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ANALYTIC SOLUTIONS OF THE CAUCHY PROBLEM FOR THE GENERALIZED TWO-COMPONENT HUNTER-SAXTON SYSTEM
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  • Journal title : Honam Mathematical Journal
  • Volume 37, Issue 1,  2015, pp.99-112
  • Publisher : The Honam Mathematical Society
  • DOI : 10.5831/HMJ.2015.37.1.99
 Title & Authors
ANALYTIC SOLUTIONS OF THE CAUCHY PROBLEM FOR THE GENERALIZED TWO-COMPONENT HUNTER-SAXTON SYSTEM
Moon, Byungsoo;
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 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;
 Language
English
 Cited by
 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. crossref(new window)

2.
M. S. Baouendi, C. Goulaouic, Sharp estimates for analytic pseudodifferential operators and applications to Cauchy problems, J. Differ. Equ. 48 (1983) 241-268. crossref(new window)

3.
R. Beals, D. H. Sattinger, J. Szmigielski, Inverse scattering solutions of the Hunter-Saxton equation, Appl. Anal. 78 (3&4) (2001), 255-269. crossref(new window)

4.
A. Bressan, A. Constantin, Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal. 37 (2005), 996-1026. crossref(new window)

5.
R. Camassa, D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993), 1661-1664. crossref(new window)

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. crossref(new window)

8.
A. Constantin, D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal.192 (2009), 165-186. crossref(new window)

9.
A. Constantin, B. Kolev, On the geometric approach to the motion of inertial mechanical systems, J. Physic A 35 (2002), R51-R79. crossref(new window)

10.
J. Escher, Non-metric two-component Euler equation on the circle, Monatsh Math. (2010), DOI 10.1007/s00605-011-0323-3. crossref(new window)

11.
A. A. Himonas, G. Misiolek, Analyticity of the Cauchy problem for an integrable evolution equation, Math. Ann. 327 (2003), 575-584. crossref(new window)

12.
J. K. Hunter, R. Saxton, Dynamics of director fields, SIAM J. Appl. Math. 51(1991), 1498-1521. crossref(new window)

13.
J. K. Hunter, Y. Zheng, On a completely integrable hyperbolic variational equation, Physica D, 79 (1994), 361-386. crossref(new window)

14.
R. Ivanov, Two-component integrable systems modelling shallow water waves: the constant vorticity case, Wave Motion, 46 (2009), 389-96. crossref(new window)

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. crossref(new window)

19.
L. Nirenberg, An abstract form of the nonlinear Cauchy-Kowalevski theorem, J. Differential Geom. 6 (1972), 561-576.

20.
T. Nishida, A note on a theorem of Nirenberg, J. Differential Geom. 12 (1977), 629-633.

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. crossref(new window)

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. crossref(new window)

26.
E. Trubowitz, The inverse problem for periodic potentials, Comm. Pure Appl. Math. 30 (1977), 321-327. crossref(new window)

27.
H. Wu, M. Wunsch, Global existence for the generalized two-component Hunter-Saxton system, J. Math. Fluid Mech. 14 (2012), 455-469. crossref(new window)

28.
M. Wunsch, On the Hunter-Saxton system, Discrete Contin. Dyn. Syst. 12 (2009), 647-656. crossref(new window)

29.
M. Wunsch, The generalized Hunter-Saxton system, SIAM J. Math. Anal. 42 (2010), 1286-1304. crossref(new window)

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. crossref(new window)

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. crossref(new window)

33.
K. Yan, Z. Yin, Analyticity of the Cauchy problem for two-component Hunter-Saxton systems, Nonlinear Analysis 75 (2012), 253-259. crossref(new window)

34.
Z. Yin, On the structure of solutions to the periodic Hunter-Saxton equation, SIAM J. Math. Anal. 36 (2004), 272-283. crossref(new window)