Long time asymptotics of the Camassa–Holm equation on the half-line
[Étude asymptotique de l’équation de Camassa–Holm sur la demi-droite pour de grandes valeurs du temps]
Annales de l'Institut Fourier, Tome 59 (2009) no. 7, pp. 3015-3056.

Nous étudions le comportement asymptotique, pour de grandes valeurs du temps t, de solutions de problèmes aux limites pour l’équation de Camassa–Holm (CH) u t -u txx +2u x +3uu x =2u x u xx +uu xxx sur la demi-droite x0. Cet article prolonge nos travaux antérieurs sur les problèmes aux limites pour l’équation de Camassa–Holm, travaux dont la clef est la formulation et l’analyse de problèmes de Riemann–Hilbert associés. Dans le quart de plan espace-temps x>0, t>0, nous distinguons des régions où les solutions ont un comportement asymptotique qualitativement différent, et nous calculons pour chacune d’elles le terme principal de l’asymptotique en termes de données spectrales associées aux valeurs initiales et au bord.

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 x u xx +uu xxx on the half-line x0. 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.

DOI : 10.5802/aif.2514
Classification : 35Q53, 37K10, 30E25, 35Q15, 37K15, 35B40
Keywords: Camassa–Holm equation, asymptotics, initial-boundary value problem, Riemann–Hilbert problem
Mot clés : équation de Camassa–Holm, asymptotique, problème aux limites, problème de Riemann–Hilbert
Boutet de Monvel, Anne 1 ; Shepelsky, Dmitry 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)
@article{AIF_2009__59_7_3015_0,
     author = {Boutet de Monvel, Anne and Shepelsky, Dmitry},
     title = {Long time asymptotics of the {Camassa{\textendash}Holm} equation on the half-line},
     journal = {Annales de l'Institut Fourier},
     pages = {3015--3056},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {7},
     year = {2009},
     doi = {10.5802/aif.2514},
     zbl = {1191.35245},
     mrnumber = {2649345},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2514/}
}
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
SP  - 3015
EP  - 3056
VL  - 59
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2514/
DO  - 10.5802/aif.2514
LA  - en
ID  - AIF_2009__59_7_3015_0
ER  - 
%0 Journal Article
%A Boutet de Monvel, Anne
%A Shepelsky, Dmitry
%T Long time asymptotics of the Camassa–Holm equation on the half-line
%J Annales de l'Institut Fourier
%D 2009
%P 3015-3056
%V 59
%N 7
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2514/
%R 10.5802/aif.2514
%G en
%F AIF_2009__59_7_3015_0
Boutet de Monvel, Anne; Shepelsky, Dmitry. Long time asymptotics of the Camassa–Holm equation on the half-line. Annales de l'Institut Fourier, Tome 59 (2009) no. 7, pp. 3015-3056. doi : 10.5802/aif.2514. http://www.numdam.org/articles/10.5802/aif.2514/

[1] Beals, Richard; Sattinger, David H.; Szmigielski, Jacek Multipeakons and the classical moment problem, Adv. Math., Volume 154 (2000) no. 2, pp. 229-257 | DOI | MR | Zbl

[2] Beals, Richard; Sattinger, David H.; Szmigielski, Jacek The string density problem and the Camassa-Holm equation, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., Volume 365 (2007) no. 1858, pp. 2299-2312 | DOI | MR | Zbl

[3] Boutet de Monvel, A.; Fokas, A. S.; Shepelsky, D. The mKdV equation on the half-line, J. Inst. Math. Jussieu, Volume 3 (2004) no. 2, pp. 139-164 | DOI | MR | Zbl

[4] Boutet de Monvel, A.; Fokas, A. S.; Shepelsky, D. Integrable nonlinear evolution equations on a finite interval, Comm. Math. Phys., Volume 263 (2006) no. 1, pp. 133-172 | DOI | MR | Zbl

[5] Boutet de Monvel, Anne; Kostenko, Aleksey; Shepelsky, Dmitry; Teschl, Gerald Long-Time Asymptotics for the Camassa–Holm Equation, SIAM J. Math. Anal., Volume 41 (2009) no. 4, pp. 1559-1588 | DOI

[6] Boutet de Monvel, Anne; Shepelsky, Dmitry Initial boundary value problem for the mKdV equation on a finite interval, Ann. Inst. Fourier (Grenoble), Volume 54 (2004) no. 5, p. 1477-1495, xv, xxi | DOI | Numdam | MR | Zbl

[7] Boutet de Monvel, Anne; Shepelsky, Dmitry Riemann-Hilbert approach for the Camassa-Holm equation on the line, C. R. Math. Acad. Sci. Paris, Volume 343 (2006) no. 10, pp. 627-632 | MR | Zbl

[8] Boutet de Monvel, Anne; Shepelsky, Dmitry The Camassa-Holm equation on the half-line: a Riemann-Hilbert approach, J. Geom. Anal., Volume 18 (2008) no. 2, pp. 285-323 | DOI | MR | Zbl

[9] Boutet de Monvel, Anne; Shepelsky, Dmitry; Baik, J.; Li, L-C.; Kriecherbauer, T.; McLaughlin, K.; Tomei, C. 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 (Contemporary Mathematics), Volume 458, Amer. Math. Soc., Providence, RI (2008), pp. 99-116 | MR

