DOI QR코드

DOI QR Code

ANALYTIC SOLUTIONS OF THE CAUCHY PROBLEM FOR THE GENERALIZED TWO-COMPONENT HUNTER-SAXTON SYSTEM

Moon, Byungsoo

  • Received : 2014.11.25
  • Accepted : 2015.01.09
  • Published : 2015.03.25

Abstract

In this paper we consider the periodic Cauchy problem for the generalized two-component Hunter-Saxton system with analytic initial data and we prove a Cauchy-Kowalevski type theorem for the generalized two-component Hunter-Saxton system, that establishes the existence and uniqueness of real analytic solutions.

Keywords

Generalized Hunter-Saxton system;Analytic solutions

References

  1. M. S. Baouendi, C. Goulaouic, Remarks on the abstract form of nonlinear Cauchy-Kowalevski theorems, Comm. Partial Differ. Equ. 2 (1977), 1151-1162. https://doi.org/10.1080/03605307708820057
  2. M. S. Baouendi, C. Goulaouic, Sharp estimates for analytic pseudodifferential operators and applications to Cauchy problems, J. Differ. Equ. 48 (1983) 241-268. https://doi.org/10.1016/0022-0396(83)90051-7
  3. R. Beals, D. H. Sattinger, J. Szmigielski, Inverse scattering solutions of the Hunter-Saxton equation, Appl. Anal. 78 (3&4) (2001), 255-269. https://doi.org/10.1080/00036810108840938
  4. A. Bressan, A. Constantin, Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal. 37 (2005), 996-1026. https://doi.org/10.1137/050623036
  5. R. Camassa, D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993), 1661-1664. https://doi.org/10.1103/PhysRevLett.71.1661
  6. R. M. Chen, Y. Liu, Wave breaking and global existence for a generalized two-component Camassa-Holm system, Int. Mat. Res. Not. 6 (2011), 1381-1416.
  7. A. Constantin, R. I. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Physics Letters A 372 (2008), 7129-7132. https://doi.org/10.1016/j.physleta.2008.10.050
  8. A. Constantin, D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal.192 (2009), 165-186. https://doi.org/10.1007/s00205-008-0128-2
  9. A. Constantin, B. Kolev, On the geometric approach to the motion of inertial mechanical systems, J. Physic A 35 (2002), R51-R79. https://doi.org/10.1088/0305-4470/35/32/201
  10. J. Escher, Non-metric two-component Euler equation on the circle, Monatsh Math. (2010), DOI 10.1007/s00605-011-0323-3. https://doi.org/10.1007/s00605-011-0323-3
  11. A. A. Himonas, G. Misiolek, Analyticity of the Cauchy problem for an integrable evolution equation, Math. Ann. 327 (2003), 575-584. https://doi.org/10.1007/s00208-003-0466-1
  12. J. K. Hunter, R. Saxton, Dynamics of director fields, SIAM J. Appl. Math. 51(1991), 1498-1521. https://doi.org/10.1137/0151075
  13. J. K. Hunter, Y. Zheng, On a completely integrable hyperbolic variational equation, Physica D, 79 (1994), 361-386. https://doi.org/10.1016/S0167-2789(05)80015-6
  14. R. Ivanov, Two-component integrable systems modelling shallow water waves: the constant vorticity case, Wave Motion, 46 (2009), 389-96. https://doi.org/10.1016/j.wavemoti.2009.06.012
  15. T. Kato, K. Masuda, Nonlinear evolution equations and analyticity I, Ann. de Inst. H. Poincare 3 (1986), 455-467.
  16. J. Lenells, O. Lechtenfeld, On the N=2 supersymmetric Camassa-Holm and Hunter-Saxton systems, J. Math. Phys. 50(2009), 1-17.
  17. J. Liu, Z. Yin, Blow-up phenomena and global existence for a periodic two-component Hunter-Saxton system, (2010): preprint, arXiv:1012.5448v3 [math.AP].
  18. B. Moon, Y. Liu, Wave breaking and global existence for the generalized periodic two-component Hunter-Saxton system, J. Differential Equations, 253 (2012), 319-355. https://doi.org/10.1016/j.jde.2012.02.011
  19. L. Nirenberg, An abstract form of the nonlinear Cauchy-Kowalevski theorem, J. Differential Geom. 6 (1972), 561-576. https://doi.org/10.4310/jdg/1214430643
  20. T. Nishida, A note on a theorem of Nirenberg, J. Differential Geom. 12 (1977), 629-633. https://doi.org/10.4310/jdg/1214434231
  21. P. Olver, P. Rosenau, Tri-Hamiltonian duality between solitons and solitary wave solutions having compact support, Phys. Rev. E (3) 53 (1996), 1900-1906.
  22. L. V. Ovsiannikov, A nonlinear Cauchy problems in a scale of Banach spaces, Dokl. Akad. Nauk SSSR 200 (1971).
  23. M. V. Pavlov, The Gurevich-Zybin system, J. Phys. A 38 (2005), 3823-3840. https://doi.org/10.1088/0305-4470/38/17/008
  24. F. Tiglay, The periodic Cauchy problem for Novikov's equation, Int. Math.Res. Not. 2011 No. 20, (2010), 4633-4648.
  25. F. Treves, An abstract nonlinear Cauchy-Kovalevska theorem, Trans. Amer. Math. Soc. 150 (1970), 77-92. https://doi.org/10.1090/S0002-9947-1970-0274911-X
  26. E. Trubowitz, The inverse problem for periodic potentials, Comm. Pure Appl. Math. 30 (1977), 321-327. https://doi.org/10.1002/cpa.3160300305
  27. H. Wu, M. Wunsch, Global existence for the generalized two-component Hunter-Saxton system, J. Math. Fluid Mech. 14 (2012), 455-469. https://doi.org/10.1007/s00021-011-0075-9
  28. M. Wunsch, On the Hunter-Saxton system, Discrete Contin. Dyn. Syst. 12 (2009), 647-656. https://doi.org/10.3934/dcdsb.2009.12.647
  29. M. Wunsch, The generalized Hunter-Saxton system, SIAM J. Math. Anal. 42 (2010), 1286-1304. https://doi.org/10.1137/090768576
  30. M. Wunsch, Weak geodesic flow on a semi-direct product and global solutions to the periodic Hunter-Saxton system, Nonlinear Analysis, Theory, Methods Appl. 74 (2011), 4951-4960. https://doi.org/10.1016/j.na.2011.04.041
  31. T. Yamanaka, Note on Kowalevskaja's system of partial differential equations, Comment. Math. Univ. St. Paul. 9 (1961), 7-10.
  32. K. Yan, Z. Yin, Analytic solutions of the Cauchy problem for two-component shallow water system, Math. Z. (2011) 269:1113-1127 , doi:10.1007/s00209-010-0775-5. https://doi.org/10.1007/s00209-010-0775-5
  33. K. Yan, Z. Yin, Analyticity of the Cauchy problem for two-component Hunter-Saxton systems, Nonlinear Analysis 75 (2012), 253-259. https://doi.org/10.1016/j.na.2011.08.029
  34. Z. Yin, On the structure of solutions to the periodic Hunter-Saxton equation, SIAM J. Math. Anal. 36 (2004), 272-283. https://doi.org/10.1137/S0036141003425672