RECURSION OPERATORS FOR KP,mKP AND HARRY DYM HIERARCHIES

In this paper, we give a unified construction of the recursion operators from the Lax representation for three integrable hierarchies: Kadomtsev–Petviashvili (KP), modified Kadomtsev–Petviashvili (mKP) and Harry Dym under n-reduction. This shows a new inherent relationship between them. To illustrate our construction, the recursion operator are calculated explicitly for 2-reduction and 3-reduction.     

A Poisson–Lie Framework for Rational Reductions of the KP Hierarchy

We give a simple proof of I. Krichever's theorem on rational reductions of the Kadomtsev–Petviashvili hierarchy by using the Poisson–Lie structure on the group of pseudo-differential symbols.     

Equations of Hurwitz schemes in the infinite Grassmannian

The main result proved in the paper is the computation of the explicit equations defining the Hurwitz schemes of coverings with punctures as subschemes of the Sato infinite Grassmannian. As an application, we characterize the existence of certain linear series on a smooth curve in terms of soliton equations. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Social paradoxes of majority rule voting and renormalization group

Real-space renormalization group ideas are used to study a voting problem in political science. A model to construct self-directed pyramidal structures from bottom up to the top is presented. Using majority rules, it is shown that a minority and even a majority can be systematically self-eliminated from top leadership, provided the hierarchy has a minimal number of levels. In some cases, 70% of the population is found to have zero representation after six hierarchical levels. Results are discussed with respect to the internal operation of political organizations.     

Compatible Dubrovin–Novikov Hamiltonian Operators,Lie Derivative,and Integrable Systems of Hydrodynamic Type

We prove that two Dubrovin–Novikov Hamiltonian operators are compatible if and only if one of these operators is the Lie derivative of the other operator along a certain vector field. We consider the class of flat manifolds, which correspond to arbitrary pairs of compatible Dubrovin–Novikov Hamiltonian operators. Locally, these manifolds are defined by solutions of a system of nonlinear equations, which is integrable by the method of the inverse scattering problem. We construct the integrable hierarchies generated by arbitrary pairs of compatible Dubrovin–Novikov Hamiltonian operators.     

On a Generalization of the Trèves Criterion for the First Integrals of the KdV Hierarchy to Higher GD Hierarchies

First, a new proof of the necessity of the Trèves criterion for a differential polynomialto be a first integral of the KdV hierarchy is given. This proof is much shorter than the original one and admits generalizations. Second, a similar criterion is suggested and its necessity proven for the next of the GD hierarchies which is generated by the third-order linear operator.     

Twistor correspondences for the soliton hierarchies

In this article we propose a new overview on the theory of integrable systems based on symmetry reduction of the anti-self-dual Yang—Mills equations and its twistor correspondence. First, the non-linear Schrödinger (NS) equations and the Korteweg de Vries (KdV) equations are shown to be symmetry reductions of the anti-self-dual Yang—Mills (ASDYM) equation with real forms of SL (2, ) as gauge groups.

We obtain a twistor correspondence between solutions of the NS and KdV equations and certain holomorphic vector bundles with a symmetry on the total space of the complex line bundle of Chern class two on the Riemann sphere. Remarkably, when the Chern class is increased, the correspondence extends to the NS and KdV hierarchies. If the symmetry condition is dropped we obtain a twistor correspondence for a hierarchy for the Bogomolny equations, which yields the KdV and NS hierarchies when the symmetry is imposed.

The inverse scattering transform is shown to be a coordinate realization of the twistor correspondence. Both the pure solitons and the solitonless cases are treated. The k-soliton solutions arise from the kth “Ward ansatze” in an analogous fashion to the monopole solutions.     

A Three-component Extension of the Camassa–Holm Hierarchy

We introduce a bi-Hamiltonian hierarchy on the loop-algebra of endowed with a suitable Poisson pair. It gives rise to the usual Camassa–Holm (CH) hierarchy by means of a bi-Hamiltonian reduction, and its first nontrivial flow provides a three-component extension of the CH equation.     

Lump solutions of the 2D Toda equation

In this research, the lump solution, which is rationally localized and decays along the directions of space variables, of a 2D Toda equation is studied. The effective method of constructing the lump solutions of this 2D Toda equation is derived, and the constraint conditions that make the lump solutions analytical and positive are obtained as well. Finally, three classes of lump solutions are constructed, 3D plots, density plots, and contour plots are given to illustrate this proposed method.