March  2021, 13(1): 73-143. doi: 10.3934/jgm.2020023

On nomalized differentials on spectral curves associated with the sinh-Gordon equation

Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, Zürich, 8057, Switzerland

 

Received  February 2020 Published  March 2021 Early access  July 2020

Fund Project: * Both authors are supported in part by the Swiss National Science Foundation

The spectral curve associated with the sinh-Gordon equation on the torus is defined in terms of the spectrum of the Lax operator appearing in the Lax pair formulation of the equation. If the spectrum is simple, it is an open Riemann surface of infinite genus. In this paper we construct normalized differentials on this curve and derive estimates for the location of their zeroes, needed for the construction of angle variables.

Citation: Thomas Kappeler, Yannick Widmer. On nomalized differentials on spectral curves associated with the sinh-Gordon equation. Journal of Geometric Mechanics, 2021, 13 (1) : 73-143. doi: 10.3934/jgm.2020023
References:
[1]

L. Ahlfors and L. Sario, Riemann surfaces, Princeton Mathematical Series, 26, Princeton University Press, 1960.

[2]

D. BättigA. M. BlochJ.-C. Guillot and T. Kappeler, On the symplectic structure of the phase space for periodic KdV, Toda, and defocusing NLS, Duke Math. J., 79 (1995), 549-604.  doi: 10.1215/S0012-7094-95-07914-9.

[3]

E. BelokolosA. BobenkoV. Matveev and V. Enolskii, Algebro-geometric principles of superposition of finite-zone solutions of integrable nonlinear equations, Uspekhi Mat. Nauk, 41 (1986), 3-42. 

[4]

M. Berti, T. Kappeler and R. Montalto, Large KAM Tori for Arbitrary Semi-linear Perturbations of the Defocusing NLS Equation, 403, Astérisque, 2018.

[5]

B. Dubrovin and I. Krichever, The Schrödinger equation in a periodic field and Riemann surfaces, Dokl. Akad. Nauk SSSR, 229 (1976), 15-18. 

[6]

B. Dubrovin and S. Novikov, A periodic problem for the Korteweg-de Vries and Sturm-Liouville equations. Their connection with algebraic geometry, Dokl. Akad. Nauk SSSR, 219 (1974), 531-534. 

[7]

L. FaddeevL. Takhtajan and V. Zakharov, Complete description of solutions of the sine-Gordon equation, Dokl. Akad. Nauk Ser. Fiz., 219 (1974), 1334-1337. 

[8]

J. Feldman, H. Knörrer and E. Trubowitz, Riemann Surfaces of Infinite Genus, CRM Monograph Series, 20, American Math. Soc., 2003.

[9]

H. Flaschka and D. McLaughlin, Canonically conjugate variables for the Korteweg-de Vries equation and the Toda lattice with periodic boundary conditions, Progr. Theoret. Phys., 55 (1976), 438-456.  doi: 10.1143/PTP.55.438.

[10]

P. Gérard and T. Kappeler, On the integrability of the Benjamin-Ono equation on the torus, preprint, arXiv: 1905.01849.

[11]

P. Gérard, T. Kappeler and P. Toplaov, Sharp well-posedness results for the Benjamin-Ono equation in $H^{s}(\mathbb T, \mathbb R)$ and qualitative properties of its solutions, preprint, arXiv: 2004.04857.

[12]

B. Grébert and T. Kappeler, The Defocusing NLS Equation and its Normal Form, EMS series of Lectures in Mathematics, European Math. Soc., 2014. doi: 10.4171/131.

[13]

T. Kappeler, P. Lohrmann and P. Topalov, On normalized differentials on families of curves of infinite genus, in Spectral Theory and Geometric Analysis, Contemp. Math, 535, American Math. Soc., 2011,109–140. doi: 10.1090/conm/535/10538.

[14]

T. Kappeler and J. Pöschel, KdV & KAM, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer Verlag, 2003. doi: 10.1007/978-3-662-08054-2.

[15]

T. Kappeler and J. Molnar, On the wellposedness of the defocusing mKdV equation below $L^2$, SIAM J. Math. Anal., 49 (2017), 2191–2219. doi: 10.1137/16M1096979.

[16]

T. Kappeler and P. Topalov, Global well-posedness of KdV in $H^{-1}(\mathbb T, \mathbb R)$, Duke Math. J., 135 (2006), 327-360.  doi: 10.1215/S0012-7094-06-13524-X.

