no. 3
Real-Valued Algebro-Geometric Solutions of the Two-Component Camassa–Holm Hierarchy
[Solutions algebro-géométriques à valeurs réelles de la hierarchie de Camassa–Holm à deux composantes]
Eckhardt, Jonathan1 ; Gesztesy, Fritz2 ; Holden, Helge3 ; Kostenko, Aleksey1 ; Teschl, Gerald1, 4
1 Faculty of Mathematics University of Vienna Oskar-Morgenstern-Platz 1 1090 Wien (Austria)
2 Department of Mathematics University of Missouri Columbia, MO 65211 (USA)
3 Department of Mathematical Sciences Norwegian University of Science and Technology NO–7491 Trondheim (Norway)
4 International Erwin Schrödinger Institute for Mathematical Physics Boltzmanngasse 9 1090 Wien (Austria)
Annales de l'Institut Fourier, Tome 67 (2017) no. 3, pp. 1185-1230.

Nous présentons une construction de la hiérarchie de l’équation de Camassa–Holm à deux composantes (CH-2) en utilisant un nouveau formalisme de courbure nulle. Nous décrivons en détail et identifions l’ensemble isospectral associé à toutes les solutions algébro-géométriques à valeur réelle, réguliéres et bornées de la n-ème équation de l’équation stationnaire de la hiérarchie CH-2 au tore 𝕋 n de dimension n. Nous utilisons des équations de type Dubrovin pour les diviseurs auxiliaires et certains aspects de la théorie spectrale et d’inversion spectrale pour les systèmes Hamiltoniens singuliers auto-adjoints. En particulier, nous utilisons la théorie de Weyl–Titchmarsh pour les systèmes (canoniques) Hamiltoniens singuliers.

Bien que nous nous concentrons principalement sur le cas des solutions algébro-géométriques stationnaires pour CH-2, nous remarquons que le cas de la solution évolutive qui dépend du temps est subordonné au cas stationnaire en ce qui concernent les questions isospectrales liées au tore.

We provide a construction of the two-component Camassa–Holm (CH-2) hierarchy employing a new zero-curvature formalism and identify and describe in detail the isospectral set associated to all real-valued, smooth, and bounded algebro-geometric solutions of the nth equation of the stationary CH-2 hierarchy as the real n-dimensional torus 𝕋 n . We employ Dubrovin-type equations for auxiliary divisors and certain aspects of direct and inverse spectral theory for self-adjoint singular Hamiltonian systems. In particular, we employ Weyl–Titchmarsh theory for singular (canonical) Hamiltonian systems.

While we focus primarily on the case of stationary algebro-geometric CH-2 solutions, we note that the time-dependent case subordinates to the stationary one with respect to isospectral torus questions.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3107
Classification : 35Q51, 35Q53, 37K15, 37K10, 37K20
Keywords: Two-component Camassa–Holm hierarchy, real-valued algebro-geometric solutions, isospectral tori, self-adjoint Hamiltonian systems, Weyl–Titchmarsh theory
Mot clés : Hiérarchie de Camassa–Holm à deux composantes, solutions algebro-géométriques à valeurs réelles, théorie isospectrale, systèmes hamiltoniens auto-adjoints, théorie de Weyl–Titchmarsh

Eckhardt, Jonathan 1 ; Gesztesy, Fritz 2 ; Holden, Helge 3 ; Kostenko, Aleksey 1 ; Teschl, Gerald 1, 4

1 Faculty of Mathematics University of Vienna Oskar-Morgenstern-Platz 1 1090 Wien (Austria)
2 Department of Mathematics University of Missouri Columbia, MO 65211 (USA)
3 Department of Mathematical Sciences Norwegian University of Science and Technology NO–7491 Trondheim (Norway)
4 International Erwin Schrödinger Institute for Mathematical Physics Boltzmanngasse 9 1090 Wien (Austria)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     journal = {Annales de l'Institut Fourier},
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Eckhardt, Jonathan; Gesztesy, Fritz; Holden, Helge; Kostenko, Aleksey; Teschl, Gerald. Real-Valued Algebro-Geometric Solutions of the Two-Component Camassa–Holm Hierarchy. Annales de l'Institut Fourier, Tome 67 (2017) no. 3, pp. 1185-1230. doi : 10.5802/aif.3107. https://aif.centre-mersenne.org/articles/10.5802/aif.3107/

