×

zbMATH — the first resource for mathematics

An asymptotic property of the Camassa-Holm equation. (English) Zbl 1292.35265
Summary: Asymptotic densities are used to study an asymptotic property of the dispersionless Camassa-Holm equation. An asymptotic density of a global solution is a weak limit of its scaled momentum density along a sequence of time increasing to infinity. For a global solution with non-negative compactly supported initial momentum density, we show that if the asymptotic density is unique, then it is a positive combination of Dirac measures supported in a bounded interval in the non-negative axis with zero as the only possible accumulation point. In other words, if the scaled momentum density does not oscillate as time goes to infinity, the solution behaves as a combination of peakons moving to the right at different speeds. In contrast to many investigations on the topic, our approach is not spectral theoretic and hence is independent of the structure of the isospectral problem associated with the equation.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Camassa, R.; Holm, D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71, 1661-1664, (1993) · Zbl 0972.35521
[2] Constantin, A.; Lannes, D., The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equation, Arch. Ration. Mech. Anal., 192, 165-186, (2009) · Zbl 1169.76010
[3] Johnson, R. S., Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455, 63-82, (2002) · Zbl 1037.76006
[4] Dai, H.-H., Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 127, 193-207, (1998) · Zbl 0910.73036
[5] Constantin, A., Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier (Grenoble), 50, 321-362, (2000) · Zbl 0944.35062
[6] Constantin, A.; Escher, J., Global existence and blow-up for a shallow water equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 26, 303-328, (1998) · Zbl 0918.35005
[7] Rodriguez-Blanco, G., On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46, 309-327, (2001) · Zbl 0980.35150
[8] Bressan, A.; Constantin, A., Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 187, 215-239, (2007) · Zbl 1105.76013
[9] Bressan, A.; Constantin, A., Global dissipative solutions of the Camassa-Holm equation, J. Anal. Appl., 5, 1-27, (2007) · Zbl 1139.35378
[10] Constantin, A., The trajectories of particles in Stokes waves, Invent. Math., 166, 523-535, (2006) · Zbl 1108.76013
[11] Constantin, A.; Escher, J., Particle trajectories in solitary water waves, Bull. Amer. Math. Soc. (N.S.), 44, 423-431, (2007) · Zbl 1126.76012
[12] Constantin, A.; Escher, J., Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math. (2), 173, 559-568, (2011) · Zbl 1228.35076
[13] Toland, J. F., Stokes waves, Topol. Methods Nonlinear Anal., 7, 1-48, (1996) · Zbl 0897.35067
[14] Camassa, R.; Holm, D.; Hyman, M., A new integrable shallow water equation, Adv. Appl. Mech., 31, 1-33, (1994) · Zbl 0808.76011
[15] Constantin, A.; Strauss, W. A., Stability of peakons, Comm. Pure Appl. Math., 53, 603-610, (2000) · Zbl 1049.35149
[16] El Dika, K.; Molinet, L., Stability of multipeakons, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26, 1517-1532, (2009) · Zbl 1171.35459
[17] El Dika, K.; Molinet, L., Stability of multi-antipeakon-peakon profile, Discrete Contin. Dyn. Syst. Ser. B, 12, 561-577, (2009) · Zbl 1180.35453
[18] Lenells, J., A variational approach to the stability of periodic peakons, J. Nonlinear Math. Phys., 11, 151-163, (2004) · Zbl 1067.35076
[19] Constantin, A., On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. Lond. Ser. A, 457, 953-970, (2001) · Zbl 0999.35065
[20] Constantin, A.; McKean, H. P., A shallow water equation on the circle, Comm. Pure Appl. Math., 52, 949-982, (1999) · Zbl 0940.35177
[21] Eckhardt, J.; Teschl, G., On the isospectral problem of the dispersionless Camassa-Holm equation, Adv. Math., 235, 469-495, (2013) · Zbl 1279.37046
[22] Holm, D. D.; Ó Náraigh, L.; Tronci, C., Singular solutions of a modified two-component Camassa-Holm equation, Phys. Rev. E (3), 79, 1, (2009), 016601. 13 pp
[23] Holm, D. D.; Schmah, T.; Stoica, C., Geometric mechanics and symmetry: from finite to infinite dimensions, (Oxford Texts in Applied and Engineering Mathematics, (2009), Oxford University Press New York) · Zbl 1175.70001
[24] McKean, H. P., Fredholm determinants and the Camassa-Holm hierarchy, Comm. Pure Appl. Math., 56, 638-680, (2003) · Zbl 1047.37047
[25] Beals, R.; Sattinger, D. H.; Szmigielski, J., Multipeakons and the classical moment problem, Adv. Math., 154, 229-257, (2000) · Zbl 0968.35008
[26] Loubet, E., Genesis of solitons arising from individual flows of the Camassa-Holm hierarchy, Comm. Pure Appl. Math., 59, 408-465, (2006) · Zbl 1106.35066
[27] Li, L.-C., Long time behavior for a class of low-regularity solutions of the Camassa-Holm equation, Comm. Math. Phys., 285, 265-291, (2009) · Zbl 1228.35205
[28] Chen, G.-Q.; Frid, H., Large-time behavior of entropy solutions of conservation laws, J. Differential Equations, 152, 308-357, (1999) · Zbl 0926.35085
[29] Iftimie, D.; Lopes, M. C.; Nussenzveig, H. J., Large time behaviour for vortex evolution in the half plane, Comm. Math. Phys., 237, 441-469, (2003) · Zbl 1037.76009
[30] Iftimie, D.; Lopes, M. C.; Nussenzveig, H. J., On the large time behaviour of two-dimensional vortex dynamics, Physica D, 179, 153-160, (2003) · Zbl 1092.76010
[31] Iftimie, D., Large time behavior in perfect incompressible flows, Sémin. Congr., 15, 119-179, (2007) · Zbl 1132.35441
[32] Guo, W.-W.; Tang, T.-M., Evolutions of the momentum density, deformation tensor and the nonlocal term of the Camassa-Holm equation, Nonlinear Anal., 88, 16-23, (2013) · Zbl 1279.35078
[33] Kang, S.-G.; Tang, T.-M., The support of the momentum density of the Camassa-Holm equation, Appl. Math. Lett., 24, 2128-2132, (2011) · Zbl 1408.35165
[34] Xin, Z.; Zhang, P., On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53, 1411-1433, (2000) · Zbl 1048.35092
[35] Xin, Z.; Zhang, P., On the uniqueness and large time behavior of the weak solutions to a shallow water equation, Comm. Partial Differential Equations, 27, 1815-1844, (2002) · Zbl 1034.35115
[36] Himonas, A. A.; Misiolek, G.; Ponce, G.; Zhou, Y., Persistence properties and unique continuation of solutions of the Camassa-Holm equation, Comm. Math. Phys., 271, 511-522, (2007) · Zbl 1142.35078
[37] Hewitt, E.; Stromberg, K., Real and abstract analysis, (1965), Springer-Verlag Berlin, Heidelberg
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.