Long time asymptotics of the Camassa–Holm equation on the half-line

Anne Boutet de Monvel[1]; Dmitry Shepelsky[2]

  • [1] Université Paris Diderot Paris 7 Institut de Mathématiques de Jussieu Site Chevaleret, Case 7012 75205 Paris Cedex 13 (France)
  • [2] Institute B. Verkin Mathematical Division 47 Lenin Avenue 61103 Kharkiv (Ukraine)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 7, page 3015-3056
  • ISSN: 0373-0956

Abstract

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We study the long-time behavior of solutions of the initial-boundary value (IBV) problem for the Camassa–Holm (CH) equation u t - u t x x + 2 u x + 3 u u x = 2 u x u x x + u u x x x on the half-line x ≥ 0 . The paper continues our study of IBV problems for the CH equation, the key tool of which is the formulation and analysis of associated Riemann–Hilbert factorization problems. We specify the regions in the quarter space-time plane x > 0 , t > 0 having qualitatively different asymptotic pictures, and give the main terms of the asymptotics in terms of spectral data associated with the initial and boundary values.

How to cite

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Boutet de Monvel, Anne, and Shepelsky, Dmitry. "Long time asymptotics of the Camassa–Holm equation on the half-line." Annales de l’institut Fourier 59.7 (2009): 3015-3056. <http://eudml.org/doc/10478>.

@article{BoutetdeMonvel2009,
abstract = {We study the long-time behavior of solutions of the initial-boundary value (IBV) problem for the Camassa–Holm (CH) equation $u_t-u_\{txx\}+2u_x+3uu_x=2u_xu_\{xx\}+uu_\{xxx\}$ on the half-line $x\ge 0$. The paper continues our study of IBV problems for the CH equation, the key tool of which is the formulation and analysis of associated Riemann–Hilbert factorization problems. We specify the regions in the quarter space-time plane $x&gt;0$, $t&gt;0$ having qualitatively different asymptotic pictures, and give the main terms of the asymptotics in terms of spectral data associated with the initial and boundary values.},
affiliation = {Université Paris Diderot Paris 7 Institut de Mathématiques de Jussieu Site Chevaleret, Case 7012 75205 Paris Cedex 13 (France); Institute B. Verkin Mathematical Division 47 Lenin Avenue 61103 Kharkiv (Ukraine)},
author = {Boutet de Monvel, Anne, Shepelsky, Dmitry},
journal = {Annales de l’institut Fourier},
keywords = {Camassa–Holm equation; asymptotics; initial-boundary value problem; Riemann–Hilbert problem; Camassa-Holm equation; nonlinear steepest descent method; Riemann-Hilbert problem},
language = {eng},
number = {7},
pages = {3015-3056},
publisher = {Association des Annales de l’institut Fourier},
title = {Long time asymptotics of the Camassa–Holm equation on the half-line},
url = {http://eudml.org/doc/10478},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Boutet de Monvel, Anne
AU - Shepelsky, Dmitry
TI - Long time asymptotics of the Camassa–Holm equation on the half-line
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 7
SP - 3015
EP - 3056
AB - We study the long-time behavior of solutions of the initial-boundary value (IBV) problem for the Camassa–Holm (CH) equation $u_t-u_{txx}+2u_x+3uu_x=2u_xu_{xx}+uu_{xxx}$ on the half-line $x\ge 0$. The paper continues our study of IBV problems for the CH equation, the key tool of which is the formulation and analysis of associated Riemann–Hilbert factorization problems. We specify the regions in the quarter space-time plane $x&gt;0$, $t&gt;0$ having qualitatively different asymptotic pictures, and give the main terms of the asymptotics in terms of spectral data associated with the initial and boundary values.
LA - eng
KW - Camassa–Holm equation; asymptotics; initial-boundary value problem; Riemann–Hilbert problem; Camassa-Holm equation; nonlinear steepest descent method; Riemann-Hilbert problem
UR - http://eudml.org/doc/10478
ER -

