Tau functions of solutions of soliton equations

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Abstract

In the holomorphic version of the inverse scattering method, we prove that the determinant of aToeplitz-type Fredholm operator arising in the solution of the inverse problem is an entire function of the spatial variablefor all potentials whose scattering data belong to a Gevrey class strictly less than 1. As a corollary, we establishthat, up to a constant factor,every local holomorphic solution of the Korteweg–de Vries equation is the second logarithmicderivative of an entire function of the spatial variable. We discuss the possible order of growth of this entire function.Analogous results are given for all soliton equations of parabolic type.

About the authors

Andrei Victorovich Domrin

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics; Institution of Russian Academy of Sciences Institute of Mathematics with Computer Center; Moscow Center for Fundamental and Applied Mathematics

Email: domrin@mi-ras.ru
Doctor of physico-mathematical sciences, Professor

References

  1. А. В. Домрин, “Мероморфное продолжение решений солитонных уравнений”, Изв. РАН. Сер. матем., 74:3 (2010), 23–44
  2. G. Wilson, “The $tau$-functions of the $mathfrak g$AKNS equations”, Integrable systems, The Verdier memorial conference (Luminy, 1991), Progr. Math., 115, Birkhäuser Boston, Boston, MA, 1993, 131–145
  3. R. Hirota, The direct method in soliton theory, Cambridge Tracts in Math., 155, Cambridge Univ. Press, Cambridge, 2004, xii+200 pp.
  4. M. Kashiwara, T. Miwa, “The $tau$ function of the Kadomtsev–Petviashvili equation. Transformation groups for soliton equations. I”, Proc. Japan Acad. Ser. A Math. Sci., 57:7 (1981), 342–347
  5. M. Sato, “Soliton equations as dynamical systems on infinite-dimensional Grassmann manifolds”, Random systems and dynamical systems (Kyoto Univ., Kyoto, 1981), RIMS Kokyuroku, 439, Kyoto Univ., Research Institute for Mathematical Sciences, Kyoto, 1981, 30–46
  6. G. Segal, G. Wilson, “Loop groups and equations of KdV type”, Inst. Hautes Etudes Sci. Publ. Math., 61 (1985), 5–65
  7. J. Dorfmeister, “Weighted $ell_1$-Grassmannians and Banach manifolds of solutions of the KP-equation and the KdV-equation”, Math. Nachr., 180 (1996), 43–73
  8. M. J. Dupre, J. F. Glazebrook, E. Previato, “Differential algebras with Banach-algebra coefficients. II: The operator cross-ratio tau-function and the Schwarzian derivative”, Complex Anal. Oper. Theory, 7:6 (2013), 1713–1734
  9. M. Cafasso, Chao-Zhong Wu, “Tau functions and the limit of block Toeplitz determinants”, Int. Math. Res. Not. IMRN, 2015:20 (2015), 10339–10366
  10. Chuu-Lian Terng, K. Uhlenbeck, “Tau functions and Virasoro actions for soliton hierarchies”, Comm. Math. Phys., 342:1 (2016), 117–150
  11. А. Ньюэлл, Солитоны в математике и физике, Мир, М., 1989, 326 с.
  12. R. Carroll, “On the determinant theme for tau functions, Grassmannians, and inverse scattering”, Inverse scattering and applications (Univ. of Massachusetts, Amherst, MA, 1990), Contemp. Math., 122, Amer. Math. Soc., Providence, RI, 1991, 23–28
  13. P. D. Lax, Functional analysis, Pure Appl. Math. (N. Y.), Wiley-Interscience [John Wiley & Sons], New York, 2002, xx+580 pp.
  14. A. Pietsch, History of Banach spaces and linear operators, Birkhäuser Boston, Inc., Boston, MA, 2007, xxiv+855 pp.
  15. I. Gohberg, S. Coldberg, N. Krupnik, Traces and determinants of linear operators, Oper. Theory Adv. Appl., 116, Birkhäuser Verlag, Basel, 2000, x+258 pp.
  16. Э. Т. Уиттекер, Дж. Н. Ватсон, Курс современного анализа, т. 1, 2, 2-е изд., Физматгиз, М., 1963, 343 с., 516 с.
  17. Т. Като, Теория возмущений линейных операторов, Мир, М., 1972, 740 с.
  18. А. Ф. Леонтьев, Целые функции. Ряды экспонент, Наука, М., 1983, 176 с.
  19. H. Widom, “On the limit of block Toeplitz determinants”, Proc. Amer. Math. Soc., 50 (1975), 167–173
  20. B. Malgrange, “Deformations isomonodromiques, forme de Liouville, fonction $tau$”, Ann. Inst. Fourier (Grenoble), 54:5 (2004), 1371–1392
  21. А. В. Домрин, “Замечания о локальном варианте метода обратной задачи рассеяния”, Комплексный анализ и приложения, Сборник статей, Тр. МИАН, 253, Наука, МАИК «Наука/Интерпериодика», М., 2006, 46–60
  22. А. В. Комлов, “О полюсах пикаровских потенциалов”, Тр. ММО, 71, УРСС, М., 2010, 270–282
  23. J. J. Duistermaat, F. A. Grünbaum, “Differential equations in the spectral parameter”, Comm. Math. Phys., 103:2 (1986), 177–240
  24. Л. А. Тахтаджян, Л. Д. Фаддеев, Гамильтонов подход в теории солитонов, Наука, М., 1986, 528 с.

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