Superposition formulas for rectangular matrix Riccati equations. (English) Zbl 0628.34010
A system of nonlinear ordinary differential equations allowing a superposition formula can be associated with every Lie group- subgroup pair \(G\supset G_ 0\). We consider the case when \(G=SL(n+k,{\mathbb{C}})\) and \(G_ 0=P(k)\) is a maximal parabolic subgroup of G, leaving a k- dimensional vector space invariant \((1\leq k\leq n)\). The nonlinear ordinary differential equations (ODE’s) in this case are rectangular matrix Riccati equations for a matrix \(W(t)\in {\mathbb{C}}^{n\times k}\). The special case \(n=rk\) \((n,r,k\in {\mathbb{N}})\) is considered and a superposition formula is obtained, expressing the general solution in terms of \(r+3\) particular solutions for \(r\geq 2\), \(k\geq 2\). For \(r=1\) (square matrix Riccati equations) five solutions are needed, for \(r=n\) (projective Riccati equations) the required number is \(n+2\).
MSC:
| 34A34 | Nonlinear ordinary differential equations and systems |
| 34A99 | General theory for ordinary differential equations |
References:
| [1] | DOI: 10.1063/1.526085 · Zbl 0586.34010 · doi:10.1063/1.526085 |
| [2] | DOI: 10.1063/1.526085 · Zbl 0586.34010 · doi:10.1063/1.526085 |
| [3] | DOI: 10.1090/S0002-9904-1964-11019-3 · Zbl 0121.38801 · doi:10.1090/S0002-9904-1964-11019-3 |
| [4] | Golubitsky M., J. Diff. Geom. 7 pp 175– (1972) |
| [5] | DOI: 10.1063/1.527381 · Zbl 0606.22016 · doi:10.1063/1.527381 |
| [6] | DOI: 10.1007/BF00419796 · Zbl 0443.34023 · doi:10.1007/BF00419796 |
| [7] | DOI: 10.1016/0167-2789(82)90058-6 · Zbl 1194.34013 · doi:10.1016/0167-2789(82)90058-6 |
| [8] | DOI: 10.1063/1.525831 · Zbl 0556.34015 · doi:10.1063/1.525831 |
| [9] | DOI: 10.1007/BF00417468 · Zbl 0605.34011 · doi:10.1007/BF00417468 |
| [10] | DOI: 10.1090/S0002-9947-1966-0204480-0 · doi:10.1090/S0002-9947-1966-0204480-0 |
| [11] | Wolf J., Mem. Am. Math. Soc. 180 pp 1– (1976) |
| [12] | DOI: 10.1016/0010-4655(84)90136-X · doi:10.1016/0010-4655(84)90136-X |
| [13] | DOI: 10.1109/TAC.1985.1103934 · Zbl 0597.93023 · doi:10.1109/TAC.1985.1103934 |
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