Solutions of the KP equation associated to rank-three commuting differential operators over a singular elliptic curve.

*(English)*Zbl 0721.35078Summary: We give a method for finding explicit solutions of the Kadomtsev- Petviashvili (KP) equation which are associated to rank-three commuting ordinary differential operators. Explicit formulae for the coefficients of rank-three commuting differential operators are used to reduce the KP equation to a nonlinear partial differential equation in only two variables. This equation, which is obtained in a similar way to the rank- two Krichever-Novikov (KN) equation, is easier to solve than KP and some solutions are given.

##### Keywords:

Kadomtsev-Petviashvili equation
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##### References:

[1] | Krichever, I.M.; Novikov, S.P., Holomorphic bundles over algebraic curves and nonlinear equations, Russ. math. surveys, 35, 53, (1980) · Zbl 0548.35100 |

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[3] | Kadomtsev, B.; Petviashvili, B., On the stability of solitary waves in weakly dispersive media, Sov. phys. dokl., 15, 539, (1970) · Zbl 0217.25004 |

[4] | Mokhov, I.O., Commuting differential operators of rank 3 corresponding to an elliptic curve, Russ. math. surveys, 39, 133, (1984) |

[5] | Latham, G.A., Ph.D. thesis, (May 1989), University of California Berkeley, CA |

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