Differential substitutions for non-Abelian equations of KdV type

V. E. Adler

Ufa mathematical journal, Tome 13 (2021) no. 2, pp. 107-114

Résumé

The work is devoted to constructing differential substitutions connecting the non-Abelian KdV equation with other third-order evolution equations. One of the main results is the construction of a non-Abelian analog of the exponential Calogero–Degasperis equation in a rational form. Some generalizations of the Schwarzian KdV equation are also obtained. Equations and differential substitutions under study contain arbitrary non-Abelian parameters. The construction method is based on the auxiliary linear problem for KdV, in which the usual spectral parameter is replaced by a non-Abelian one. The wave function, corresponding to a fixed value of this parameter, also satisfies a certain evolution equation. Passing to the left and right logarithmic derivatives of the wave function leads one to two versions of the modified KdV equation. In addition, a gauge transformation of the original linear problem leads to a linear problem for one of these versions, mKdV-2. After that, the described procedure is repeated, and the resulting evolution equation for the wave function contains already two arbitrary non-Abelian parameters. For the logarithmic derivative, we obtain an analog of the Calogero–Degasperis equation, which is thus a second modification of the KdV equation. Combining the found Miura-type transformations with discrete symmetries makes it possible to obtain chains of Bäcklund transformations for the modified equations.

Détail

Keywords: non-Abelian equation
Mots-clés : Lax pair, Miura transformation.

Bibliographie

[1] M. Wadati, T. Kamijo, “On the extension of inverse scattering method”, Prog. Theor. Phys., 52 (1974), 397–414 | DOI | MR | Zbl

[2] F. Calogero, A. Degasperis, “Nonlinear evolution equations solvable by the inverse spectral transform. II”, Il Nuovo Cim., 39B:1 (1977), 1–54 | MR

[3] L. Martinez Alonso, E. Olmedilla, “Trace identities in the inverse scattering transform method associated with matrix Schrödinger operators”, J. Math. Phys., 23:11 (1982), 2116–2121 | DOI | MR | Zbl

[4] V. A. Marchenko, Nonlinear equations and operator algebras, Reidel, Boston, 1988 | MR | Zbl

[5] V. M. Goncharenko, A. P. Veselov, “Monodromy of the matrix Schrödinger equations and Darboux transformations”, J. Phys. A, 31:23 (1998), 5315–5326 | DOI | MR | Zbl

[6] A. A. Suzko, “Intertwining technique for the matrix Schrödinger equation”, Phys. Lett. A, 335:2–3 (2005), 88–102 | DOI | MR | Zbl

[7] F. A. Khalilov, E. Ya. Khruslov, “Matrix generalization of the modified Korteweg-de Vries equation”, Inverse Problems, 6:2 (1990), 193–204 | DOI | MR | Zbl

[8] Q. P. Liu, C. Athorne, “Comment on ‘Matrix generalization of the modified Korteweg-de Vries equation’”, Inverse Problems, 7:5 (1991), 783–785 | DOI | MR | Zbl

[9] B. A. Kupershmidt, KP or mKP. Noncommutative mathematics of Lagrangian, Hamiltonian, and integrable systems, AMS, Providence, RI, 2000 | MR | Zbl

[10] V. E. Adler, V. V. Sokolov, “Non-Abelian evolution systems with conservation laws”, Math. Phys. Anal. Geom., 24 (2021), 7 | DOI | MR | Zbl

[11] S. I. Svinolupov, V. V. Sokolov, “Evolution equations with nontrivial conservation laws”, Funct. Anal. Appl., 16:4 (1982), 317–319 | DOI | MR

[12] A. V. Mikhailov, A. B. Shabat, V. V. Sokolov, “The symmetry approach to classification of integrable equations”, What is integrability?, ed. V.E. Zakharov, Springer-Verlag, 1991, 115–184 | DOI | MR | Zbl

[13] S. I. Svinolupov, V. V. Sokolov, “Deformations of Jordan triple systems and integrable equations”, Theor. Math. Phys., 108:3 (1996), 1160–1163 | DOI | MR | Zbl

[14] V. E. Adler, V. V. Sokolov, “On matrix Painlevé II equations”, Theor. Math. Phys., 207:2 (2021), 560–571 | DOI | MR | Zbl

[15] S. Carillo, M. L. Schiavo, E. Porten, C. Schiebold, “A novel noncommutative KdV-type equation, its recursion operator, and solitons”, J. Math. Phys., 59:4 (2018), 043501 | DOI | MR | Zbl

[16] S. I. Svinolupov, V. V. Sokolov, R. I. Yamilov, “On Bäcklund transformations for integrable evolution equations”, Sov. Math. Dokl., 28 (1983), 165–168 | MR | Zbl

[17] I. M. Krichever, S. P. Novikov, “Holomorphic bundles over algebraic curves and nonlinear equations”, Russ. Math. Surveys, 35:6 (1980), 53–79 | DOI | MR | Zbl

[18] F. Calogero, A. Degasperis, “Reduction technique for matrix nonlinear evolution equations solvable by the spectral transform”, J. Math. Phys., 22:1 (1981), 23–31 | DOI | MR | Zbl

[19] A. S. Fokas, “A symmetry approach to exactly solvable evolution equations”, J. Math. Phys., 21:6 (1980), 1318–1325 | DOI | MR | Zbl

[20] R. I. Yamilov, “Invertible changes of variables generated by Bäcklund transformations”, Theor. Math. Phys., 85:3 (1990), 1269–1275 | DOI | MR | Zbl

[21] R. I. Yamilov, “On the construction of Miura type transformations by others of this kind”, Phys. Lett. A, 173:1 (1993), 53–57 | DOI | MR

[22] A. B. Borisov, S. A. Zykov, “The dressing chain of discrete symmetries and proliferation of nonlinear equations”, Theor. Math. Phys., 115:2 (1998), 530–541 | DOI | MR | Zbl

[23] S. Ya. Startsev, “Differential substitutions of the Miura transformation type”, Theor. Math. Phys., 116:3 (1998), 1001–1010 | DOI | MR | Zbl

[24] S. I. Svinolupov, “Jordan algebras and integrable systems”, Funct. Anal. Appl., 27:4 (1993), 257–265 | DOI | MR | Zbl

[25] A. P. Veselov, A. B. Shabat, “Dressing chains and the spectral theory of the Schrödinger operators”, Funct. Anal. Appl., 27:2 (1993), 81–96 | DOI | MR | Zbl