References

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  1. Richard Beals, David H. Sattinger, Jacek Szmigielski, Multipeakons and the classical moment problem, Adv. Math. 154 (2000), 229-257 Zbl0968.35008MR1784675
  2. Richard Beals, David H. Sattinger, Jacek Szmigielski, The string density problem and the Camassa-Holm equation, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 365 (2007), 2299-2312 Zbl1152.35468MR2329150
  3. A. Boutet de Monvel, A. S. Fokas, D. Shepelsky, The mKdV equation on the half-line, J. Inst. Math. Jussieu 3 (2004), 139-164 Zbl1057.35050MR2055707
  4. A. Boutet de Monvel, A. S. Fokas, D. Shepelsky, Integrable nonlinear evolution equations on a finite interval, Comm. Math. Phys. 263 (2006), 133-172 Zbl1131.37064MR2207326
  5. Anne Boutet de Monvel, Aleksey Kostenko, Dmitry Shepelsky, Gerald Teschl, Long-Time Asymptotics for the Camassa–Holm Equation, SIAM J. Math. Anal. 41 (2009), 1559-1588 Zbl1204.37073
  6. Anne Boutet de Monvel, Dmitry Shepelsky, Initial boundary value problem for the mKdV equation on a finite interval, Ann. Inst. Fourier (Grenoble) 54 (2004), 1477-1495, xv, xxi Zbl1137.35419MR2127855
  7. Anne Boutet de Monvel, Dmitry Shepelsky, Riemann-Hilbert approach for the Camassa-Holm equation on the line, C. R. Math. Acad. Sci. Paris 343 (2006), 627-632 Zbl1110.35056MR2271736
  8. Anne Boutet de Monvel, Dmitry Shepelsky, The Camassa-Holm equation on the half-line: a Riemann-Hilbert approach, J. Geom. Anal. 18 (2008), 285-323 Zbl1157.37334MR2393262
  9. Anne Boutet de Monvel, Dmitry Shepelsky, Long-time asymptotics of the Camassa–Holm equation on the line, Proceedings of the Conference on Integrable Systems, Random Matrices, and Applications: A conference in honor of Percy Deift’s 60th birthday 458 (2008), 99-116, Amer. Math. Soc., Providence, RI Zbl05310327MR2411903
  10. Anne Boutet de Monvel, Dmitry Shepelsky, Riemann-Hilbert problem in the inverse scattering for the Camassa-Holm equation on the line, Probability, geometry and integrable systems 55 (2008), 53-75, Cambridge Univ. Press, Cambridge Zbl1157.35447MR2407592
  11. Anne Boutet de Monvel, Dmitry Shepelsky, A class of linearizable problems for the Camassa–Holm equation on the half-line, (2009) Zbl1079.35086
  12. Roberto Camassa, Darryl D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993), 1661-1664 Zbl0972.35521MR1234453
  13. Roberto Camassa, Darryl D. Holm, James M. Hyman, A new integrable shallow water equation, Hutchinson, John W. et al. (eds.), Advances in Applied Mechanics. Vol. 31, Boston, MA: Academic Press, p. 1-33 (1994) Zbl0808.76011
  14. Roberto Camassa, Jingfang Huang, Long Lee, Integral and integrable algorithms for a nonlinear shallow-water wave equation, J. Comput. Phys. 216 (2006), 547-572 Zbl1220.76016MR2235383
  15. A. Constantin, H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math. 52 (1999), 949-982 Zbl0940.35177MR1686969
  16. Adrian Constantin, On the scattering problem for the Camassa-Holm equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457 (2001), 953-970 Zbl0999.35065MR1875310
  17. Adrian Constantin, Vladimir S. Gerdjikov, Rossen I. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems 22 (2006), 2197-2207 Zbl1105.37044MR2277537
  18. Adrian Constantin, David Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal. 192 (2009), 165-186 Zbl1169.76010MR2481064
  19. Adrian Constantin, Jonatan Lenells, On the inverse scattering approach to the Camassa-Holm equation, J. Nonlinear Math. Phys. 10 (2003), 252-255 Zbl1038.35067MR1990677
  20. Adrian Constantin, Walter A. Strauss, Stability of a class of solitary waves in compressible elastic rods, Phys. Lett. A 270 (2000), 140-148 Zbl1115.74339MR1763691
  21. P. Deift, S. Venakides, X. Zhou, The collisionless shock region for the long-time behavior of solutions of the KdV equation, Comm. Pure Appl. Math. 47 (1994), 199-206 Zbl0797.35143MR1263128
  22. P. Deift, X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. (2) 137 (1993), 295-368 Zbl0771.35042MR1207209
  23. P. A. Deift, A. R. Its, X. Zhou, Long-time asymptotics for integrable nonlinear wave equations, Important developments in soliton theory (1993), 181-204, Springer, Berlin Zbl0926.35132MR1280475
  24. P. A. Deift, X. Zhou, Long-time asymptotics for integrable systems. Higher order theory, Comm. Math. Phys. 165 (1994), 175-191 Zbl0812.35122MR1298946
  25. A. S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London Ser. A 453 (1997), 1411-1443 Zbl0876.35102MR1469927
  26. A. S. Fokas, Integrable nonlinear evolution equations on the half-line, Comm. Math. Phys. 230 (2002), 1-39 Zbl1010.35089MR1930570
  27. A. S. Fokas, A. R. Its, An initial-boundary value problem for the Korteweg-de Vries equation, Math. Comput. Simulation 37 (1994), 293-321 Zbl0832.35125MR1308105
  28. A. S. Fokas, A. R. Its, L.-Y. Sung, The nonlinear Schrödinger equation on the half-line, Nonlinearity 18 (2005), 1771-1822 Zbl1181.37095MR2150354
  29. Athanassios S. Fokas, Alexander R. Its, Andrei A. Kapaev, Victor Yu. Novokshenov, Painlevé transcendents, 128 (2006), American Mathematical Society, Providence, RI Zbl1111.34001MR2264522
  30. Alexander R. Its, The Riemann-Hilbert problem and integrable systems, Notices Amer. Math. Soc. 50 (2003), 1389-1400 Zbl1053.34081MR2011605
  31. R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech. 455 (2002), 63-82 Zbl1037.76006MR1894796
  32. R. S. Johnson, On solutions of the Camassa-Holm equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459 (2003), 1687-1708 Zbl1039.76006MR1997519
  33. Jonatan Lenells, The scattering approach for the Camassa-Holm equation, J. Nonlinear Math. Phys. 9 (2002), 389-393 Zbl1014.35082MR1931996
  34. Shixiang Ma, Shijin Ding, On the initial boundary value problem for a shallow water equation, J. Math. Phys. 45 (2004), 3479-3497 Zbl1071.35102MR2081640
  35. Yoshimasa Matsuno, Parametric representation for the multisoliton solution of the Camassa-Holm equation, J. Phys. Soc. Japan 74 (2005), 1983-1987 Zbl1076.35102MR2164341

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