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  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, doi: 10.3934/jgm.2020023
References:
[1]

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

[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.  Google Scholar

[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.   Google Scholar

[4]

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

[5]

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

[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.   Google Scholar

[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.   Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[19]

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

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[28]

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

[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.   Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.   Google Scholar

[4]

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

[5]

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

[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.   Google Scholar

[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.   Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[19]

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

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[28]

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

[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.   Google Scholar

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 $\sqrt[c]{\Delta ^2-1}$
[1]

Goong Chen, Zhonghai Ding, Shujie Li. On positive solutions of the elliptic sine-Gordon equation. Communications on Pure & Applied Analysis, 2005, 4 (2) : 283-294. doi: 10.3934/cpaa.2005.4.283

[2]

Qin Sheng, David A. Voss, Q. M. Khaliq. An adaptive splitting algorithm for the sine-Gordon equation. Conference Publications, 2005, 2005 (Special) : 792-797. doi: 10.3934/proc.2005.2005.792

[3]

Christopher K. R. T. Jones, Robert Marangell. The spectrum of travelling wave solutions to the Sine-Gordon equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 925-937. doi: 10.3934/dcdss.2012.5.925

[4]

Cornelia Schiebold. Noncommutative AKNS systems and multisoliton solutions to the matrix sine-gordon equation. Conference Publications, 2009, 2009 (Special) : 678-690. doi: 10.3934/proc.2009.2009.678

[5]

Carl-Friedrich Kreiner, Johannes Zimmer. Heteroclinic travelling waves for the lattice sine-Gordon equation with linear pair interaction. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 915-931. doi: 10.3934/dcds.2009.25.915

[6]

Shuang Yang, Yangrong Li. Forward controllability of a random attractor for the non-autonomous stochastic sine-Gordon equation on an unbounded domain. Evolution Equations & Control Theory, 2020, 9 (3) : 581-604. doi: 10.3934/eect.2020025

[7]

Sara Cuenda, Niurka R. Quintero, Angel Sánchez. Sine-Gordon wobbles through Bäcklund transformations. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1047-1056. doi: 10.3934/dcdss.2011.4.1047

[8]

Igor Chueshov, Peter E. Kloeden, Meihua Yang. Synchronization in coupled stochastic sine-Gordon wave model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 2969-2990. doi: 10.3934/dcdsb.2016082

[9]

Yangrong Li, Shuang Yang. Backward compact and periodic random attractors for non-autonomous sine-Gordon equations with multiplicative noise. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1155-1175. doi: 10.3934/cpaa.2019056

[10]

Ivan Christov, C. I. Christov. The coarse-grain description of interacting sine-Gordon solitons with varying widths. Conference Publications, 2009, 2009 (Special) : 171-180. doi: 10.3934/proc.2009.2009.171

[11]

V. V. Chepyzhov, M. I. Vishik, W. L. Wendland. On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 27-38. doi: 10.3934/dcds.2005.12.27

[12]

Aslihan Demirkaya, Panayotis G. Kevrekidis, Milena Stanislavova, Atanas Stefanov. Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation. Conference Publications, 2015, 2015 (special) : 359-368. doi: 10.3934/proc.2015.0359

[13]

Hironobu Sasaki. Remark on the scattering problem for the Klein-Gordon equation with power nonlinearity. Conference Publications, 2007, 2007 (Special) : 903-911. doi: 10.3934/proc.2007.2007.903

[14]

Karen Yagdjian. The semilinear Klein-Gordon equation in de Sitter spacetime. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 679-696. doi: 10.3934/dcdss.2009.2.679

[15]

Satoshi Masaki, Jun-ichi Segata. Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1595-1611. doi: 10.3934/cpaa.2018076

[16]

Elena Kopylova. On dispersion decay for 3D Klein-Gordon equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5765-5780. doi: 10.3934/dcds.2018251

[17]

Stefano Pasquali. A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3573-3594. doi: 10.3934/dcdsb.2017215

[18]

Chi-Kun Lin, Kung-Chien Wu. On the fluid dynamical approximation to the nonlinear Klein-Gordon equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2233-2251. doi: 10.3934/dcds.2012.32.2233

[19]

Hironobu Sasaki. Small data scattering for the Klein-Gordon equation with cubic convolution nonlinearity. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 973-981. doi: 10.3934/dcds.2006.15.973

[20]

Jun Yang. Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2359-2388. doi: 10.3934/dcds.2014.34.2359

2019 Impact Factor: 0.649

Metrics

  • PDF downloads (13)
  • HTML views (23)
  • Cited by (0)

Other articles
by authors

[Back to Top]