We present a Riemann–Hilbert problem formalism for the initial value problem for the Camassa–Holm equation on the line (CH). We show that: (i) for all , the solution of this problem can be obtained in a parametric form via the solution of some associated Riemann–Hilbert problem; (ii) for large time, it develops into a train of smooth solitons; (iii) for small ω, this soliton train is close to a train of peakons, which are piecewise smooth solutions of the CH equation for .
Nous étudions par la méthode de « Riemann–Hilbert » le problème de Cauchy pour l'équation de Camassa–Holm (CH) sur la droite : . Nous obtenons que : (i) pour tout , la solution du problème de Cauchy s'exprime de façon paramétrique en termes de la solution d'un problème de Riemann–Hilbert associé ; (ii) cette solution a pour asymptotique, pour t grand, un train de solitons lisses ; (iii) pour , ce train de solitons tend vers un train de « peakons », solutions lisses par morceaux de l'équation CH pour .
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Anne Boutet de Monvel 1; Dmitry Shepelsky 2
@article{CRMATH_2006__343_10_627_0, author = {Anne Boutet de Monvel and Dmitry Shepelsky}, title = {Riemann{\textendash}Hilbert approach for the {Camassa{\textendash}Holm} equation on the line}, journal = {Comptes Rendus. Math\'ematique}, pages = {627--632}, publisher = {Elsevier}, volume = {343}, number = {10}, year = {2006}, doi = {10.1016/j.crma.2006.10.014}, language = {en}, }
Anne Boutet de Monvel; Dmitry Shepelsky. Riemann–Hilbert approach for the Camassa–Holm equation on the line. Comptes Rendus. Mathématique, Volume 343 (2006) no. 10, pp. 627-632. doi : 10.1016/j.crma.2006.10.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.10.014/
[1] Direct and Inverse Scattering on the Line, Mathematical Surveys and Monographs, vol. 28, American Mathematical Society, Providence, RI, 1988
[2] Multipeakons and a theorem of Stieltjes, Inverse Problems, Volume 15 (1999), p. L1-L4
[3] An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., Volume 71 (1993), pp. 1661-1664
[4] On the scattering problem for the Camassa–Holm equation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., Volume 457 (2001), pp. 953-970
[5] Inverse scattering transform for the Camassa–Holm equation (arXiv:) | arXiv
[6] A shallow water equation on the circle, Comm. Pure Appl. Math., Volume 52 (1999), pp. 949-982
[7] Stability of peakons, Comm. Pure Appl. Math., Volume 53 (2000), pp. 603-610
[8] A steepest descent method for oscillatory Riemann–Hilbert problem. Asymptotics for the MKdV equation, Ann. of Math. (2), Volume 137 (1993), pp. 295-368
[9] Parametric representation for the multisoliton solution of the Camassa–Holm equation, J. Phys. Soc. Japan, Volume 74 (2005), pp. 1983-1987
[10] Fredholm determinants and the Camassa–Holm hierarchy, Comm. Pure Appl. Math., Volume 56 (2003), pp. 638-680
[11] On the Camassa–Holm equation and a direct method of solution I. Bilinear form and solitary waves, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., Volume 460 (2004), pp. 2929-2957
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