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1995 Global existence of small solutions to the Davey-Stewartson and the Ishimori systems
Nakao Hayashi, Jean-Claude Saut
Differential Integral Equations 8(7): 1657-1675 (1995).

Abstract

We study the initial-value problems for the Davey-Stewartson systems and the Ishimori equations. Elliptic-hyperbolic and hyperbolic-elliptic cases were treated by the inverse scattering techniques ([2--4, 10, 13--15, 32] for the Davey-Stewartson systems and [28, 29, 33] for the Ishimori equations). Elliptic-elliptic and hyperbolic-elliptic cases were studied (in [16, 17] for the Davey-Stewartson systems and [31] for the Ishimori equations) without the use of the inverse scattering techniques. Existence of a weak solution to the Davey-Stewartson systems for the elliptic-hyperbolic case is also obtained in [16] with a smallness condition on the data in $L^2$ and a blow-up result was also obtained for the elliptic-elliptic case. By using the sharp smoothing property of solutions to the linear Schrödinger equations the local existence of a unique solution to the Davey-Stewartson systems for the elliptic-hyperbolic and hyperbolic-hyperbolic cases was established in [30] in the usual Sobolev spaces with a smallness condition on the data. We prove the local existence of a unique solution to the Davey-Stewartson systems for the elliptic-hyperbolic and hyperbolic-hyperbolic cases in some analytic function spaces without a smallness condition on the data. Furthermore we prove existence of global small solutions of these equations for the elliptic-hyperbolic and hyperbolic-hyperbolic cases in some analytic function spaces.

Citation

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Nakao Hayashi. Jean-Claude Saut. "Global existence of small solutions to the Davey-Stewartson and the Ishimori systems." Differential Integral Equations 8 (7) 1657 - 1675, 1995.

Information

Published: 1995
First available in Project Euclid: 12 May 2013

zbMATH: 0827.35120
MathSciNet: MR1347974

Subjects:
Primary: 35Q55

Rights: Copyright © 1995 Khayyam Publishing, Inc.

JOURNAL ARTICLE
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Vol.8 • No. 7 • 1995
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