论文信息 - The initial value problem for a third order dispersive equation on the two-dimensional torus

The initial value problem for a third order dispersive equation on the two-dimensional torus

We present the necessary and sufficient conditions for the L 2 -wellposedness of the initial problem for a third order linear dispersive equation on the two-dimensional torus. Birkhoff's method of asymptotic solutions is used to prove necessity. Some properties of a system for quadratic algebraic equations associated to the principal symbol play a crucial role in proving sufficiency.

H. Chihara

[1] Herbert Koch,et al. Dispersion Estimates for Third Order Equations in Two Dimensions , 2003 .

[2] W. Ichinose. On $L^2$ well posedness of the Cauchy problem for Schrödinger type equations on the Riemannian manifold and the Maslov theory , 1988 .

[3] S. J. Hogan. The fourth-order evolution equation for deep-water gravity-capillary waves , 1985, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[4] K. Dysthe,et al. Note on a modification to the nonlinear Schrödinger equation for application to deep water waves , 1979, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[5] H. Chihara. The initial value problem for Schrödinger equations on the torus , 2002 .

[6] S. Tarama. Remarks on $L^2$-wellposed Cauchy problem for some dispersive equations , 1997 .

[7] R. Curtain,et al. The Cauchy Problem , 1995 .

[8] S. Tarama. On the wellposed Cauchy problem for some dispersive equations , 1995 .

[9] V. S. Vladimirov. The Cauchy Problem , 1986 .

[10] C. Tsukamoto. Integrability of Infinitesimal Zoll Deformations , 1984 .