[10] Boutet de Monvel, Anne; Shepelsky, Dmitry Riemann-Hilbert problem in the inverse scattering for the Camassa-Holm equation on the line, Probability, geometry and integrable systems (Math. Sci. Res. Inst. Publ.), Volume 55, Cambridge Univ. Press, Cambridge, 2008, pp. 53-75 | MR | Zbl

[11] Boutet de Monvel, Anne; Shepelsky, Dmitry A class of linearizable problems for the Camassa–Holm equation on the half-line (2009) (In preparation) | Zbl

[12] 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 | MR | Zbl

[13] Camassa, Roberto; Holm, Darryl D.; Hyman, James M. 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 | Zbl

[14] Camassa, Roberto; Huang, Jingfang; Lee, Long Integral and integrable algorithms for a nonlinear shallow-water wave equation, J. Comput. Phys., Volume 216 (2006) no. 2, pp. 547-572 | DOI | MR

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

[16] Constantin, Adrian On the scattering problem for the Camassa-Holm equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., Volume 457 (2001) no. 2008, pp. 953-970 | DOI | MR | Zbl

[17] Constantin, Adrian; Gerdjikov, Vladimir S.; Ivanov, Rossen I. Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, Volume 22 (2006) no. 6, pp. 2197-2207 | DOI | MR | Zbl

[18] Constantin, Adrian; Lannes, David The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., Volume 192 (2009) no. 1, pp. 165-186 | DOI | MR | Zbl

[19] Constantin, Adrian; Lenells, Jonatan On the inverse scattering approach to the Camassa-Holm equation, J. Nonlinear Math. Phys., Volume 10 (2003) no. 3, pp. 252-255 | DOI | MR | Zbl

[20] Constantin, Adrian; Strauss, Walter A. Stability of a class of solitary waves in compressible elastic rods, Phys. Lett. A, Volume 270 (2000) no. 3-4, pp. 140-148 | DOI | MR | Zbl

[21] Deift, P.; Venakides, S.; Zhou, X. The collisionless shock region for the long-time behavior of solutions of the KdV equation, Comm. Pure Appl. Math., Volume 47 (1994) no. 2, pp. 199-206 | DOI | MR | Zbl

[22] Deift, P.; Zhou, X. A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. (2), Volume 137 (1993) no. 2, pp. 295-368 | DOI | MR | Zbl

[23] Deift, P. A.; Its, A. R.; Zhou, X. Long-time asymptotics for integrable nonlinear wave equations, Important developments in soliton theory (Springer Ser. Nonlinear Dynam.), Springer, Berlin, 1993, pp. 181-204 | MR | Zbl

[24] Deift, P. A.; Zhou, X. Long-time asymptotics for integrable systems. Higher order theory, Comm. Math. Phys., Volume 165 (1994) no. 1, pp. 175-191 | DOI | MR | Zbl

[25] Fokas, A. S. A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London Ser. A, Volume 453 (1997) no. 1962, pp. 1411-1443 | DOI | MR | Zbl

[26] Fokas, A. S. Integrable nonlinear evolution equations on the half-line, Comm. Math. Phys., Volume 230 (2002) no. 1, pp. 1-39 | DOI | MR | Zbl

[27] Fokas, A. S.; Its, A. R. An initial-boundary value problem for the Korteweg-de Vries equation, Math. Comput. Simulation, Volume 37 (1994) no. 4-5, pp. 293-321 Solitons, nonlinear wave equations and computation (New Brunswick, NJ, 1992) | DOI | MR | Zbl

[28] Fokas, A. S.; Its, A. R.; Sung, L.-Y. The nonlinear Schrödinger equation on the half-line, Nonlinearity, Volume 18 (2005) no. 4, pp. 1771-1822 | DOI | MR

[29] Fokas, Athanassios S.; Its, Alexander R.; Kapaev, Andrei A.; Novokshenov, Victor Yu. Painlevé transcendents, Mathematical Surveys and Monographs, 128, American Mathematical Society, Providence, RI, 2006 (The Riemann-Hilbert approach) | MR | Zbl

[30] Its, Alexander R. The Riemann-Hilbert problem and integrable systems, Notices Amer. Math. Soc., Volume 50 (2003) no. 11, pp. 1389-1400 | MR | Zbl

[31] Johnson, R. S. Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., Volume 455 (2002), pp. 63-82 | DOI | MR | Zbl

[32] Johnson, R. S. On solutions of the Camassa-Holm equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., Volume 459 (2003) no. 2035, pp. 1687-1708 | DOI | MR | Zbl

[33] Lenells, Jonatan The scattering approach for the Camassa-Holm equation, J. Nonlinear Math. Phys., Volume 9 (2002) no. 4, pp. 389-393 | DOI | MR | Zbl

[34] Ma, Shixiang; Ding, Shijin On the initial boundary value problem for a shallow water equation, J. Math. Phys., Volume 45 (2004) no. 9, pp. 3479-3497 | DOI | MR | Zbl

[35] Matsuno, Yoshimasa Parametric representation for the multisoliton solution of the Camassa-Holm equation, J. Phys. Soc. Japan, Volume 74 (2005) no. 7, pp. 1983-1987 | DOI | MR | Zbl

Cité par Sources :