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A class of multidimensional integrable hierarchies and their reductions. (English. Russian original) Zbl 1177.37064
Theor. Math. Phys. 160, No. 1, 887-893 (2009); translation from Teor. Mat. Fiz. 160, No. 1, 15-22 (2009).
Author’s abstract: We consider a class of multidimensional integrable hierarchies connected with the commutativity of general (unreduced) \((N+1)\)-dimensional vector fields containing a derivative with respect to a spectral variable. These hierarchies are determined by a generating equation, equivalent to the Lax-Sato form. We present a dressing scheme based on a nonlinear vector Riemann problem for this class. As characteristic examples, we consider the hierarchies connected with the Manakov-Santini equation and the Dunajski system.

MSC:
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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[1] L. V. Bogdanov, V. S. Dryuma, and S. V. Manakov, ”On the dressing method for Dunajski anti-self-duality equation,” arXiv:nlin/0612046v1 (2006). · Zbl 1125.37047
[2] L. V. Bogdanov, V. S. Dryuma, and S. V. Manakov, J. Phys. A, 40, 14383–14393 (2007). · Zbl 1125.37047 · doi:10.1088/1751-8113/40/48/005
[3] S. V. Manakov and P. M. Santini, JETP Lett., 83, 462–466 (2006). · doi:10.1134/S0021364006100080
[4] S. V. Manakov and P. M. Santini, Theor. Math. Phys., 152, 1004–1011 (2007). · Zbl 1131.37061 · doi:10.1007/s11232-007-0084-2
[5] S. V. Manakov and P. M. Santini, J. Phys. A, 41, 055204 (2008).
[6] M. Dunajski, Proc. Roy. Soc. London Ser. A, 458, 1205–1222 (2002). · Zbl 1006.53040 · doi:10.1098/rspa.2001.0918
[7] K. Takasaki, J. Math. Phys., 30, 1515–1521 (1989). · Zbl 0683.53017 · doi:10.1063/1.528283
[8] K. Takasaki, J. Math. Phys., 31, 1877–1888 (1990). · Zbl 0718.53050 · doi:10.1063/1.528686
[9] M. V. Pavlov, J. Math. Phys., 44, 4134–4156 (2003). · Zbl 1062.37078 · doi:10.1063/1.1597946
[10] M. Dunajski, J. Geom. Phys., 51, 126–137 (2004). · Zbl 1110.53032 · doi:10.1016/j.geomphys.2004.01.004
[11] L. Martínez Alonso and A. B. Shabat, Phys. Lett. A, 300, 58–64 (2002). · Zbl 0997.37045 · doi:10.1016/S0375-9601(02)00703-X
[12] L. Martínez Alonso and A. B. Shabat, Theor. Math. Phys., 140, 1073–1085 (2004). · Zbl 1178.37067 · doi:10.1023/B:TAMP.0000036538.41884.57
[13] K. Takasaki, Phys. Lett. B, 285, 187–190 (1992). · doi:10.1016/0370-2693(92)91450-N
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