Picard.

A systematic investigation of certain higher order or deformed soliton equations with (1 + 1) dimensions, from the point complete integrability, is presented. Following procedure Ablowitz, Kaup, Newell and Segur (AKNS) we find that version Nonlinear Schrodinger equation, Hirota equation AKNS admit Lax pairs. We report each identified possesses Painleve property for partial differential admits t...

The trace variational identity is generalized to zero curvature equations associated with non-semi-simple Lie algebras or, equivalently, Lie algebras possessing degenerate Killing forms. An application of the resulting generalized variational identity to a class of semi-direct sums of Lie algebras in the AKNS case furnishes Hamiltonian and quasi-Hamiltonian structures of the associated integrab...

The commutator of enlarged vector fields was explicitly computed for integrable coupling systems associated with semidirect sums of Lie algebras. An algebraic structure of zero curvature representations is then established for such integrable coupling systems. As an application example of this algebraic structure, the commutation relations of Lax operators corresponding to the enlarged isospect...

Bi-integrable couplings of soliton equations are presented through introducing non-semisimple matrix Lie algebras on which there exist non-degenerate, symmetric and ad-invariant bilinear forms. The corresponding variational identity yields Hamiltonian structures of the resulting bi-integrable couplings. An application to the AKNS spectral problem gives bi-integrable couplings with Hamiltonian s...

We provide an overview of elliptic algebro-geometric solutions of the KdV and AKNS hierarchies, with special emphasis on Floquet theoretic and spectral theoretic methods. Our treatment includes an effective characterization of all stationary elliptic KdV and AKNS solutions based on a theory developed by Hermite and Picard.

An explicit reciprocal transformation between a 2-component generalization of the CamassaHolm equation, called the 2-CH system, and the first negative flow of the AKNS hierarchy is established, this transformation enables one to obtain solutions of the 2-CH system from those of the first negative flow of the AKNS hierarchy. Interesting examples of peakon and multi-kink solutions of the 2-CH sys...

In this work, a novel integrable evolution system in the sense of Lax’s scheme associated with mixed spectral Ablowitz-Kaup-Newell-Segur (AKNS) matrix problem is first derived. Then, time dependences scattering data corresponding to AKNS are given inverse analysis. Based on data, reconstruction potentials carried out, and finally analytical solutions four arbitrary functions derived formulated....

In this paper the Galilean, scaling and translational self–similarity conditions for the AKNS hierarchy are analysed geometrically in terms of the infinite dimensional Grassmannian. The string equations found recently by non–scaling limit analysis of the one–matrix model are shown to correspond to the Galilean self–similarity condition for this hierarchy. We describe, in terms of the initial da...

We develop an alternative systematic approach to the AKNS hierarchy based on elementary algebraic methods. In particular, we recursively construct Lax pairs for the entire AKNS hierarchy by introducing a fundamental polynomial formalism and establish the basic algebro-geometric setting including associated BurchnallChaundy curves, Baker-Akhiezer functions, trace formulas, Dubrovin-type equation...