We study the long-time behavior of solutions of the initial-boundary value (IBV) problem for the Camassa–Holm (CH) equation on the half-line . 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 , 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.
Nous étudions le comportement asymptotique, pour de grandes valeurs du temps , de solutions de problèmes aux limites pour l’équation de Camassa–Holm (CH) sur la demi-droite . 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 , , 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.
Revised:
Accepted:
DOI: 10.5802/aif.2514
Classification: 35Q53, 37K10, 30E25, 35Q15, 37K15, 35B40
Keywords: Camassa–Holm equation, asymptotics, initial-boundary value problem, Riemann–Hilbert problem
@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 = {https://aif.centre-mersenne.org/articles/10.5802/aif.2514/} }
TY - JOUR TI - Long time asymptotics of the Camassa–Holm equation on the half-line JO - Annales de l'Institut Fourier PY - 2009 DA - 2009/// SP - 3015 EP - 3056 VL - 59 IS - 7 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2514/ UR - https://zbmath.org/?q=an%3A1191.35245 UR - https://www.ams.org/mathscinet-getitem?mr=2649345 UR - https://doi.org/10.5802/aif.2514 DO - 10.5802/aif.2514 LA - en ID - AIF_2009__59_7_3015_0 ER -
Boutet de Monvel, Anne; Shepelsky, Dmitry. Long time asymptotics of the Camassa–Holm equation on the half-line. Annales de l'Institut Fourier, Volume 59 (2009) no. 7, pp. 3015-3056. doi : 10.5802/aif.2514. https://aif.centre-mersenne.org/articles/10.5802/aif.2514/
[1] Multipeakons and the classical moment problem, Adv. Math., Tome 154 (2000) no. 2, pp. 229-257 | Article | MR: 1784675 | Zbl: 0968.35008
[2] The string density problem and the Camassa-Holm equation, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., Tome 365 (2007) no. 1858, pp. 2299-2312 | Article | MR: 2329150 | Zbl: 1152.35468
[3] The mKdV equation on the half-line, J. Inst. Math. Jussieu, Tome 3 (2004) no. 2, pp. 139-164 | Article | MR: 2055707 | Zbl: 1057.35050
[4] Integrable nonlinear evolution equations on a finite interval, Comm. Math. Phys., Tome 263 (2006) no. 1, pp. 133-172 | Article | MR: 2207326 | Zbl: 1131.37064
[5] Long-Time Asymptotics for the Camassa–Holm Equation, SIAM J. Math. Anal., Tome 41 (2009) no. 4, pp. 1559-1588 | Article
[6] Initial boundary value problem for the mKdV equation on a finite interval, Ann. Inst. Fourier (Grenoble), Tome 54 (2004) no. 5, p. 1477-1495, xv, xxi | Article | Numdam | MR: 2127855 | Zbl: 1137.35419
[7] Riemann-Hilbert approach for the Camassa-Holm equation on the line, C. R. Math. Acad. Sci. Paris, Tome 343 (2006) no. 10, pp. 627-632 | MR: 2271736 | Zbl: 1110.35056
[8] The Camassa-Holm equation on the half-line: a Riemann-Hilbert approach, J. Geom. Anal., Tome 18 (2008) no. 2, pp. 285-323 | Article | MR: 2393262 | Zbl: 1157.37334
[9] 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) Tome 458 (2008), pp. 99-116 | MR: 2411903 | Zbl: pre05310327
[10] 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.) Tome 55, Cambridge Univ. Press, Cambridge, 2008, pp. 53-75 | MR: 2407592 | Zbl: 1157.35447
[11] A class of linearizable problems for the Camassa–Holm equation on the half-line (2009) (In preparation) | Zbl: 1079.35086
[12] An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., Tome 71 (1993) no. 11, pp. 1661-1664 | Article | MR: 1234453 | Zbl: 0972.35521
[13] 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: 0808.76011
[14] Integral and integrable algorithms for a nonlinear shallow-water wave equation, J. Comput. Phys., Tome 216 (2006) no. 2, pp. 547-572 | Article | MR: 2235383 | Zbl: pre05046930
