## Abstract

We study the initial value problem of the Kadomtsev-Petviashvili I (KPI) equation with initial data u(x_{1}, x_{2}, 0) = u_{1} (x_{1}) + u_{2} (x_{1}, x_{2}), where u _{1} (x_{1}) is the one-soliton solution of the Korteweg-de Vries equation evaluated at zero time and u_{2}(x_{1}, x _{2}) decays sufficiently rapidly on the (x_{1}, x _{2})-plane. This involves the analysis of the nonstationary Schrödinger equation (with time replaced by x_{2}) with potential u(x_{1}, x_{2}, 0). We introduce an appropriate sectionally analytic eigenfunction in the complex k-plane where k is the spectral parameter. This eigenfunction has the novelty that in addition to the usual jump across the real k-axis, it also has a jump across a segment of the imaginary k-axis. We show that this eigenfunction can be reconstructed through a linear integral equation uniquely defined in terms of appropriate scattering data. In turn, these scattering data are uniquely constructed in terms of u_{1}(x _{1}) and u_{2}(x_{1}, x_{2}). This result implies that the solution of the KPI equation can be obtained through the above linear integral equation where the scattering data have a simple t-dependence.

Original language | English |
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Pages (from-to) | 771-783 |

Number of pages | 13 |

Journal | Nonlinearity |

Volume | 16 |

Issue number | 2 |

DOIs | |

Publication status | Published - Mar 2003 |