Article: Variational identities and applications to Hamiltonian structures of soliton equations Authors: Ma, WX Journal: NONLINEAR ANAL-THEOR METH APP, Volume: 71, Issue: 12, Page: E1716-E1726, Year: DEC 15 2009 Addresses: * Zhejiang Normal Univ, Dept Math, Jinhua 321004, Peoples R China. * Zhejiang Normal Univ, Dept Math, Jinhua 321004, Peoples R China.

Wen-Xiu Ma talks with ScienceWatch.com and answers a few questions about this month's Fast Breaking Paper paper in the field of Mathematics.

Why do you think your paper is highly cited?

This introductory report, initially a one-hour invited talk given at the 5th World Congress of Nonlinear Analysts, involves a number of exciting new developments in the Hamiltonian theory of integrable couplings that have taken place over the last few years.

Does it describe a new discovery, methodology, or synthesis of knowledge?

"Through extensive research, I successfully developed, with my fellow collaborators, a genuine tool to generate the required Hamiltonian structures from the discovery of variational identities hidden in matrix spectral problems."

The Hamiltonian theory provides a unified description of the nature of integrability that dynamical systems possess, and it allows us to link together diverse integrable characteristics such as conservation laws and symmetries. The paper reviews the state of the art in the modern Hamiltonian theory and soliton theory and explains how Lie algebras yield Hamiltonian structures of soliton equations, particularly integrable couplings, arising as compatibility conditions of matrix spectral problems.

Would you summarize the significance of your paper in layman's terms?

This paper presents a thorough overview of uses of variational identities in the Hamiltonian theory of soliton equations. The algebraic study of integrable couplings advances the complete classification of integrable equations from a Lie algebraic point of view. The paper also outlines a few new ideas for possible progress and specific recommendations for future research directions in soliton theory.

How did you become involved in this research, and how would you describe the particular challenges, setbacks, and successes that you've encountered along the way?

I have been working in this field since I was in graduate school. While studying the Hamiltonian theory of equations in classical mechanics and KdV-type equations in soliton theory, I continually faced the fundamental problem of how to construct Hamiltonian structures for given equations to explore integrable characteristics that the equations carry.

Through extensive research, I successfully developed, with my fellow collaborators, a genuine tool to generate the required Hamiltonian structures from the discovery of variational identities hidden in matrix spectral problems. The resulting theory unifies soliton equations associated with two distinct categories of Lie algebras: semisimple and non-semisimple Lie algebras.

Where do you see your research leading in the future?

Variational identities provide ways to search for Hamiltonian structures of soliton equations. But it is still unclear how to classify all integrable couplings full of block structures observed. In particular, due to complications of bi-integrable couplings and tri-integrable couplings, there is even a lack of information from variational identities about their Hamiltonian structures. I will certainly continue to study in this field, together with my colleagues and students.

Wen-Xiu Ma
Professor of Mathematics
Department of Mathematics and Statistics
University of South Florida
Tampa, FL, USA

KEYWORDS: VARIATIONAL IDENTITIES, HAMILTONIAN STRUCTURES, ZERO CURVATURE EQUATIONS, INTEGRABLE COUPLINGS, LIE ALGEBRAS, SEMIDIRECT SUMS, EVOLUTION EQUATIONS, SPECTRAL PROBLEMS, TRACE IDENTITY, SYSTEMS, HIERARCHY, REPRESENTATIONS.