Algebro-geometric solutions for the two-component Hunter-Saxton hierarchy

References

• M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, The inverse scattering transform–Fourier analysis for nonlinear problems, Stud. Appl. Math. 53 (1974) 249–315.
   Web of Science ®Google Scholar
• M. S. Alber, R. Camassa, Y. N. Fedorov, D. D. Holm, and J. E. Marsden, The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDEs of shallow water and Dym type, Commun. Math. Phys. 221 (2001) 197–227.
   Web of Science ®Google Scholar
• M. S. Alber, and Y. N. Fedorov, Wave solutions of evolution equations and Hamiltonian flows on non-linear subvarieties of generalized Jacobians, J. Phys. A. 33 (2000) 8409–8425.
   Google Scholar
• M. S. Alber, and Y. N. Fedorov, Algebraic geometrical solutions for certain evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians, Inverse Problems. 17 (2001) 1017–1042.
   Web of Science ®Google Scholar
• R. Beals, D. Sattinger, and J. Szmigielski, Inverse scattering solutions of the Hunter-Saxton equations, Appl. Anal. 78 (2001) 255–269.
   Google Scholar
• E. D. Belokolos, A. I. Bobenko, V. Z. Enol'skii, A. R. Its, and V. B. Matveev, Algebro-Geometric Approach to Nolinear Integrable Equations, (Springer, Berlin, 1994).
   Google Scholar
• R. Camassa, and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993) 1661–1664.
   PubMed Web of Science ®Google Scholar
• M. Chen, S. Q. Liu, and Y. Zhang, A two-component generalization of the Camassa-Holm equation and its solutions, Lett. Math. Phys. 75 (2006) 1–15.
   Web of Science ®Google Scholar
• A. Constantin, and R. I. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A 372 (2008) 7129–7132.
   Web of Science ®Google Scholar
• A. Constantin, and B. Kolev, On the geometric approach to the motion of inertial mechanical systems, J. Phys. A 35 (2002) R51–R79.
   Google Scholar
• A. Constantin, and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal. 192 (2009) 165–186.
   Web of Science ®Google Scholar
• A. Constantin, and H. P. McKean, A shallow water equation on the circle, Commun. Pure Appl. Math. 52 (1999) 949–982.
   Web of Science ®Google Scholar
• H. H. Dai, and M. Pavlov, Transformations for the Camassa-Holm equation, its high-frequency limit and the Sinh-Gordon equation, J. P. Soc. Japan, 67 (1998) 3655–3657.
   Web of Science ®Google Scholar
• B. A. Dubrovin, Completely integrable Hamiltonian systems associated with matrix operators and Abelian varieties, Funct. Anal. Appl. 11 (1977) 265–277.
   Google Scholar
• B. A. Dubrovin, Theta functions and nonlinear equations, Russian Math. Surv. 36 (1981) 11–92.
   Web of Science ®Google Scholar
• B. A. Dubrovin, Matrix finite-zone operators, Revs. Sci. Technol. 23 (1983) 20–50.
   Google Scholar
• J. Escher, O. Lechtenfeld, and Z. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete Contin. Dyn. Syst. 19 (2007) 493–513.
   Web of Science ®Google Scholar
• A. S. Fokas, and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Phys. D 4 (1981) 47–66.
   Web of Science ®Google Scholar
• F. Gesztesy, and H. Holden, Algebro-geometric solutions of the Camassa-Holm hierarchy, Rev. Mat. Iberoam. 19 (2003) 73–142.
   Web of Science ®Google Scholar
• F. Gesztesy, and H. Holden, Real-valued algebro-geometric solutions of the Camassa-Holm hierarchy, Philos. Trans. R. Soc. Lond. Ser. A. 366 (2008) 1025–1054.
   PubMed Web of Science ®Google Scholar
• F. Gesztesy, and H. Holden, Soliton Equations and Their Algebro-Geometric Solutions. vol. I: (1+1)-Dimensional Continuous Models. (Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2003).
   Google Scholar
• F. Gesztesy, and R. Ratneseelan, An alternative approach to algebro-geometric solutions of the AKNS hierarchy, Rev. Math. Phys. 10 (1998) 345–391.
   Web of Science ®Google Scholar
• C. Guan, and Z. Yin, Global weak solutions and smooth solutions for a two-component Hunter-Saxton system, J. Math. Phys. 52 (2011) 103707.