[1] Alber, Mark S.; Camassa, Roberto; Fedorov, Yuri N.; Holm, Darryl D.; Marsden, Jerrold E. The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDE’s of shallow water and Dym type, Comm. Math. Phys., Volume 221 (2001) no. 1, pp. 197-227 | DOI

[2] Alber, Mark S.; Fedorov, Yuri N. Algebraic geometrical solutions for certain evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians, Inverse Problems, Volume 17 (2001) no. 4, pp. 1017-1042 Special issue to celebrate Pierre Sabatier’s 65th birthday (Montpellier, 2000) | DOI

[3] Aratyn, Henrik; Gomes, Jose Francisco; Zimerman, Abraham H. On a negative flow of the AKNS hierarchy and its relation to a two-component Camassa–Holm equation, SIGMA Symmetry Integrability Geom. Methods Appl., Volume 2 (2006), Paper 070, 12 pages | DOI

[4] Atkinson, Frederik V. Discrete and Continuous Boundary Problems, Mathematics in Science and Engineering, 8, Academic Press, New York-London, 1964, xiv+570 pages

[5] Camassa, Roberto; Holm, Darryl D. An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., Volume 71 (1993) no. 11, pp. 1661-1664 | DOI

[6] Chen, Caixia; Wen, Shu Wave breaking phenomena and global solutions for a generalized periodic two-component Camassa–Holm system, Discrete Contin. Dyn. Syst., Volume 32 (2012) no. 10, pp. 3459-3484 | DOI

[7] Chen, Defu; Li, Yongsheng; Yan, Wei On well-posedness of two-component Camassa–Holm system in the critical Besov space, Nonlinear Anal., Volume 120 (2015), pp. 285-298 | DOI

[8] Chen, Ming; Liu, Si-Qi; Zhang, Youjin A two-component generalization of the Camassa–Holm equation and its solutions, Lett. Math. Phys., Volume 75 (2006) no. 1, pp. 1-15 | DOI

[9] Clark, Steve; Gesztesy, Fritz Weyl-Titchmarsh M-function asymptotics, local uniqueness results, trace formulas, and Borg-type theorems for Dirac operators, Trans. Amer. Math. Soc., Volume 354 (2002) no. 9, p. 3475-3534 (electronic) | DOI

[10] Constantin, Adrian; Ivanov, Rossen I. On an integrable two-component Camassa–Holm shallow water system, Phys. Lett. A, Volume 372 (2008) no. 48, pp. 7129-7132 | DOI

[11] Constantin, Adrian; McKean, Henry P. A shallow water equation on the circle, Comm. Pure Appl. Math., Volume 52 (1999) no. 8, pp. 949-982 | DOI

[12] Eckhardt, Jonathan The inverse spectral transform for the conservative Camassa–Holm flow with decaying initial data (https://arxiv.org/abs/1510.04916, to appear in Arch. Rat. Mech. Anal.)

[13] Eckhardt, Jonathan; Kostenko, Aleksey An isospectral problem for global conservative multi-peakon solutions of the Camassa–Holm equation, Commun. Math. Phys., Volume 329 (2014) no. 3, pp. 893-918 | DOI

[14] Eckhardt, Jonathan; Kostenko, Aleksey Quadratic operator pencils associated with the conservative Camassa–Holm flow (2014) (https://arxiv.org/abs/1406.3703, to appear in Bull. Soc. Math. Fr.)

[15] Eckhardt, Jonathan; Kostenko, Aleksey The inverse spectral problem for indefinite strings, Invent. Math., Volume 204 (2016) no. 3, pp. 939-977 | DOI

[16] Escher, Joachim; Kohlmann, Martin; Lenells, Jonatan The geometry of the two-component Camassa–Holm and Degasperis-Procesi equations, J. Geom. Phys., Volume 61 (2011) no. 2, pp. 436-452 | DOI

[17] Escher, Joachim; Lechtenfeld, Olaf; Yin, Zhaoyang Well-posedness and blow-up phenomena for the 2-component Camassa–Holm equation, Discrete Contin. Dyn. Syst., Volume 19 (2007) no. 3, pp. 493-513 | DOI

