A survey of the origins and physical importance of soliton equations

1985 ◽  
Vol 315 (1533) ◽  
pp. 335-365 ◽  
Keyword(s):  
Significant Role ◽  
Integrable Systems ◽  
Applied Mathematics ◽  
Theoretical Physics ◽  
Completely Integrable Systems ◽  
Soliton Equations ◽  
Completely Integrable ◽  
Physical Importance ◽  
The Subject

An introduction to the subject is given in an elementary way for the non-specialist, outlining why many completely integrable systems, although special, play a significant role in wave motions in applied mathematics and theoretical physics.

scholarly journals Dubrovin equations and integrable systems on hyperelliptic curves

2002 ◽  
Vol 91 (1) ◽  
pp. 91
Author(s):  
Fritz Gesztesy ◽  
Helge Holden
Keyword(s):  
Integrable Systems ◽  
Hyperelliptic Curve ◽  
Integrable Hierarchies ◽  
Hyperelliptic Curves ◽  
Completely Integrable Systems ◽  
Trace Formulas ◽  
Soliton Equations ◽  
Completely Integrable ◽  
Arbitrary Genus ◽  
Dubrovin Equations

We introduce the most general version of Dubrovin-type equations for divisors on a hyperelliptic curve $\mathcal K_g$ of arbitrary genus $g\in\boldsymbol N$, and provide a new argument for linearizing the corresponding completely integrable flows. Detailed applications to completely integrable systems, including the KdV, AKNS, Toda, and the combined sine-Gordon and mKdV hierarchies, are made. These investigations uncover a new principle for $1+1$-dimensional integrable soliton equations in the sense that the Dubrovin equations, combined with appropriate trace formulas, encode all hierarchies of soliton equations associated with hyperelliptic curves. In other words, completely integrable hierarchies of soliton equations determine Dubrovin equations and associated trace formulas and, vice versa, Dubrovin-type equations combined with trace formulas permit the construction of hierarchies of soliton equations.

scholarly journals Transverse Hilbert schemes and completely integrable systems

2017 ◽  
Vol 4 (1) ◽  
pp. 263-272 ◽  
Author(s):  
Niccolò Lora Lamia Donin
Keyword(s):  
Integrable Systems ◽  
Hilbert Scheme ◽  
Symplectic Form ◽  
Initial Surface ◽  
Hilbert Schemes ◽  
Completely Integrable Systems ◽  
Completely Integrable ◽  
Symplectic Surface ◽  
Complex Symplectic ◽  
Minimum Polynomial

Abstract In this paper we consider a special class of completely integrable systems that arise as transverse Hilbert schemes of d points of a complex symplectic surface S projecting onto ℂ via a surjective map p which is a submersion outside a discrete subset of S. We explicitly endow the transverse Hilbert scheme Sp[d] with a symplectic form and an endomorphism A of its tangent space with 2-dimensional eigenspaces and such that its characteristic polynomial is the square of its minimum polynomial and show it has the maximal number of commuting Hamiltonians.We then provide the inverse construction, starting from a 2ddimensional holomorphic integrable system W which has an endomorphism A: TW → TW satisfying the above properties and recover our initial surface S with W ≌ Sp[d].

scholarly journals GEORGE KEITH BATCHELOR 8 March 1920–30 March 2000 Founding Editor, Journal of Fluid Mechanics, 1956

2000 ◽  
Vol 421 ◽  
pp. 1-14 ◽  
Author(s):  
HERBERT E. HUPPERT
Keyword(s):  
Fluid Mechanics ◽  
International Organizations ◽  
Solid Mechanics ◽  
Applied Mathematics ◽  
Fluid Flows ◽  
Theoretical Physics ◽  
Founding Editor ◽  
Number Of Factors ◽  
Quantitative Understanding ◽  
The Subject

George Batchelor was one of the giants of fluid mechanics in the second half of the twentieth century. He had a passion for physical and quantitative understanding of fluid flows and a single-minded determination that fluid mechanics should be pursued as a subject in its own right. He once wrote that he ‘spent a lifetime happily within its boundaries’. Six feet tall, thin and youthful in appearance, George's unchanging attire and demeanour contrasted with his ever-evolving scientific insights and contributions. His strongly held and carefully articulated opinions, coupled with his forthright objectivity, shone through everything he undertook.George's pervasive influence sprang from a number of factors. First, he conducted imaginative, ground-breaking research, which was always based on clear physical thinking. Second, he founded a school of fluid mechanics, inspired by his mentor G. I. Taylor, that became part of the world renowned Department of Applied Mathematics and Theoretical Physics (DAMTP) of which he was the Head from its inception in 1959 until he retired from his Professorship in 1983. Third, he established this Journal in 1956 and actively oversaw all its activities for more than forty years, until he relinquished his editorship at the end of 1998. Fourth, he wrote the monumental textbook An Introduction to Fluid Dynamics, which first appeared in 1967, has been translated into four languages and has been relaunched this year, the year of his death. This book, which describes the fundamentals of the subject and discusses many applications, has been closely studied and frequently cited by generations of students and research workers. It has already sold over 45 000 copies. And fifth, but not finally, he helped initiate a number of international organizations (often European), such as the European Mechanics Committee (now Society) and the biennial Polish Fluid Mechanics Meetings, and contributed extensively to the running of IUTAM, the International Union of Theoretical and Applied Mechanics. The aim of all of these associations is to foster fluid (and to some extent solid) mechanics and to encourage the development of the subject.

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