Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Relativistic Toda-type systems
We establish the pluri-Lagrangian structure for families of Bäcklund transformations of relativistic Toda-type systems. The key idea is a novel embedding of these discrete-time (one-dimensional) systems into certain two-dimensional (2D) pluri-Lagrangian lattice systems. This embedding allows us to i...
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Published in | Journal of physics. A, Mathematical and theoretical Vol. 48; no. 8; pp. 85203 - 28 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
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27.02.2015
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Abstract | We establish the pluri-Lagrangian structure for families of Bäcklund transformations of relativistic Toda-type systems. The key idea is a novel embedding of these discrete-time (one-dimensional) systems into certain two-dimensional (2D) pluri-Lagrangian lattice systems. This embedding allows us to identify the corner equations (which are the main building blocks of the multi-time Euler-Lagrange equations) with local superposition formulae for Bäcklund transformations. These superposition formulae, in turn, are key ingredients necessary to understand and to prove commutativity of the multi-valued Bäcklund transformations. Furthermore, we discover a 2D generalization of the spectrality property known for families of Bäcklund transformations. This result produces a family of local conservations laws for 2D pluri-Lagrangian lattice systems, with densities being derivatives of the discrete 2-form with respect to the Bäcklund (spectral) parameter. Thus, a relation of the pluri-Lagrangian structure with more traditional integrability notions is established. |
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AbstractList | We establish the pluri-Lagrangian structure for families of Backlund transformations of relativistic Toda-type systems. The key idea is a novel embedding of these discrete-time (one-dimensional) systems into certain twodimensional (2D) pluri-Lagrangian lattice systems. This embedding allows us to identify the corner equations (which are the main building blocks of the multi-time Euler-Lagrange equations) with local superposition formulae for Backlund transformations. These superposition formulae, in turn, are key ingredients necessary to understand and to prove commutativity of the multivalued Backlund transformations. Furthermore, we discover a 2D generalization of the spectrality property known for families of Backlund transformations. This result produces a family of local conservations laws for 2D pluri-Lagrangian lattice systems, with densities being derivatives of the discrete 2-form with respect to the Backlund (spectral) parameter. Thus, a relation of the pluri-Lagrangian structure with more traditional integrability notions is established. We establish the pluri-Lagrangian structure for families of Bäcklund transformations of relativistic Toda-type systems. The key idea is a novel embedding of these discrete-time (one-dimensional) systems into certain two-dimensional (2D) pluri-Lagrangian lattice systems. This embedding allows us to identify the corner equations (which are the main building blocks of the multi-time Euler-Lagrange equations) with local superposition formulae for Bäcklund transformations. These superposition formulae, in turn, are key ingredients necessary to understand and to prove commutativity of the multi-valued Bäcklund transformations. Furthermore, we discover a 2D generalization of the spectrality property known for families of Bäcklund transformations. This result produces a family of local conservations laws for 2D pluri-Lagrangian lattice systems, with densities being derivatives of the discrete 2-form with respect to the Bäcklund (spectral) parameter. Thus, a relation of the pluri-Lagrangian structure with more traditional integrability notions is established. |
Author | Petrera, Matteo Suris, Yuri B Boll, Raphael |
Author_xml | – sequence: 1 givenname: Raphael surname: Boll fullname: Boll, Raphael email: boll@math.tu-berlin.de organization: Technische Universität Berlin Institut für Mathematik, MA 7-2, Straße des 17. Juni 136, D-10623 Berlin, Germany – sequence: 2 givenname: Matteo surname: Petrera fullname: Petrera, Matteo email: petrera@math.tu-berlin.de organization: Technische Universität Berlin Institut für Mathematik, MA 7-2, Straße des 17. Juni 136, D-10623 Berlin, Germany – sequence: 3 givenname: Yuri B surname: Suris fullname: Suris, Yuri B email: suris@math.tu-berlin.de organization: Technische Universität Berlin Institut für Mathematik, MA 7-2, Straße des 17. Juni 136, D-10623 Berlin, Germany |
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Cites_doi | 10.1016/S0375-9601(02)00287-6 10.1007/BF02634155 10.1090/gsm/098 10.1080/00036810903397495 10.1088/1751-8113/45/11/115201 10.1088/1751-8113/44/36/365203 10.3934/jgm.2013.5.365 10.1007/BF02467062 10.1155/S1073792802110075 10.1007/s11232-012-0138-y 10.1112/plms/s3-61.3.546 10.1007/978-3-0348-8016-9 10.1155/S107379280413273X 10.1088/1751-8113/43/7/072003 10.1007/BF02634053 10.1088/0305-4470/31/9/012 10.1098/rspa.2013.0550 10.1007/s00220-002-0762-8 10.1088/1751-8113/46/27/275204 10.1007/s11005-010-0381-9 10.1007/s00220-014-2240-5 10.2307/2946637 |
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References | 22 24 25 Boll R (15) 2012; 45 Rudin W (23) 1969 Boll R (11) 2013; 46 Bobenko A I (8) 2008; 98 10 12 14 Lobb S (19) 2010; 43 Lobb S (18) 2009; 42 16 Yoo-Kong S (26) 2011; 44 Adler V E (2) 2003; 233 1 Boll R (13) 2014 3 Lobb S (20) 2009; 42 4 5 6 7 9 Kuznetsov V B (17) 1998; 31 21 |
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Snippet | We establish the pluri-Lagrangian structure for families of Bäcklund transformations of relativistic Toda-type systems. The key idea is a novel embedding of... We establish the pluri-Lagrangian structure for families of Backlund transformations of relativistic Toda-type systems. The key idea is a novel embedding of... |
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SubjectTerms | Density Derivatives Euler-Lagrange equations integrable systems Lagrangian mechanics Lattices Mathematical analysis relativistic Toda systems Spectra Transformations Transformations (mathematics) Two dimensional |
Title | Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Relativistic Toda-type systems |
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