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Definition of integrable. First Known Use of integrable circa , in the meaning defined above. Learn More about integrable. Resources for integrable Time Traveler! An explanation of this surprising non-existence result is as follows. Consider a three-point Lagrangian two-form 4. Then also the corner equations become regular perturbations of E i and E ij : the first turns into. These equations can be written as quotients of three-leg forms of multi-affine quad-equations:.
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Equation 5. On the other hand, the three-leg form 5. In this paper, we formulated the notion of consistent pluri-Lagrangian systems, which, in our view, should be treated as an answer to the question posed in the title. However, much research is still to be done to justify this proposal.
In particular, the relation to more common notions of integrability has only been demonstrated for one-dimensional systems [ 11 ]. We expect similar results for two-dimensional systems, as well. Details will be given in a forthcoming publication [ 28 ]. The new definition seems to be capable of being put at the basis of the classification task. Classifying all consistent systems of corner equations within a certain ansatz for the discrete two-form is therefore an extremely important problem.
It would also be very useful to elaborate on the precise relations of the new notion with its predecessors mentioned in the introduction theory of pluriharmonic functions; Z -invariant models of statistical mechanics and their quasiclassical limit; variational symmetries.
Proc Math Phys Eng Sci. PMID: Juni , Berlin, Germany.
Received Aug 15; Accepted Nov All rights reserved. Abstract We propose a notion of a pluri-Lagrangian problem, which should be understood as an analogue of multi-dimensional consistency for variational systems. Keywords: discrete integrable systems, Euler—Lagrange equations, variational systems, multi-dimensional consistency, discrete forms.
Introduction In the last decade, a new understanding of integrability of discrete systems as their multi-dimensional consistency has been a major breakthrough [ 1 , 2 ]. Open in a separate window. Figure 1.
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Local move of a quad-surface involving one three-dimensional cube. By definition, a pluriharmonic function of several complex variables minimizes the Dirichlet functional along any holomorphic curve in its domain. Differential equations governing pluriharmonic functions and maps are heavily overdetermined. Therefore, it is not surprising that they belong to the theory of integrable systems. This concept is based on invariance of the partition function of solvable models under elementary local transformations of the underlying planar graph.
It is well known e. On the other hand, the classical mechanical analogue of the partition function is the action functional. This makes the relation of Z -invariance to the concept of closedness of the Lagrangian two-form rather natural, at least at the heuristic level. Moreover, this relation has been made mathematically precise for a number of models, through the quasiclassical limit, in the work of Bazhanov et al.
This is elucidated in Yu [ 23 ]. Motivation The motivation for this study comes from the desire to answer the following two long-standing questions. Figure 2. Pluri-Lagrangian systems Definition 3. Definition 3. Figure 3. Lemma 3. Figure 4. Remark 3. Theorem 3. System of corner equations for a three-point two-form corresponding to integrable quad-equations The main class of examples we consider in this paper is characterized by the following ansatz for the discrete two-form :.
Figure 5. Figure 6. Figure 7. Theorem 4. Proposition 4. H1, discrete KdV equation :. H3, Hirota equation :. Corollary 4. System of corner equations for a three-point two-form not corresponding to quad-equations In this section, we show that the general three-point ansatz 4.
Theorem 5. Proposition 5. Conclusion In this paper, we formulated the notion of consistent pluri-Lagrangian systems, which, in our view, should be treated as an answer to the question posed in the title. Footnotes 1 A similar claim, along with a definition similar to our definition 3. References 1. Integrable systems on quad-graphs. Notices , — doi Nijhoff FW. Lax pair for the Adler lattice Krichever—Novikov system.
A , 49—58 doi Classification of integrable equations on quad-graphs. The consistency approach. Lobb S, Nijhoff FW. Lagrangian multiforms and multidimensional consistency. A 42 , doi On the Lagrangian structure of integrable quad-equations. Boll R, Suris YuB. On the Lagrangian structure of 3D consistent systems of asymmetric quad-equations.
A 45 , doi On the Lagrangian formulation of multidimensionally consistent systems. A , — doi A 43 , doi Lagrangian multiform structure for the lattice KP system. Discrete-time Calogero—Moser system and Lagrangian 1-form structure. A 44 , doi Suris YuB. Variational formulation of commuting Hamiltonian flows: multi-time Lagrangian 1-forms. A 31 , — doi In [ 12 ], we also noticed that all the flows generated by such symmetries. In [ 19 — 21 ], its recursion operator is constructed and thus an infinite number of generalized symmetries can be constructed showing that the equation is integrable by a spectral transform or linearizable.
In Section 2 , we will show that up to multiplication by a function and redefinition of the parameters there is only one quad graph equation possessing the two-periodic Klein symmetry. We will provide then the appropriate connection formulae. In Section 3 , we will present some concluding remarks. The explicit identification of the coefficients of such equations is given in Table 1.
The reader can refer to Appendix A for the explicit expressions of these equations or to [ 11 ] for a complete derivation following the prescription of [ 6 ]. However, we have not been able to have a complete direct proof in the general two-periodic case due to computational complexities.
In all generality, we have two possible connection formulae 21, B. Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide. Sign In or Create an Account.
Sign In. Advanced Search. Article Navigation. Close mobile search navigation Article Navigation. Volume 2. Article Contents. Appendix A. Appendix B. A two-periodic generalization of the Q V equation G. Oxford Academic. Google Scholar.