   Web of Science ®Google Scholar
• C. Guan, and Z. Yin, Global weak solutions for a periodic two-component Hunter-Saxton system, Q. Appl. math. 2 (2012) 285–297.
   Google Scholar
• G. L. Gui, and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system, J. Funct. Anal. 258 (2010) 4251–4278.
   Web of Science ®Google Scholar
• D. D. Holm, L. Ó Náraigh, and C. Tronci, Singular solutions of a modified two-component Camassa-Holm equation, Phys. Rev. E 79 (2009) 016601.
   Web of Science ®Google Scholar
• Y. Hou, E. G. Fan, and Z. J. Qiao, The algebro-geometric solutions for the modified Camassa-Holm hierarchy, preprint, (2012) arXiv: 1205.6062.
   Google Scholar
• Y. Hou, E. G. Fan, and P. Zhao, Algebro-geometirc solutions for the Hunter-Saxton hierarchy, Z. Angew. Math. Phys. (2013) Springer Basel DOI 10.1007/s00033-013-0339-8.
   Web of Science ®Google Scholar
• Y. Hou, P. Zhao, E. G. Fan, and Z. J. Qiao, Algebro-geometric solutions for Degasperis-Procesi hierarchy, SIAM J. Math. Anal. 45 (2013) 1216–1266.
   Web of Science ®Google Scholar
• J. K. Hunter, and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math. 51 (1991) 1498–1521.
   Web of Science ®Google Scholar
• J. K. Hunter, and Y. X. Zheng, On a completely integrable nonlinear hyperbolic variational equation, Phys. D 79 (1994) 361–386.
   Google Scholar
• R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech. 457 (2002) 63–82.
   Web of Science ®Google Scholar
• B. Khesin, and G. Misiolek, Euler equations on homogeneous spaces and Virasoro orbits, Adv. Math. 176 (2003) 116–144.
   Web of Science ®Google Scholar
• J. Lenells, Weak geodesic flow and global solutions of the Hunter-Saxton equation, Discret. Contin. Dyn. Syst. 18 (2007) 643–656.
   Web of Science ®Google Scholar
• J. Lenells, The Hunter-Saxton equation describes the geodesic flow on a sphere, J. Geom. Phys. 57 (2007) 2049–2064.
   Web of Science ®Google Scholar
• J. Liu, and Z. Yin, Blow-up phenomena and global existence for a periodic Hunter-Saxton system, Preprint, (2011) arXiv:1012.5448v3.
   Google Scholar
• V. B. Matveev, and M. I. Yavor, Solutions presque périodiques et á N-solitons de l'équation hydrodynamique non linéaire de Kaup, Ann. Inst. H. Poincaré Sect. A. 31 (1979) 25–41.
   Google Scholar
• S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons, the Inverse Scattering Methods. (Concultants Bureau, New York, 1984).
   Google Scholar
• P. Olver, and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E 53 (1996) 1900–1906.
   PubMed Web of Science ®Google Scholar
• M. V. Pavlov, The Gurevich-Zybin system, J. Phys. A 38 (2005) 3823–3840.
   Google Scholar
• E. G. Reyes, The soliton content of the Camassa-Holm and Hunter-Saxton equations. In: Nikitin, A.G., Boyko, V.M., Popovych, R.O. (eds.), Proceedings of the Fourth International Conference on Symmetry in Nonlinear Mathematical Physics, Proceedings of the Institute of Mathematics of the NAS of Ukraine, vol. 43, pp. 201–208. Kyiv (2002)
   Google Scholar
• J. F. Song, and C. Z. Qu, Geometric integrability of two-component Camassa-Holm and Hunter-Saxton systems, Commun. Theor. Phys. 55 (2011) 955–959.
   Web of Science ®Google Scholar
• H. Wu, and M. Wunsch, Global existence for the generalized two-component Hunter-Saxton system, J. Math. Fluid Mech. 14 (2012) 455–469.
   Web of Science ®Google Scholar
• M. Wunsch, On the Hunter-Saxton system, Discret Contin. Dyn. Syst. Ser. B 12 (2009) 647–656.
   Web of Science ®Google Scholar
• P. Zhang, and Y. Liu, Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system, Int. Math. Res. Not. 2010 (2010) 1981–2021.
   Google Scholar