The $\text{m}$KdV-type equations related to $A_5^{(1)}$ and $A_5^{(2)}$ Kac–Moody algebras

V. S. Gerdjikov   ; D. M. Mladenov   ; A. A. Stefanov   ; S. K. Varbev

Teoretičeskaâ i matematičeskaâ fizika, Tome 207 (2021) no. 2, pp. 237-260

Résumé

We explain the details of the algebraic constructions on nontrivial examples of the mKdV equations related to the $A_5^{(1)}$ and $A_5^{(2)}$ Kac–Moody algebras. Several types of recursion operators appear naturally in formulating the equations and their Hamiltonian structures. We next introduce the resolvent of the Lax operator and demonstrate that it generates the hierarchy of the Lax representations and also the hierarchy of conservation laws of these equations.

Détail

Keywords: mKdV equation, recursion operator, Kac–Moody algebra, hierarchy of integrable equations.

Bibliographie

[1] M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, “The inverse scattering transform–Fourier analysis for nonlinear problems”, Stud. Appl. Math., 53:4 (1974), 249–315 | DOI | MR

[2] D. J. Kaup, “Closure of the squared Zakharov–Shabat eigenstates”, J. Math. Anal. Appl., 54:3 (1976), 849–864 | DOI | MR

[3] V. S. Gerdjikov, E. Kh. Khristov, “On the evolution equations, solvable through the inverse scattering method. I. Spectral theory”, Bulgar. J. Phys., 7:1 (1980), 28–41 ; “On the evolution equations, solvable through the inverse scattering method. II. Hamiltonian structure and Bäcklund transformations”, 7:2, 119–133 | MR | MR

[4] V. S. Gerdjikov, P. P. Kulish, “The generating operator for the $n \times n$ linear system”, Phys. D, 3:3 (1981), 549–564 | DOI | MR

[5] V. S. Gerdjikov, “Algebraic and analytic aspects of $N$-wave type equations”, The Legacy of the Inverse Scattering Transform in Applied Mathematics (South Hadley, MA, USA, June 17–21, 2001), Contemporary Mathematics, 301, eds. J. Bona, R. Choudhury, D. Kaup, AMS, Providence, RI, 2002, 35–68 | DOI | MR | Zbl

[6] V. S. Gerdjikov, A. B. Yanovski, “CBC systems with Mikhailov reductions by Coxeter automorphism: I. Spectral theory of the recursion operators”, Stud. Appl. Math., 134:2 (2015), 145–180 | DOI | MR

[7] A. V. Mikhailov, “Ob integriruemosti dvumernogo obobscheniya tsepochki Toda”, Pisma v ZhETF, 30:7 (1979), 443–448

[8] A. V. Mikhailov, “The reduction problem and the inverse scattering method”, Phys. D, 3:1–2 (1981), 73–117 | DOI

[9] V. S. Gerdjikov, “$Z_N$-reductions and new integrable versions of derivative nonlinear Schrödinger equations”, Nonlinear Evolution Equations: Integrability and Spectral Methods, eds. A. P. Fordy, A. Degasperis, M. Lakshmanan, Manchester Univ. Press, Manchester, UK, 1981, 367–379

[10] V. S. Gerdjikov, “Derivative nonlinear Schrödinger equations with ${\mathbb Z}_N$ and $\mathbb D_N $-reductions”, Romanian J. Phys., 58:5–6 (2013), 573–582 | MR

[11] D. J. Kaup, A. C. Newell, “Soliton equations, singular dispersion relations and moving eigenvalues”, Adv. Math., 31:1 (1979), 67–100 | DOI | MR

[12] V. G. Drinfeld, V. V. Sokolov, “Uravneniya tipa Kortevega–de Friza i prostye algebry Li”, Dokl. AN SSSR, 258:1 (1981), 11–16 | MR | Zbl

[13] V. G. Drinfeld, V. V. Sokolov, “Algebry Li i uravneniya tipa Kortevega–de Friza”, Itogi nauki i tekhn. Ser. Sovrem. probl. matem. Nov. dostizh., 24 (1984), 81–180 | DOI | MR | Zbl

