04135nam a22005415i 4500 978-981-99-9738-1 DE-He213 20251030111714.0 cr nn 008mamaa 240226s2024 si | s |||| 0|eng d 9789819997381 978-981-99-9738-1 10.1007/978-981-99-9738-1 doi QC19.2-20.85 PHU bicssc SCI040000 bisacsh PHU thema Kotani, Shinichi. author. aut http://id.loc.gov/vocabulary/relators/aut 72778 Korteweg–de Vries Flows with General Initial Conditions [electronic resource] / by Shinichi Kotani. 1st ed. 2024. Singapore : Springer Nature Singapore : Imprint: Springer, 2024. X, 162 p. 3 illus. online resource. text txt rdacontent computer c rdamedia online resource cr rdacarrier text file PDF rda Mathematical Physics Studies, 2352-3905 Introduction -- Sato’s Theory -- KdV Flow I: Reflectionless Case -- KdV Flow II: Extension -- Applications -- Further Topics -- Appendix. 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 Schrödinger operators or an advanced Fourier analysis. Although algebraic approaches have been discovered by Hirota–Sato 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 author’s method is to represent the tau functions introduced by Hirota–Sato and developed by Segal–Wilson later in terms of the Weyl–Titchmarsh functions (WT functions, in short) for the underlying Schrödinger 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 Schrödinger 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 Schrödinger operators have dense point spectrum. This provides a mathematical foundation for the study of the soliton turbulence problem initiated by Zakharov, which was the author’s 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 reader’s benefit, the author has included here (1) a basic knowledge of direct and inverse spectral problem for 1D Schrödinger operators, including the notion of the WT functions; (2) Sato’s Grassmann manifold method revised by Segal–Wilson; and (3) basic results of ergodic Schrödinger operators. Mathematical physics. Probabilities. 65724 Functional analysis. Mathematical Physics. Probability Theory. 67428 Functional Analysis. SpringerLink (Online service) Springer Nature eBook Printed edition: 9789819997374 Printed edition: 9789819997398 Printed edition: 9789819997404 Mathematical Physics Studies, 2352-3905 72779 https://doi.org/10.1007/978-981-99-9738-1 ZDB-2-SMA ZDB-2-SXMS 58227 58227