[18] Falqui, Gregorio On a Camassa–Holm type equation with two dependent variables, J. Phys. A, Math. Theor., Volume 39 (2006) no. 2, pp. 327-342 | DOI

[19] Fu, Ying; Qu, Changzheng Well posedness and blow-up solution for a new coupled Camassa–Holm equations with peakons, J. Math. Phys., Volume 50 (2009) no. 1, 012906, 25 pages | DOI

[20] Gesztesy, Fritz; Holden, Helge Algebro-geometric solutions of the Camassa–Holm hierarchy, Rev. Mat. Iberoam., Volume 19 (2003) no. 1, pp. 73-142 | DOI

[21] Gesztesy, Fritz; Holden, Helge Soliton Equations and Their Algebro-Geometric Solutions. Vol. I, Cambridge Studies in Advanced Mathematics, 79, Cambridge University Press, Cambridge, 2003, xii+505 pages (1+1)-dimensional continuous models | DOI

[22] Gesztesy, Fritz; Holden, Helge Real-valued algebro-geometric solutions of the Camassa–Holm hierarchy, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., Volume 366 (2008) no. 1867, pp. 1025-1054 | DOI

[23] Gesztesy, Fritz; Tsekanovskii, Eduard On matrix-valued Herglotz functions, Math. Nachr., Volume 218 (2000), pp. 61-138 | DOI

[24] Grunert, Katrin; Holden, Helge; Raynaud, Xavier Global solutions for the two-component Camassa–Holm system, Comm. Partial Differential Equations, Volume 37 (2012) no. 12, pp. 2245-2271 | DOI

[25] Grunert, Katrin; Holden, Helge; Raynaud, Xavier Periodic conservative solutions for the two-component Camassa–Holm system, Spectral analysis, differential equations and mathematical physics: a festschrift in honor of Fritz Gesztesy’s 60th birthday (Proc. Sympos. Pure Math.), Volume 87, Amer. Math. Soc., Providence, RI, 2013, pp. 165-182 | DOI

[26] Grunert, Katrin; Holden, Helge; Raynaud, Xavier Global dissipative solutions of the two-component Camassa–Holm system for initial data with nonvanishing asymptotics, Nonlinear Anal., Real World Appl., Volume 17 (2014), pp. 203-244 | DOI

[27] Grunert, Katrin; Holden, Helge; Raynaud, Xavier A continuous interpolation between conservative and dissipative solutions for the two-component Camassa–Holm system, Forum Math. Sigma, Volume 3 (2015) (Art. ID e1, 73 p.) | DOI

[28] Guan, Chunxia; Yin, Zhaoyang Global existence and blow-up phenomena for an integrable two-component Camassa–Holm shallow water system, J. Differ. Equations, Volume 248 (2010) no. 8, pp. 2003-2014 | DOI

[29] Guan, Chunxia; Yin, Zhaoyang Global weak solutions for a two-component Camassa–Holm shallow water system, J. Funct. Anal., Volume 260 (2011) no. 4, pp. 1132-1154 | DOI

[30] Hinton, Don B.; Shaw, J. K. On Titchmarsh–Weyl M(λ)-functions for linear Hamiltonian systems, J. Differ. Equations, Volume 40 (1981) no. 3, pp. 316-342 | DOI

[31] Hinton, Don B.; Shaw, J. K. Hamiltonian systems of limit point or limit circle type with both endpoints singular, J. Differ. Equations, Volume 50 (1983) no. 3, pp. 444-464 | DOI

[32] Hinton, Don B.; Shaw, J. K. On boundary value problems for Hamiltonian systems with two singular points, SIAM J. Math. Anal., Volume 15 (1984) no. 2, pp. 272-286 | DOI

[33] Holm, Darryl D.; Ivanov, Rossen I. Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples, J. Phys. A, Math. Theor., Volume 43 (2010) no. 49, 492001, 20 pages | DOI

[34] Holm, Darryl D.; Ivanov, Rossen I. Two-component CH system: inverse scattering, peakons and geometry, Inverse Problems, Volume 27 (2011) no. 4, 045013, 19 pages | DOI

