Recursion operators and hierarchies of \(\text{mKdV}\) equations related to the Kac–Moody algebras \(D_4^{(1)}\), \(D_4^{(2)}\), and \(D_4^{(3)}\)

Abstract

We construct three nonequivalent gradings in the algebra \(D_4\simeq so(8)\). The first is the standard grading obtained with the Coxeter automorphism \(C_1=S_{\alpha_2}S_{\alpha_1}S_{\alpha_3}S_{\alpha_4}\) using its dihedral realization. In the second, we use \(C_2=C_1R\), where \(R\) is the mirror automorphism. The third is \(C_3=S_{\alpha_2}S_{\alpha_1}T\), where \(T\) is the external automorphism of order 3. For each of these gradings, we construct a basis in the corresponding linear subspaces \( \mathfrak{g} ^{(k)}\), the orbits of the Coxeter automorphisms, and the related Lax pairs generating the corresponding modified Korteweg–de Vries (mKdV) hierarchies. We find compact expressions for each of the hierarchies in terms of recursion operators. Finally, we write the first nontrivial mKdV equations and their Hamiltonians in explicit form. For \(D_4^{(1)}\), these are in fact two mKdV systems because the exponent 3 has the multiplicity two in this case. Each of these mKdV systems consists of four equations of third order in \( \partial _x\). For \(D_4^{(2)}\), we have a system of three equations of third order in \( \partial _x\). For \(D_4^{(3)}\), we have a system of two equations of fifth order in \( \partial _x\).

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Figure 1.

Notes

  1. 1.

    Not every Coxeter automorphism can be realized in this way. For example, Coxeter automorphisms of the form \(C(X)=cF(X) c^{-1}\), where \(c\) is a diagonal matrix and \(F\) is some properly chosen outer automorphism, can never be constructed from a Weyl group element.

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Acknowledgments

One of the authors (V. S. G.) is grateful to Professors A. V. Mikhailov and V. S. Novikov for the useful discussions and comments. Two of the authors (V. S. G. and A. A. S.) are grateful to the organizing committee of the IXth International Conference “Solitons, Collapses, and Turbulence: Achievements, Developments, and Prospects,” held in Yaroslavl, Russia, 5–9 August 2019, and dedicated to Vladimir Zakharov’s 80th birthday, for the support and hospitality.

Funding

This research was supported by the Bulgarian Science Foundation (Grant No. NTS-Russia 02/101 from 23 October 2017) and the Russian Foundation for Basic Research (Grant No. 18-51-18007).

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Gerdjikov, V.S., Stefanov, A.A., Iliev, I.D. et al. Recursion operators and hierarchies of \(\text{mKdV}\) equations related to the Kac–Moody algebras \(D_4^{(1)}\), \(D_4^{(2)}\), and \(D_4^{(3)}\). Theor Math Phys 204, 1110–1129 (2020). https://doi.org/10.1134/S0040577920090020

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Keywords

  • mKdV equation
  • recursion operator
  • Kac–Moody algebra
  • hierarchy of integrable equations