KdV hierarchy via Abelian coverings and operator identities

Eichinger, B.; VandenBoom, T.; Yuditskii, P.

doi:10.1090/btran/30

KdV hierarchy via Abelian coverings and operator identities
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by B. Eichinger, T. VandenBoom and P. Yuditskii HTML | PDF

Trans. Amer. Math. Soc. Ser. B 6 (2019), 1-44

Abstract:

We establish precise spectral criteria for potential functions $V$ of reflectionless Schrödinger operators $L_V = -\partial _x^2 + V$ to admit solutions to the Korteweg–de Vries (KdV) hierarchy with $V$ as an initial value. More generally, our methods extend the classical study of algebro-geometric solutions for the KdV hierarchy to noncompact Riemann surfaces by defining generalized Abelian integrals and analogues of the Baker-Akhiezer function on infinitely connected domains with a uniformly thick boundary satisfying a fractional moment condition.

References

Ilia Binder, David Damanik, Michael Goldstein, and Milivoje Lukic, Almost periodicity in time of solutions of the KdV equation, Duke Math. J. 167 (2018), no. 14, 2633–2678. MR 3859361, DOI 10.1215/00127094-2018-0015
P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. (2) 137 (1993), no. 2, 295–368. MR 1207209, DOI 10.2307/2946540
David Damanik and Peter Yuditskii, Counterexamples to the Kotani-Last conjecture for continuum Schrödinger operators via character-automorphic Hardy spaces, Adv. Math. 293 (2016), 738–781. MR 3474334, DOI 10.1016/j.aim.2016.02.023
B. A. Dubrovin, A periodic problem for the Korteweg-de Vries equation in a class of short-range potentials, Funkcional. Anal. i Priložen. 9 (1975), no. 3, 41–51 (Russian). MR 0486780
B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern geometry—methods and applications. Part II, Graduate Texts in Mathematics, vol. 104, Springer-Verlag, New York, 1985. The geometry and topology of manifolds; Translated from the Russian by Robert G. Burns. MR 807945, DOI 10.1007/978-1-4612-1100-6
B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, Integrable systems. I [ MR0842910 (87k:58112)], Dynamical systems, IV, Encyclopaedia Math. Sci., vol. 4, Springer, Berlin, 2001, pp. 177–332. MR 1866633, DOI 10.1007/978-3-662-06791-8_{3}
I. E. Egorova, The Cauchy problem for the KdV equation with almost periodic initial data whose spectrum is nowhere dense, Spectral operator theory and related topics, Adv. Soviet Math., vol. 19, Amer. Math. Soc., Providence, RI, 1994, pp. 181–208. MR 1298446, DOI 10.1007/bf02230779
C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, A method of solving the Korteweg-de Vries equation, Phys. Rev. Lett. 19 (1967), 1095–1097.
John B. Garnett, Bounded analytic functions, 1st ed., Graduate Texts in Mathematics, vol. 236, Springer, New York, 2007. MR 2261424
John B. Garnett and Donald E. Marshall, Harmonic measure, New Mathematical Monographs, vol. 2, Cambridge University Press, Cambridge, 2005. MR 2150803, DOI 10.1017/CBO9780511546617
Fritz Gesztesy and Helge Holden, Soliton equations and their algebro-geometric solutions. Vol. I, Cambridge Studies in Advanced Mathematics, vol. 79, Cambridge University Press, Cambridge, 2003. $(1+1)$-dimensional continuous models. MR 1992536, DOI 10.1017/CBO9780511546723
Morisuke Hasumi, Hardy classes on infinitely connected Riemann surfaces, Lecture Notes in Mathematics, vol. 1027, Springer-Verlag, Berlin, 1983. MR 723502, DOI 10.1007/BFb0071447
Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496, DOI 10.1007/978-1-4419-8638-2
Adolf Hurwitz, Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen, Die Grundlehren der mathematischen Wissenschaften, Band 3, Springer-Verlag, Berlin-New York, 1964 (German). Herausgegeben und ergänzt durch einen Abschnitt über geometrische Funktionentheorie von R. Courant. Mit einem Anhang von H. Röhrl; Vierte vermehrte und verbesserte Auflage. MR 0173749
Paul Koosis, The logarithmic integral. I, Cambridge Studies in Advanced Mathematics, vol. 12, Cambridge University Press, Cambridge, 1998. Corrected reprint of the 1988 original. MR 1670244
