On characteristic integrals of Toda field theories

Nie, Zhaogang

doi:10.1080/14029251.2014.894724

Cited by 11 publications

(4 citation statements)

References 21 publications

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“…These systems are also known in literature as generalized Toda lattices corresponding to the Cartan matrices of simple Lie algebras. Explicit formulas for characteristic integrals in terms of wronskians were found in [21] for Toda lattices of series A-D. Another approach that allows to obtain generating function for characteristic integrals was developed in [22] for the A-series Toda lattices and in [19] for lattices of the series A-C.…”

Section: Darboux Integrability and Characteristic Algebrasmentioning

confidence: 99%

Integral preserving discretization of 2D Toda lattices

Smirnov

2023

J. Phys. A: Math. Theor.

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There are different methods of discretizing integrable systems. We consider semi-discrete analog of two-dimensional Toda lattices associated to the Cartan matrices of simple Lie algebras that was proposed by Habibullin in 2011. This discretization is based on the notion of Darboux integrability. Generalized Toda lattices are known to be Darboux integrable in the continuous case (that is, they admit complete families of characteristic integrals in both directions). We prove that semi-discrete analogs of Toda lattices associated to the Cartan matrices of all simple Lie algebras are Darboux integrable. By examining the properties of Habibullin's discretization we show that if a function is a characteristic integral for a generalized Toda lattice in the continuous case, then the same function is a characteristic integral in the semi-discrete case as well. We consider characteristic algebras of such integral-preserving discretizations of Toda lattices to prove the existence of complete families of characteristic integrals in the second direction.

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Section: Darboux Integrability and Characteristic Algebrasmentioning

confidence: 99%

Integral preserving discretization of 2D Toda lattices

Smirnov

2023

J. Phys. A: Math. Theor.

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“…It is a standard fact (cf. [23,30]) that p k 's are all apparent singularities of (1.11) if it comes from a solution (U, V) of (1.9); see Section 2 for a brief explanation. Thanks to this fact, we will prove Theorem 1.4 in Section 5 by showing that at least one of ω k 2 's can not be apparent of (1.11) under the assumption of Theorem 1.4.…”

Section: Have Only Finitely Many Solutions?mentioning

confidence: 99%

“…Let (U, V) be a solution of the Toda system (1.9), as in [23,30] we consider the following linear differential operator…”

Section: The Associated Linear Odementioning

confidence: 99%

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On number and evenness of solutions of the $SU(3)$ Toda system on flat tori with non-critical parameters

Chen,

Lin

2021

Preprint

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We study the SU(3) Toda system with singular sources, where E τ := C/(Z + Zτ) with Im τ > 0 is a flat torus, δ p k is the Dirac measure at p k , and n i,k ∈ Z ≥0 satisfy ∑ k n 1,k ≡ ∑ k n 2,k mod 3. This is known as the non-critical case and it follows from a general existence result of [3] that solutions always exist. In this paper we prove that (i) The system has at mostWe have several examples to indicate that this upper bound should be sharp. Our proof presents a nice combination of the apriori estimates from analysis and the classical Bézout theorem from algebraic geometry.(ii) For m = 0 and p 0 = 0, the system has even solutions if and only if at least one of {n 1,0 , n 2,0 } is even. Furthermore, if n 1,0 is odd, n 2,0 is even and n 1,0 < n 2,0 , then except for finitely many τ's modulo SL(2, Z) action, the system has exactly n 1,0 +1 2 even solutions. Differently from [3], our proofs are based on the integrability of the Toda system, and also imply a general non-existence result for even solutions of the Toda system with four singular sources.

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Classification of solutions to Toda systems of types C and B with singular sources

Nie

2016

Calc. Var.

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Abstract. In this paper, the classification in [LWY12] of solutions to Toda systems of type A with singular sources is generalized to Toda systems of types C and B. Like in the A case, the solution space is shown to be parametrized by the abelian subgroup and a subgroup of the unipotent subgroup in the Iwasawa decomposition of the corresponding complex simple Lie group. The method is by studying the Toda systems of types C and B as reductions of Toda systems of type A with symmetries. The theories of Toda systems as integrable systems as developed in [Lez80, LS92, Nie12, Nie14], in particular the W -symmetries and the iterated integral solutions, play essential roles in this work, together with certain characterizing properties of minors of symplectic and orthogonal matrices.

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On characteristic integrals of Toda field theories

Abstract: Characteristic integrals of Toda field theories associated to general simple Lie algebras are constructed using systematic techniques, and complete mathematical proofs are provided. Plenty of examples illustrating the results are presented in explicit forms.

Cited by 11 publications

References 21 publications

Integral preserving discretization of 2D Toda lattices

Integral preserving discretization of 2D Toda lattices

On number and evenness of solutions of the $SU(3)$ Toda system on flat tori with non-critical parameters

Classification of solutions to Toda systems of types C and B with singular sources

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