## Abstract

We study the initial value problem for the elliptic-hyperbolic Davey-Stewartson system {i∂_{t}u + △u = c_{1}\u\^{2}u + c_{2}u∂_{xi}φ (t, x) ∈ ℝ^{3} (∂_{x1}^{2} - ∂_{x2}^{2})φ = ∂_{x1}|u|^{2} u = u(t, x) φ = φ(t, x) u(0, x) = φ(x) where △ = ∂_{x1}^{2} + ∂_{x2}^{2}, c_{1}, c_{2} ∈ ℝ, u is a complex valued function and φ is a real valued function. When (c_{1},c_{2}) = (-1, 2) the above system is called a DSI equation in the inverse scattering literature. Our purpose in this paper is to prove global existence of small solutions to this system in the usual weighted Sobolev space H^{3,0} ∩ H^{0,3}, where H^{m,l} = {f ∈ L^{2}; ∥(1 - ∂_{x1}^{2} - ∂_{x2}^{2})^{m/2}(1 + x_{1}^{2} + x_{2}^{2})^{l/2}f∥_{L2} < ∞}. Furthermore, we prove L∞ time decay estimates of solutions to the system such that ∥u(t)∥_{L∞} ≤ C(1 + |t|)^{-1}.

Original language | English |
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Pages (from-to) | 1387-1409 |

Number of pages | 23 |

Journal | Nonlinearity |

Volume | 9 |

Issue number | 6 |

DOIs | |

Publication status | Published - 1996 |