Svinolupov, S. I. Jordan algebras and integrable systems. (English. Russian original) Zbl 0840.35099 Funct. Anal. Appl. 27, No. 4, 257-265 (1993); translation from Funkts. Anal. Prilozh. 27, No. 4, 40-53 (1993). We construct Jordan analogs of the scalar mKdV and sG equations. It is shown that every system \[ w^i_t= w^i_{xxx}- 6a^i_{jk} w^j w^k_x,\quad i= 1,\dots, N\tag{1} \] admits the differential substitution \[ w^i= u^i_x+ a^i_{jk} u^j u^k,\quad i= 1,\dots, N,\tag{2} \] which is a natural generalization of the Miura map. By substituting (2) into (1) we obtain the Jordan mKdV system \[ u^i_t= u^i_{xxx}- 6a^i_{nm} a^n_{jk} u^j u^k u^m_x,\quad i= 1,\dots, N. \] We also construct Jordan analogs of the scalar pmKdV equation. Cited in 1 ReviewCited in 19 Documents MSC: 35Q53 KdV equations (Korteweg-de Vries equations) 17C20 Simple, semisimple Jordan algebras 58J70 Invariance and symmetry properties for PDEs on manifolds Keywords:Korteweg-de Vries equation; modified Koteweg-de Vries equation; sine-Gordon equation; potential mKdV equation; Jordan analogs; Miura map × Cite Format Result Cite Review PDF References: [1] V. I. Arnold, ”Remarks on perturbation theory for problems of Mathieu type,” Usp. Mat. Nauk,38, No. 4, 189–203 (1983). [2] V. I. Arnold, ”Small denominators I. Mappings of the circumference onto itself,” Trans. Amer. Math. Soc.,46, 213–284 (1965). · Zbl 0152.41905 [3] O. G. Galkin, ”Phase-locking for Mathieu-type vector fields on the torus,” Funkts. Anal. Prilozhen.,26, No. 1, 1–8 (1992). · Zbl 0828.20025 · doi:10.1007/BF01077066 [4] A. Khinchin, Continued fractions, Groningen, Nordhorff (1963). · Zbl 0117.28601 [5] C. Baesens, J. Guckenheimer, S. Kim, and R. S. MacKay, ”Three coupled oscillators: mode-locking, global bifurcations and toroidal chaos,” Phys. D,49, 387–475 (1991). · Zbl 0734.58036 · doi:10.1016/0167-2789(91)90155-3 [6] R. E. Ecke, J. D. Farmer, and D. K. Umberger, ”Scaling of the Arnold tongues,” Nonlinearity,2, 175–196 (1989). · Zbl 0689.58017 · doi:10.1088/0951-7715/2/2/001 [7] J. Franks and M. Misiurewicz, ”Rotation sets of toral flows,” Proc. Amer. Math. Soc.,109, 243–249 (1990). · Zbl 0701.57016 · doi:10.1090/S0002-9939-1990-1021217-5 [8] O. G. Galkin, ”Resonance regions for Mathieu type dynamical systems on a torus,” Phys. D,39, 287–298 (1989). · Zbl 0695.58025 · doi:10.1016/0167-2789(89)90011-0 [9] C. Grebogi, E. Ott, and J. A. Yorke, ”Attractors on ann-torus: quasiperiodicity versus chaos,” Phys. D,15, 354–373 (1985). · Zbl 0577.58023 · doi:10.1016/S0167-2789(85)80004-X [10] G. R. Hall, ”Resonance zones in two-parameter families of circle homeomorphisms,” SIAM J. Math. Anal.,15, 1075–1081 (1984). · Zbl 0554.58040 · doi:10.1137/0515083 [11] S. Kim, R. S. MacKay, and J. Guckenheimer, ”Resonance regions for families of torus maps,” Nonlinearity,2, 391–404 (1989). · Zbl 0678.58034 · doi:10.1088/0951-7715/2/3/001 [12] M. Misiurewicz and K. Ziemian, ”Rotation sets for maps of tori,” J. London Math. Soc.,40, 490–506 (1989). · Zbl 0663.58022 · doi:10.1112/jlms/s2-40.3.490 [13] S. Newhouse, J. Palis, and F. Takens, ”Bifurcations and stability of families of diffeomorphisms,” Publ. Math. IHES,57, 5–72 (1983). · Zbl 0518.58031 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.