Kortewegde Vries Flows with General Initial Conditions

Large numbers of studies of the KdV equation have appeared since the pioneering paper by Gardner, Greene, Kruskal, and Miura in 1967. Most of those works have employed the inverse spectral method for 1D Schrdinger operators or an advanced Fourier analysis. Although algebraic approaches have been discovered by HirotaSato and Marchenko independently, those have not been fully investigated and analyzed. The present book offers a new approach to the study of the KdV equation, which treats decaying initial data and oscillating data in a unified manner. The authors method is to represent the tau functions introduced by HirotaSato and developed by SegalWilson later in terms of the WeylTitchmarsh functions (WT functions, in short) for the underlying Schrdinger operators. The main result is stated by a class of WT functions satisfying some of the asymptotic behavior along a curve approaching the spectrum of the Schrdinger operators at + in an order of -(n-1/2)for the nth KdV equation. This class contains many oscillating potentials (initial data) as well as decaying ones. Especially bounded smooth ergodic potentials are included, and under certain conditions on the potentials, the associated Schrdinger operators have dense point spectrum. This provides a mathematical foundation for the study of the soliton turbulence problem initiated by Zakharov, which was the authors motivation for extending the class of initial data in this book. A large class of almost periodic potentials is also included in these ergodic potentials. P. Deift has conjectured that any solutions to the KdV equation starting from nearly periodic initial data are almost periodic in time. Therefore, our result yields a foundation for this conjecture. For the readers benefit, the author has included here (1) a basic knowledge of direct and inverse spectral problem for 1D Schrdinger operators, including the notion of the WT functions; (2)Satos Grassmann manifold method revised by SegalWilson; and (3) basic results of ergodic Schrdinger operators.