S. Kotani, Construction of KdV flow I. Tau function via Weyl function, arXiv e-prints, arXiv:1803.03056, 2018.
S. Kotani, KdV flow on generalized reflectionless potentials, Zh. Mat. Fiz. Anal. Geom. 4 (2008), no. 4, 490–528, 574 (English, with English and Ukrainian summaries). MR 2485241
Igor Moiseevich Krichever, Algebraic curves and nonlinear difference equations, Uspekhi Mat. Nauk 33 (1978), no. 4(202), 215–216 (Russian). MR 510681
Peter D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467–490. MR 235310, DOI 10.1002/cpa.3160210503
Vladimir A. Marchenko, Sturm-Liouville operators and applications, Operator Theory: Advances and Applications, vol. 22, Birkhäuser Verlag, Basel, 1986. Translated from the Russian by A. Iacob. MR 897106, DOI 10.1007/978-3-0348-5485-6
H. P. McKean and E. Trubowitz, Hill’s operator and hyperelliptic function theory in the presence of infinitely many branch points, Comm. Pure Appl. Math. 29 (1976), no. 2, 143–226. MR 427731, DOI 10.1002/cpa.3160290203
David Mumford, Tata lectures on theta. II, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2007. Jacobian theta functions and differential equations; With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura; Reprint of the 1984 original. MR 2307768, DOI 10.1007/978-0-8176-4578-6
N. K. Nikol′skiĭ, Treatise on the shift operator, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 273, Springer-Verlag, Berlin, 1986. Spectral function theory; With an appendix by S. V. Hruščev [S. V. Khrushchëv] and V. V. Peller; Translated from the Russian by Jaak Peetre. MR 827223, DOI 10.1007/978-3-642-70151-1
S. P. Novikov, A periodic problem for the Korteweg-de Vries equation. I, Funkcional. Anal. i Priložen. 8 (1974), no. 3, 54–66 (Russian). MR 0382878
Leonid Pastur and Alexander Figotin, Spectra of random and almost-periodic operators, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 297, Springer-Verlag, Berlin, 1992. MR 1223779, DOI 10.1007/978-3-642-74346-7
Alexei Poltoratski and Christian Remling, Reflectionless Herglotz functions and Jacobi matrices, Comm. Math. Phys. 288 (2009), no. 3, 1007–1021. MR 2504863, DOI 10.1007/s00220-008-0696-x
Christian Remling, The absolutely continuous spectrum of Jacobi matrices, Ann. of Math. (2) 174 (2011), no. 1, 125–171. MR 2811596, DOI 10.4007/annals.2011.174.1.4
Walter Rudin, Fourier analysis on groups, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1990. Reprint of the 1962 original; A Wiley-Interscience Publication. MR 1038803, DOI 10.1002/9781118165621
Takahiro Shiota, Characterization of Jacobian varieties in terms of soliton equations, Invent. Math. 83 (1986), no. 2, 333–382. MR 818357, DOI 10.1007/BF01388967
Mikhail Sodin and Peter Yuditskii, Almost periodic Sturm-Liouville operators with Cantor homogeneous spectrum, Comment. Math. Helv. 70 (1995), no. 4, 639–658. MR 1360607, DOI 10.1007/BF02566026
Mikhail Sodin and Peter Yuditskii, Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions, J. Geom. Anal. 7 (1997), no. 3, 387–435. MR 1674798, DOI 10.1007/BF02921627
M. L. Sodin and P. M. Yuditskiĭ, Infinite-dimensional real problem of Jacobi inversion and Hardy spaces of character-automorphic functions, Dokl. Akad. Nauk 335 (1994), no. 2, 161–163 (Russian); English transl., Russian Acad. Sci. Dokl. Math. 49 (1994), no. 2, 287–290. MR 1288838
A. Volberg and P. Yuditskii, Kotani-Last problem and Hardy spaces on surfaces of Widom type, Invent. Math. 197 (2014), no. 3, 683–740. MR 3251833, DOI 10.1007/s00222-013-0495-7
Alexander Volberg and Peter Yuditskii, Mean type of functions of bounded characteristic and Martin functions in Denjoy domains, Adv. Math. 290 (2016), 860–887. MR 3451940, DOI 10.1016/j.aim.2015.12.012
Peter Yuditskii, Two remarks on Fuchsian groups of Widom type, Operator theory, system theory and related topics (Beer-Sheva/Rehovot, 1997) Oper. Theory Adv. Appl., vol. 123, Birkhäuser, Basel, 2001, pp. 527–537. MR 1821928
Peter Yuditskii, On the direct Cauchy theorem in Widom domains: positive and negative examples, Comput. Methods Funct. Theory 11 (2011), no. 2, [On table of contents: 2012], 395–414. MR 2858955, DOI 10.1007/BF03321869