[14] V. E. Zakharov, A. V. Mikhailov, “Relyativistski-invariantnye dvumernye modeli teorii polya, integriruemye metodom obratnoi zadachi”, ZhETF, 74:6 (1978), 1953–1973 | MR

[15] F. Calogero, F. Degasperis, Spectral Transform and Solitons, v. 1, Lecture Notes in Computer Science, 144, Tools to Solve and Investigate Nonlinear Evolution Equations, North Holland, Amsterdam, New York, 1982 | MR

[16] L. A. Takhtadzhan, D. V. Fadeev, Gamiltonov podkhod v teorii solitonov, Nauka, M., 1986 | DOI | MR | MR

[17] V. E. Zakharov, S. V. Manakov, S. P. Novikov, L. P. Pitaevskii, Teoriya solitonov. Metod obratnoi zadachi, Nauka, M., 1980 | MR | MR | Zbl

[18] V. S. Gerdjikov, D. M. Mladenov, A. A. Stefanov, S. K. Varbev, “Integrable equations and recursion operators related to the affine Lie algebras $A^{(1)}_r$”, J. Math. Phys., 56:5 (2015), 052702, 18 pp., arXiv: 1411.0273 | DOI | MR

[19] V. S. Gerdjikov, D. M. Mladenov, A. A. Stefanov, S. K. Varbev, “MKdV-type of equations related to $B_2^{(1)}$ and $A_4^{(2)}$ algebra”, Nonlinear Mathematical Physics and Natural Hazards, Springer Proceedings in Physics, 163, eds. B. Aneva, M. Kouteva-Guentcheva, Springer, Cham, 2015, 59–69 | DOI

[20] V. S. Gerdjikov, D. M. Mladenov, A. A. Stefanov, S. K. Varbev, “MKdV-type of equations related to $\mathfrak{sl}(N,\mathbb{C})$ algebra”, Mathematics in Industry, ed. A. Slavova, Cambridge Scholars Publ., Cambridge, 2014, 335–344

[21] V. S. Gerdjikov, D. M. Mladenov, A. A. Stefanov, S. K. Varbev, “On a one-parameter family of mKdV equations related to the $\mathfrak{so}(8)$ Lie algebra”, Mathematics in Industry, ed. A. Slavova, Cambridge Scholars Publ., Cambridge, 2014, 345–354

[22] V. S. Gerdjikov, D. M. Mladenov, A. A. Stefanov, S. K. Varbev, “On mKdV equations related to the affine Kac–Moody algebra $A^{(2)}_5$”, J. Geom. Symmetry Phys., 39 (2015), 17–31 | DOI | MR | Zbl

[23] V. S. Gerdzhikov, A. A. Stefanov, I. D. Iliev, G. P. Boyadzhiev, A. O. Smirnov, V. B. Matveev, M. V. Pavlov, “Operatory rekursii i ierarkhii modifitsirovannykh uravnenii Kortevega–de Friza, svyazannye s algebrami Katsa–Mudi $D_4^{(1)}$, $D_4^{(2)}$ i $D_4^{(3)}$”, TMF, 204:3 (2020), 332–354 | DOI | DOI | MR

[24] V. S. Gerdjikov, G. Vilasi, A. B. Yanovski, Integrable Hamiltonian Hierarchies. Spectral and Geometric Methods, Lecture Notes in Physics, 748, Springer, Berlin, Heidelberg, New York, 2008 | DOI | MR

[25] V. S. Gerdjikov, A. B. Yanovski, “On soliton equations with $\mathbb{Z}_{h}$ and $\mathbb{D}_{h}$ reductions: conservation laws and generating operators”, J. Geom. Symmetry Phys., 31 (2013), 57–92 | DOI | MR

[26] S. Helgasson, Differential Geometry, Lie Groups and Symmetric Spaces, Graduate Studies in Mathematics, 34, AMS, Providence, RI, 2012 | MR

[27] V. Kats, Beskonechnomernye algebry Li, Mir, M., 1993 | MR | Zbl

[28] R. Carter, Lie Algebras of Finite and Affine Type, Cambridge Studies in Advanced Mathematics, 96, Cambridge Univ. Press, Cambridge, 2005 | DOI | MR

