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020 _a9789819997381
_9978-981-99-9738-1
024 7 _a10.1007/978-981-99-9738-1
_2doi
050 4 _aQC19.2-20.85
072 7 _aPHU
_2bicssc
072 7 _aSCI040000
_2bisacsh
072 7 _aPHU
_2thema
100 1 _aKotani, Shinichi.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
_972778
245 1 0 _aKorteweg–de Vries Flows with General Initial Conditions
_h[electronic resource] /
_cby Shinichi Kotani.
250 _a1st ed. 2024.
264 1 _aSingapore :
_bSpringer Nature Singapore :
_bImprint: Springer,
_c2024.
300 _aX, 162 p. 3 illus.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aMathematical Physics Studies,
_x2352-3905
505 0 _aIntroduction -- Sato’s Theory -- KdV Flow I: Reflectionless Case -- KdV Flow II: Extension -- Applications -- Further Topics -- Appendix.
520 _aLarge 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.
650 0 _aMathematical physics.
650 0 _aProbabilities.
_965724
650 0 _aFunctional analysis.
650 1 4 _aMathematical Physics.
650 2 4 _aProbability Theory.
_967428
650 2 4 _aFunctional Analysis.
710 2 _aSpringerLink (Online service)
773 0 _tSpringer Nature eBook
776 0 8 _iPrinted edition:
_z9789819997374
776 0 8 _iPrinted edition:
_z9789819997398
776 0 8 _iPrinted edition:
_z9789819997404
830 0 _aMathematical Physics Studies,
_x2352-3905
_972779
856 4 0 _uhttps://doi.org/10.1007/978-981-99-9738-1
912 _aZDB-2-SMA
912 _aZDB-2-SXMS
999 _c58227
_d58227