[17]

T. Kappeler and P. Topalov, On normalized differentials on hyperelliptic curves of infinite genus, J. Differential Geom., 105 (2017), 209-248.  doi: 10.4310/jdg/1486522814.

[18]

T. Kappeler and P. Topalov, On an Arnold-Liouville type theorem for the focusing NLS equation, Integrable Systems and Algebraic Geometry, Vol. 1, LMS Lecture Notes Series, 458, Cambridge University Press, 2020, 265–290.

[19]

T. Kappeler and P. Topalov, Arnold-Liouville theorem for integrable PDEs: A case study of the focusing NLS equation, preprint, arXiv: 2002.11638.

[20]

T. Kappeler and Y. Widmer, On spectral properties of the L operator in the Lax pair of the sine-Gordon equation, Journal of Math. Physics, Analysis, Geometry, 14 (2018), 452-509.  doi: 10.15407/mag14.04.452.

[21]

S. Kuksin, Analysis of Hamiltonian PDEs, Oxford University Press, 2000.

[22]

P. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math., 21 (1968), 467-490.  doi: 10.1002/cpa.3160210503.

[23]

H. McKean, The sine-Gordon and sinh-Gordon equations on the circle, Comm. Pure Appl. Math., 34 (1981), 197-257.  doi: 10.1002/cpa.3160340204.

[24]

H. P. McKean and P. van Moerbeke, The spectrum of Hill's equation, Invent. Math., 30 (1975), 217-274.  doi: 10.1007/BF01425567.

[25]

H. 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), 143-226.  doi: 10.1002/cpa.3160290203.

[26]

H. McKean and K. Vaninsky, Action-angle variables for the cubic Schrödinger equation, Comm. Pure Appl. Math., 50 (1997), 489-562.  doi: 10.1002/(SICI)1097-0312(199706)50:6<489::AID-CPA1>3.0.CO;2-4.

[27]

S. Novikov, S. Manakov, L. Pitaevskii and V. Zakharov, Theory of Solitons. The Inverse Scattering Method, Contemporary Soviet Mathematics, Consultants Bureau [Plenum], 1984.

[28]

A. Veselov and S. Novikov, Poisson brackets and complex tori, Trudy Mat. Inst. Steklov., 165 (1984), 49-61. 

[29]

V. Zakharov and A. Shabat, A scheme for integrating nonlinear equations of mathematical physics by the method of the inverse scattering problem I, Functional Anal. Appl., 8 (1974), 226-235. 

show all references

References:
[1]

L. Ahlfors and L. Sario, Riemann surfaces, Princeton Mathematical Series, 26, Princeton University Press, 1960.

[2]

D. BättigA. M. BlochJ.-C. Guillot and T. Kappeler, On the symplectic structure of the phase space for periodic KdV, Toda, and defocusing NLS, Duke Math. J., 79 (1995), 549-604.  doi: 10.1215/S0012-7094-95-07914-9.

[3]

E. BelokolosA. BobenkoV. Matveev and V. Enolskii, Algebro-geometric principles of superposition of finite-zone solutions of integrable nonlinear equations, Uspekhi Mat. Nauk, 41 (1986), 3-42. 

[4]

M. Berti, T. Kappeler and R. Montalto, Large KAM Tori for Arbitrary Semi-linear Perturbations of the Defocusing NLS Equation, 403, Astérisque, 2018.

[5]

B. Dubrovin and I. Krichever, The Schrödinger equation in a periodic field and Riemann surfaces, Dokl. Akad. Nauk SSSR, 229 (1976), 15-18. 

[6]

B. Dubrovin and S. Novikov, A periodic problem for the Korteweg-de Vries and Sturm-Liouville equations. Their connection with algebraic geometry, Dokl. Akad. Nauk SSSR, 219 (1974), 531-534. 

[7]

L. FaddeevL. Takhtajan and V. Zakharov, Complete description of solutions of the sine-Gordon equation, Dokl. Akad. Nauk Ser. Fiz., 219 (1974), 1334-1337. 

[8]

J. Feldman, H. Knörrer and E. Trubowitz, Riemann Surfaces of Infinite Genus, CRM Monograph Series, 20, American Math. Soc., 2003.

[9]

H. Flaschka and D. McLaughlin, Canonically conjugate variables for the Korteweg-de Vries equation and the Toda lattice with periodic boundary conditions, Progr. Theoret. Phys., 55 (1976), 438-456.  doi: 10.1143/PTP.55.438.

