Multi-Hamiltonian structure for the finite defocusing Ablowitz-Ladik equation
We study the Poisson structure associated to the defocusing Ablowitz‐Ladik equation from a functional‐analytical point of view by reexpressing the Poisson bracket in terms of the associated Carathéodory function. Using this expression, we are able to introduce a family of compatible Poisson brackets...
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| Veröffentlicht in: | Communications on pure and applied mathematics 2009-02, Vol.62 (2), p.147-182 |
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| container_title | Communications on pure and applied mathematics |
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| creator | Gekhtman, Michael Nenciu, Irina |
| description | We study the Poisson structure associated to the defocusing Ablowitz‐Ladik equation from a functional‐analytical point of view by reexpressing the Poisson bracket in terms of the associated Carathéodory function. Using this expression, we are able to introduce a family of compatible Poisson brackets that form a multi‐Hamiltonian structure for the Ablowitz‐Ladik equation. Furthermore, we show using some of these new Poisson brackets that the Geronimus relations between orthogonal polynomials on the unit circle and those on the interval define an algebraic and symplectic mapping between the Ablowitz‐Ladik and Toda hierarchies. © 2008 Wiley Periodicals, Inc. |
| doi_str_mv | 10.1002/cpa.20255 |
| format | Article |
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Using this expression, we are able to introduce a family of compatible Poisson brackets that form a multi‐Hamiltonian structure for the Ablowitz‐Ladik equation. Furthermore, we show using some of these new Poisson brackets that the Geronimus relations between orthogonal polynomials on the unit circle and those on the interval define an algebraic and symplectic mapping between the Ablowitz‐Ladik and Toda hierarchies. © 2008 Wiley Periodicals, Inc.</description><identifier>ISSN: 0010-3640</identifier><identifier>EISSN: 1097-0312</identifier><identifier>DOI: 10.1002/cpa.20255</identifier><identifier>CODEN: CPAMAT</identifier><language>eng</language><publisher>Hoboken: Wiley Subscription Services, Inc., A Wiley Company</publisher><subject>Algebra ; Applied mathematics ; Difference and functional equations, recurrence relations ; Exact sciences and technology ; Finite differences and functional equations ; Fourier analysis ; Mathematical analysis ; Mathematical functions ; Mathematics ; Nonlinear equations ; Numerical analysis ; Numerical analysis. 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Pure Appl. Math</addtitle><description>We study the Poisson structure associated to the defocusing Ablowitz‐Ladik equation from a functional‐analytical point of view by reexpressing the Poisson bracket in terms of the associated Carathéodory function. Using this expression, we are able to introduce a family of compatible Poisson brackets that form a multi‐Hamiltonian structure for the Ablowitz‐Ladik equation. Furthermore, we show using some of these new Poisson brackets that the Geronimus relations between orthogonal polynomials on the unit circle and those on the interval define an algebraic and symplectic mapping between the Ablowitz‐Ladik and Toda hierarchies. © 2008 Wiley Periodicals, Inc.</description><subject>Algebra</subject><subject>Applied mathematics</subject><subject>Difference and functional equations, recurrence relations</subject><subject>Exact sciences and technology</subject><subject>Finite differences and functional equations</subject><subject>Fourier analysis</subject><subject>Mathematical analysis</subject><subject>Mathematical functions</subject><subject>Mathematics</subject><subject>Nonlinear equations</subject><subject>Numerical analysis</subject><subject>Numerical analysis. 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Scientific computation</topic><topic>Poisson distribution</topic><topic>Polynomials</topic><topic>Sciences and techniques of general use</topic><topic>Special functions</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Gekhtman, Michael</creatorcontrib><creatorcontrib>Nenciu, Irina</creatorcontrib><collection>Istex</collection><collection>Pascal-Francis</collection><collection>CrossRef</collection><collection>ProQuest Computer Science Collection</collection><jtitle>Communications on pure and applied mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Gekhtman, Michael</au><au>Nenciu, Irina</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Multi-Hamiltonian structure for the finite defocusing Ablowitz-Ladik equation</atitle><jtitle>Communications on pure and applied mathematics</jtitle><addtitle>Comm. Pure Appl. 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| subjects | Algebra Applied mathematics Difference and functional equations, recurrence relations Exact sciences and technology Finite differences and functional equations Fourier analysis Mathematical analysis Mathematical functions Mathematics Nonlinear equations Numerical analysis Numerical analysis. Scientific computation Poisson distribution Polynomials Sciences and techniques of general use Special functions |
| title | Multi-Hamiltonian structure for the finite defocusing Ablowitz-Ladik equation |
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