[29] E. B. Dynkin, “Struktura poluprostykh algebr Li”, UMN, 2:4(20) (1947), 59–127 | MR

[30] V. S. Gerdjikov, A. B.Yanovski, “Completeness of the eigenfunctions for the Caudrey–Beals–Coifman system”, J. Math. Phys., 35:7 (1994), 3687–3725 | DOI | MR

[31] V. S. Gerdjikov, “Generalised Fourier transforms for the soliton equations. Gauge-covariant formulation”, Inverse Problems, 2:1 (1986), 51–74 | DOI | MR

[32] A. B. Shabat, “Obratnaya zadacha rasseyaniya”, Differents. uravneniya, 15:10 (1979), 1824–1834 | MR | Zbl

[33] R. Beals, R. R. Coifman, “Inverse scattering and evolution equations”, Commun. Pure Appl. Math., 38:1 (1985), 29–42 | DOI | MR

[34] A. Yanovski, “Recursion operators and expansions over adjoint solutions for the Caudrey–Beals–Coifman system with $\mathbb{Z}_p$ reductions of Mikhailov type”, J. Geom. Symmetry Phys., 30 (2013), 105–120 | DOI | MR

[35] V. E. Zakharov, A. B. Shabat, “Skhema integrirovaniya nelineinykh uravnenii matematicheskoi fiziki metodom obratnoi zadachi rasseyaniya. I”, Funkts. analiz i ego pril., 8:3 (1974), 43–53 ; “Интегрирование нелинейных уравнений математической физики методом обратной задачи рассеяния. II”, 13:3 (1979), 13–22 | DOI | MR | Zbl | DOI | MR | Zbl

[36] L. A. Dickey, Soliton equations and Hamiltonian systems, Advanced Series in Mathematical Physics, 12, World Sci., Singapore, 1991 | DOI | MR

[37] A. De Sole, M. Jibladze, V. G. Kac, D. Valeri, Integrability of classical affine $W$-algebras, arXiv: 2007.01244

[38] A. De Sole, V. G. Kac, D. Valeri, “Classical $\mathcal W$-algebras and generalized Drinfeld–Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras”, Commun. Math. Phys., 323:2 (2013), 663–711, arXiv: ; “Classical affine $\mathcal W$-algebras for $\mathfrak{gl}(N)$ and associated integrable Hamiltonian hierarchies”, 348:1 (2016), 265–319, arXiv: 1207.62861509.06878 | DOI | MR | DOI | MR

[39] N. J. Burroughs, M. F. de Groot, T. J. Hollowood, J. L. Miramontes, “Generalized Drinfel'd–Sokolov hierarchies. II. The Hamiltonian structures”, Commun. Math. Phys., 153:1 (1993), 187–215 | DOI | MR

[40] M. F. de Groot, T. J. Hollowood, J. L. Miramontes, “Generalized Drinfel'd–Sokolov hierarchies”, Commun. Math. Phys., 145:1 (1992), 57–84 | DOI | MR

[41] V. S. Gerdjikov, R. Ivanov, A. Stefanov, “Riemann–Hilbert problem, integrability and reductions,”, J. Geom. Mech., 11:2 (2019), 167–185, arXiv: 1902.10276 | DOI | MR

[42] A. V. Mikhailov, V. V. Sokolov, A. B. Shabat, “The symmetry approach to classification of integrable equations”, What is Integrability?, ed. V. E. Zakharov, Springer, Berlin, 1991, 113–184 | DOI | MR

[43] A. V. Mikhailov, J. P. Wang, V. S. Novikov, “Partially integrable nonlinear equations with one high symmetry”, J. Phys. A: Math. Gen., 38:20 (2005), L337–L341 | DOI | MR

[44] A. V. Mikhailov, V. S. Novikov, J. P. Wang, “Symbolic representation and classification of integrable systems”, Algebraic Theory of Differential Equations, London Mathematical Society Lecture Note Series, 357, eds. M. A. H. MacCallum, A. Mikhailov, Cambridge Univ. Press, Cambridge, 2008, 156–216 | DOI | MR

[45] G. Zhao, Integrability of two-component systems of partial differential equations, PhD thesis, Loughborough University, Loughborough, UK, unpublished