[10]

P. Gérard and T. Kappeler, On the integrability of the Benjamin-Ono equation on the torus, preprint, arXiv: 1905.01849.

[11]

P. Gérard, T. Kappeler and P. Toplaov, Sharp well-posedness results for the Benjamin-Ono equation in $H^{s}(\mathbb T, \mathbb R)$ and qualitative properties of its solutions, preprint, arXiv: 2004.04857.

[12]

B. Grébert and T. Kappeler, The Defocusing NLS Equation and its Normal Form, EMS series of Lectures in Mathematics, European Math. Soc., 2014. doi: 10.4171/131.

[13]

T. Kappeler, P. Lohrmann and P. Topalov, On normalized differentials on families of curves of infinite genus, in Spectral Theory and Geometric Analysis, Contemp. Math, 535, American Math. Soc., 2011,109–140. doi: 10.1090/conm/535/10538.

[14]

T. Kappeler and J. Pöschel, KdV & KAM, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer Verlag, 2003. doi: 10.1007/978-3-662-08054-2.

[15]

T. Kappeler and J. Molnar, On the wellposedness of the defocusing mKdV equation below $L^2$, SIAM J. Math. Anal., 49 (2017), 2191–2219. doi: 10.1137/16M1096979.

[16]

T. Kappeler and P. Topalov, Global well-posedness of KdV in $H^{-1}(\mathbb T, \mathbb R)$, Duke Math. J., 135 (2006), 327-360.  doi: 10.1215/S0012-7094-06-13524-X.

[17]

T. Kappeler and P. Topalov, On normalized differentials on hyperelliptic curves of infinite genus, J. Differential Geom., 105 (2017), 209-248.  doi: 10.4310/jdg/1486522814.

[18]

T. Kappeler and P. Topalov, On an Arnold-Liouville type theorem for the focusing NLS equation, Integrable Systems and Algebraic Geometry, Vol. 1, LMS Lecture Notes Series, 458, Cambridge University Press, 2020, 265–290.

[19]

T. Kappeler and P. Topalov, Arnold-Liouville theorem for integrable PDEs: A case study of the focusing NLS equation, preprint, arXiv: 2002.11638.

[20]

T. Kappeler and Y. Widmer, On spectral properties of the L operator in the Lax pair of the sine-Gordon equation, Journal of Math. Physics, Analysis, Geometry, 14 (2018), 452-509.  doi: 10.15407/mag14.04.452.

[21]

S. Kuksin, Analysis of Hamiltonian PDEs, Oxford University Press, 2000.

[22]

P. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math., 21 (1968), 467-490.  doi: 10.1002/cpa.3160210503.

[23]

H. McKean, The sine-Gordon and sinh-Gordon equations on the circle, Comm. Pure Appl. Math., 34 (1981), 197-257.  doi: 10.1002/cpa.3160340204.

[24]

H. P. McKean and P. van Moerbeke, The spectrum of Hill's equation, Invent. Math., 30 (1975), 217-274.  doi: 10.1007/BF01425567.

[25]

H. 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), 143-226.  doi: 10.1002/cpa.3160290203.

[26]

H. McKean and K. Vaninsky, Action-angle variables for the cubic Schrödinger equation, Comm. Pure Appl. Math., 50 (1997), 489-562.  doi: 10.1002/(SICI)1097-0312(199706)50:6<489::AID-CPA1>3.0.CO;2-4.

[27]

S. Novikov, S. Manakov, L. Pitaevskii and V. Zakharov, Theory of Solitons. The Inverse Scattering Method, Contemporary Soviet Mathematics, Consultants Bureau [Plenum], 1984.

[28]

A. Veselov and S. Novikov, Poisson brackets and complex tori, Trudy Mat. Inst. Steklov., 165 (1984), 49-61. 

[29]

V. Zakharov and A. Shabat, A scheme for integrating nonlinear equations of mathematical physics by the method of the inverse scattering problem I, Functional Anal. Appl., 8 (1974), 226-235. 

Figure 1.  Illustration of the domains $ D_n $, $ D_{-n} $, $ -D_{-n} $, $ -D_n $ for $ n = 1, 2 $
Figure 2.  Distribution of periodic eigenvalues
Figure 3.  Illustration of the sign of $\sqrt[c]{\Delta ^2-1}$
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