[15] A shallow water equation on the circle, Comm. Pure Appl. Math., Tome 52 (1999) no. 8, pp. 949-982 | Article | MR: 1686969 | Zbl: 0940.35177
[16] On the scattering problem for the Camassa-Holm equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., Tome 457 (2001) no. 2008, pp. 953-970 | Article | MR: 1875310 | Zbl: 0999.35065
[17] Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, Tome 22 (2006) no. 6, pp. 2197-2207 | Article | MR: 2277537 | Zbl: 1105.37044
[18] The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., Tome 192 (2009) no. 1, pp. 165-186 | Article | MR: 2481064 | Zbl: 1169.76010
[19] On the inverse scattering approach to the Camassa-Holm equation, J. Nonlinear Math. Phys., Tome 10 (2003) no. 3, pp. 252-255 | Article | MR: 1990677 | Zbl: 1038.35067
[20] Stability of a class of solitary waves in compressible elastic rods, Phys. Lett. A, Tome 270 (2000) no. 3-4, pp. 140-148 | Article | MR: 1763691 | Zbl: 1115.74339
[21] The collisionless shock region for the long-time behavior of solutions of the KdV equation, Comm. Pure Appl. Math., Tome 47 (1994) no. 2, pp. 199-206 | Article | MR: 1263128 | Zbl: 0797.35143
[22] A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. (2), Tome 137 (1993) no. 2, pp. 295-368 | Article | MR: 1207209 | Zbl: 0771.35042
[23] Long-time asymptotics for integrable nonlinear wave equations, Important developments in soliton theory (Springer Ser. Nonlinear Dynam.), Springer, Berlin, 1993, pp. 181-204 | MR: 1280475 | Zbl: 0926.35132
[24] Long-time asymptotics for integrable systems. Higher order theory, Comm. Math. Phys., Tome 165 (1994) no. 1, pp. 175-191 | Article | MR: 1298946 | Zbl: 0812.35122
[25] A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London Ser. A, Tome 453 (1997) no. 1962, pp. 1411-1443 | Article | MR: 1469927 | Zbl: 0876.35102
[26] Integrable nonlinear evolution equations on the half-line, Comm. Math. Phys., Tome 230 (2002) no. 1, pp. 1-39 | Article | MR: 1930570 | Zbl: 1010.35089
[27] An initial-boundary value problem for the Korteweg-de Vries equation, Math. Comput. Simulation, Tome 37 (1994) no. 4-5, pp. 293-321 (Solitons, nonlinear wave equations and computation (New Brunswick, NJ, 1992)) | Article | MR: 1308105 | Zbl: 0832.35125
[28] The nonlinear Schrödinger equation on the half-line, Nonlinearity, Tome 18 (2005) no. 4, pp. 1771-1822 | Article | MR: 2150354 | Zbl: pre02201258
[29] Painlevé transcendents, Mathematical Surveys and Monographs, Tome 128, American Mathematical Society, Providence, RI, 2006 (The Riemann-Hilbert approach) | MR: 2264522 | Zbl: 1111.34001
[30] The Riemann-Hilbert problem and integrable systems, Notices Amer. Math. Soc., Tome 50 (2003) no. 11, pp. 1389-1400 | MR: 2011605 | Zbl: 1053.34081
[31] Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., Tome 455 (2002), pp. 63-82 | Article | MR: 1894796 | Zbl: 1037.76006
[32] On solutions of the Camassa-Holm equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., Tome 459 (2003) no. 2035, pp. 1687-1708 | Article | MR: 1997519 | Zbl: 1039.76006
[33] The scattering approach for the Camassa-Holm equation, J. Nonlinear Math. Phys., Tome 9 (2002) no. 4, pp. 389-393 | Article | MR: 1931996 | Zbl: 1014.35082
[34] On the initial boundary value problem for a shallow water equation, J. Math. Phys., Tome 45 (2004) no. 9, pp. 3479-3497 | Article | MR: 2081640 | Zbl: 1071.35102
[35] Parametric representation for the multisoliton solution of the Camassa-Holm equation, J. Phys. Soc. Japan, Tome 74 (2005) no. 7, pp. 1983-1987 | Article | MR: 2164341 | Zbl: 1076.35102
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