[35] Hou, Yu; Fan, Engui Algebro-geometric solutions for the two-component Camassa–Holm hierarchy (https://arxiv.org/abs/1406.6153)

[36] Hu, Qiaoyi; Yin, Zhaoyang Well-posedness and blow-up phenomena for a periodic two-component Camassa–Holm equation, Proc. Roy. Soc. Edinburgh Sect. A, Volume 141 (2011) no. 1, pp. 93-107 | DOI

[37] Ionescu-Kruse, Delia Variational derivation of two-component Camassa–Holm shallow water system, Appl. Anal., Volume 92 (2013) no. 6, pp. 1241-1253 | DOI

[38] Ivanov, Rossen I. Extended Camassa–Holm hierarchy and conserved quantities, Z. Nat.forsch., A: Phys. Sci., Volume 61 (2006) no. 3-4, pp. 133-138

[39] Kogan, V. I.; Rofe-Beketov, Fedor S. On square-integrable solutions of symmetric systems of differential equations of arbitrary order, Proc. R. Soc. Edinb., Sect. A, Volume 74 (1974/75), p. 5-40 (1976) | DOI

[40] Kohlmann, Martin On a two-component π-Camassa–Holm system, J. Geom. Phys., Volume 62 (2012) no. 4, pp. 832-838 | DOI

[41] Kohlmann, Martin The two-component Camassa–Holm system in weighted L p spaces, ZAMM, Z. Angew. Math. Mech., Volume 94 (2014) no. 3, pp. 264-272 | DOI

[42] Kohlmann, Martin Curvature computations for a two-component Camassa–Holm equation with vorticity (2015) (https://arxiv.org/abs/1509.00694, to appear in Appl. Anal.)

[43] Lesch, Matthias; Malamud, Mark On the deficiency indices and self-adjointness of symmetric Hamiltonian systems, J. Differ. Equations, Volume 189 (2003) no. 2, pp. 556-615 | DOI

[44] Levitan, Boris M. Inverse Sturm–Liouville Problems, VSP, Zeist, 1987, x+240 pages (Translated from the Russian by O. Efimov)

[45] Liu, Xingxing Stability of the train of N solitary waves for the two-component Camassa–Holm shallow water system, J. Differ. Equations, Volume 260 (2016) no. 12, pp. 8403-8427 | DOI

[46] Ma, Caochuan; Alsaedi, Ahmed; Hayat, Tasawar; Zhou, Yong On blow-up of solutions to the two-component π-Camassa–Holm system, J. Math. Anal. Appl., Volume 426 (2015) no. 2, pp. 1026-1039 | DOI

[47] Mohajer, Keivan A note on traveling wave solutions to the two component Camassa–Holm equation, J. Nonlinear Math. Phys., Volume 16 (2009) no. 2, pp. 117-125 | DOI

[48] Moon, Byungsoo Global solutions to a special case of the generalized weakly dissipative periodic two-component Camassa–Holm system, Nonlinear Anal., Volume 117 (2015), pp. 38-46 | DOI

[49] Mustafa, Octavian G. On smooth traveling waves of an integrable two-component Camassa–Holm shallow water system, Wave Motion, Volume 46 (2009) no. 6, pp. 397-402 | DOI

[50] Olver, Peter J.; Rosenau, Philip Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, Volume 53 (1996) no. 2, pp. 1900-1906 | DOI

[51] Thompson, Ryan C. The periodic Cauchy problem for the 2-component Camassa–Holm system, Differ. Integral Equ., Volume 26 (2013) no. 1-2, pp. 155-182

[52] Wen, Zhenshu Several new types of bounded wave solutions for the generalized two-component Camassa–Holm equation, Nonlinear Dynam., Volume 77 (2014) no. 3, pp. 849-857 | DOI

[53] Xia, Baoqiang; Qiao, Zhijun; Zhou, Ruguang A synthetical two-component model with peakon solutions, Stud. Appl. Math., Volume 135 (2015) no. 3, pp. 248-276 | DOI

[54] Yan, Kai; Qiao, Zhijun; Yin, Zhaoyang Qualitative analysis for a new integrable two-component Camassa–Holm system with peakon and weak kink solutions, Comm. Math. Phys., Volume 336 (2015) no. 2, pp. 581-617 | DOI

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