## Abstract

We study the large time asymptotic behavior of solutions to the generalized Kadomtsev-Petviashvili (KP) equations { u_{t} + u_{xxx} + σ∂^{-1}_{x}u_{yy} = -(u^{ρ})_{x}, (t,x,y) ∈ R × R^{2}, u(0,x,y) = u_{0}(x,y), (x,y) ∈ R^{2}, (KP) where σ = 1 or σ = -1. When ρ = 2 and σ = -1, (KP) is known as the KPI equation, while ρ = 2, σ = +1 corresponds to the KPII equation. The KP equation models the propagation along the x-axis of nonlinear dispersive long waves on the surface of a fluid, when the variation along the y-axis proceeds slowly [10]. The case ρ = 3, σ = -1 has been found in the modeling of sound waves in antiferromagnetics [15]. We prove that if ρ ≥ 3 is an integer and the initial data are sufficiently small, then the solution u of (KP) satisfies the following estimates: ∥u(t)∥∞ ≤ C(1 + |t|)^{-1}(log(2 + |t|))^{κ}, ∥u_{x}(t)∞ ≤ C(1 + |t|)^{-1} for all t ∈ R, where κ = 1 if ρ = 3 and κ = 0 if ρ ≥ 4. We also find the large time asymptotics for the solution.

Original language | English |
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Pages (from-to) | 577-590 |

Number of pages | 14 |

Journal | Communications in Mathematical Physics |

Volume | 201 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1999 Apr |

Externally published | Yes |

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics