Completion of the Ablowitz-Kaup-Newell-Segur integrable coupling

Integrable couplings are associated with non-semisimple Lie algebras. In this paper, we propose a new method to generate new integrable systems through making perturbation in matrix spectral problems for integrable couplings, which is called the `completion process of integrable couplings'. As an example, the idea of construction is applied to the Ablowitz-Kaup-Newell-Segur integrable coupling. Each equation in the resulting hierarchy has a bi-Hamiltonian structure furnished by the component-trace identity

Dressing method and quadratic bundles related to symmetric spaces. Vanishing boundary conditions

We consider quadratic bundles related to Hermitian symmetric spaces of the type SU(m+n)/S(U(m)x U(n)). The simplest representative of the corresponding integrable hierarchy is given by a multi-component Kaup-Newell derivative nonlinear Schroedinger equation which serves as a motivational example for our general considerations. We extensively discuss how one can apply Zakharov-Shabat's dressing procedure to derive reflectionless potentials obeying zero boundary conditions. Those could be used for one to construct fast decaying solutions to any nonlinear equation belonging to the same hierarchy. One can distinguish between generic soliton type solutions and rational solutions.Comment: 18 page

On a Lagrangian reduction and a deformation of completely integrable systems

We develop a theory of Lagrangian reduction on loop groups for completely integrable systems after having exchanged the role of the space and time variables in the multi-time interpretation of integrable hierarchies. We then insert the Sobolev norm $H^1$ in the Lagrangian and derive a deformation of the corresponding hierarchies. The integrability of the deformed equations is altered and a notion of weak integrability is introduced. We implement this scheme in the AKNS and SO(3) hierarchies and obtain known and new equations. Among them we found two important equations, the Camassa-Holm equation, viewed as a deformation of the KdV equation, and a deformation of the NLS equation

Integrable discretization of the vector/matrix nonlinear Schr\"odinger equation and the associated Yang-Baxter map

The action of a B\"acklund-Darboux transformation on a spectral problem associated with a known integrable system can define a new discrete spectral problem. In this paper, we interpret a slightly generalized version of the binary B\"acklund-Darboux (or Zakharov-Shabat dressing) transformation for the nonlinear Schr\"odinger (NLS) hierarchy as a discrete spectral problem, wherein the two intermediate potentials appearing in the Darboux matrix are considered as a pair of new dependent variables. Then, we associate the discrete spectral problem with a suitable isospectral time-evolution equation, which forms the Lax-pair representation for a space-discrete NLS system. This formulation is valid for the most general case where the two dependent variables take values in (rectangular) matrices. In contrast to the matrix generalization of the Ablowitz-Ladik lattice, our discretization has a rational nonlinearity and admits a Hermitian conjugation reduction between the two dependent variables. Thus, a new proper space-discretization of the vector/matrix NLS equation is obtained; by changing the time part of the Lax pair, we also obtain an integrable space-discretization of the vector/matrix modified KdV (mKdV) equation. Because B\"acklund-Darboux transformations are permutable, we can increase the number of discrete independent variables in a multi-dimensionally consistent way. By solving the consistency condition on the two-dimensional lattice, we obtain a new Yang-Baxter map of the NLS type, which can be considered as a fully discrete analog of the principal chiral model for projection matrices.Comment: 33 pages; (v2) minor corrections (v3) added one paragraph on a space-discrete matrix KdV equation at the end of section

Stationary and 2+1 dimensional integrable reductions of AKNS hierarchy

Thesis (Master)--Izmir Institute of Technology, Mathematics, Izmir, 2004Includes bibliographical references (leaves: 72-82)Text in English; Abstract: Turkish and Englishvi, 84, leavesThe main concepts of the soliton theory and infinite dimensional Hamiltonian Systems, including AKNS (Ablowitz, Kaup, Newell, Segur) integrable hierarchy of nonlinear evolution equations are introduced.By integro-differential recursion operator for this hierarchy, several reductions to KDV, MKdV, mixed KdV/MKdV and Reaction-Diffusion system are constructed.The stationary reduction of the fifth order KdV is related to finite-dimensional integrable system of Henon-Heiles type.Different integrable extensions of Henon-Heiles model are found with corresponding separation of variables in Hamilton-Jacobi theory.Using the second and the third members of AKNS hierarchy, new method to solve 2+1 dimensional Kadomtsev-Petviashvili(KP-II) equation is proposed.By the Hirota bilinear method, one and two soliton solutions of KP-II are constructed and the resonance character of their mutual interactions are studied.By our bilinear form we first time created new four virtual soliton resonance solution for KPII.Finally, relations of our two soliton solution with degenerate four soliton solution in canonical Hirota form of KPII are established

Integrable discretization of recursion operators and unified bilinear forms to soliton hierarchies

In this paper, we give a procedure of how to discretize the recursion operators by considering unified bilinear forms of integrable hierarchies. As two illustrative examples, the unified bilinear forms of the AKNS hierarchy and the KdV hierarchy are presented from their recursion operators. Via the compatibility between soliton equations and their auto-B\"acklund transformations, the bilinear integrable hierarchies are discretized and the discrete recursion operators are obtained. The discrete recursion operators converge to the original continuous forms after a standard limit.Comment: 11Page