×

Traveling wave solutions of the Camassa-Holm and Korteweg-de Vries equations. (English) Zbl 1069.35072

Summary: We show that the smooth traveling waves of the Camassa-Holm equation naturally correspond to traveling waves of the Korteweg-de Vries equation.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Beals R, Inverse Problems 15 pp L1– (1999) · Zbl 0923.35154 · doi:10.1088/0266-5611/15/1/001
[2] Beals R, Adv. Math. 40 pp 190– (1998) · Zbl 0919.35118 · doi:10.1006/aima.1998.1768
[3] Camassa R, Phys. Rev. Lett. 71 pp 1661– (1993) · Zbl 0972.35521 · doi:10.1103/PhysRevLett.71.1661
[4] Camassa R, Adv. Appl. Mech. 31 pp 1– (1994) · doi:10.1016/S0065-2156(08)70254-0
[5] Colliander J, J. Amer. Math. Soc. 16 pp 705– (2003) · Zbl 1025.35025 · doi:10.1090/S0894-0347-03-00421-1
[6] Constantin A, J. Differen-tial Equations 141 pp 218– (1997) · Zbl 0889.35022 · doi:10.1006/jdeq.1997.3333
[7] Constantin A, J. Funct. Anal. 155 pp 352– (1998) · Zbl 0907.35009 · doi:10.1006/jfan.1997.3231
[8] Constantin A, Ann. Inst. Fourier (Grenoble) 50 pp 321– (2000) · doi:10.5802/aif.1757
[9] Constantin A, Proc. Roy. Soc. London 457 pp 953– (2001) · Zbl 0999.35065 · doi:10.1098/rspa.2000.0701
[10] Constantin A, Applied Mathematics Letters 14 pp 789– (2001) · Zbl 0985.35069 · doi:10.1016/S0893-9659(01)00045-3
[11] Constantin A, Acta Mathematica 181 pp 229– (1998) · Zbl 0923.76025 · doi:10.1007/BF02392586
[12] Constantin A, Annali Sc. Norm. Sup. Pisa 26 pp 303– (1998)
[13] Constantin A, Comm. Pure Appl. Math 51 pp 475– (1998) · Zbl 0934.35153 · doi:10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5
[14] Constantin A, Math. Z. 233 pp 75– (2000) · Zbl 0954.35136 · doi:10.1007/PL00004793
[15] Constantin A, Indiana Univ. Math. J. 47 pp 1527– (1998) · Zbl 0930.35133 · doi:10.1512/iumj.1998.47.1466
[16] Constantin A, J. Phys. A 35 pp R51– (2002) · Zbl 1039.37068 · doi:10.1088/0305-4470/35/32/201
[17] Constantin A, Comment. Math. Helv. 78 pp 787– (2003) · Zbl 1037.37032 · doi:10.1007/s00014-003-0785-6
[18] Constantin A, J. Nonlinear Math. Phys. 10 pp 252– (2003) · Zbl 1038.35067 · doi:10.2991/jnmp.2003.10.3.1
[19] Constantin A, Comm. Pure Appl. Math. 52 pp 949– (1999) · Zbl 0940.35177 · doi:10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D
[20] Constantin A, Comm. Math. Phys. 211 pp 45– (2000) · Zbl 1002.35101 · doi:10.1007/s002200050801
[21] Constantin A, Comm. Pure Appl. Math. 53 pp 603– (2000) · Zbl 1049.35149 · doi:10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L
[22] Constantin A, J. Nonlinear Sci. 12 pp 415– (2002) · Zbl 1022.35053 · doi:10.1007/s00332-002-0517-x
[23] Dai HH, Acta Mech. 127 pp 193– (1998) · Zbl 0910.73036 · doi:10.1007/BF01170373
[24] Danchin R, Differential Integral Equations 14 pp 953– (2001)
[25] Drazin PG, Solitons: an Introduction (1989) · Zbl 0661.35001 · doi:10.1017/CBO9781139172059
[26] Dullin HR, Phys. Rev. Lett. 87 pp 194501– (2001) · doi:10.1103/PhysRevLett.87.194501
[27] Dullin HR, Fluid Dyn. Res. 33 pp 73– (2003) · Zbl 1032.76518 · doi:10.1016/S0169-5983(03)00046-7
[28] Fokas AS, Physica D 87 pp 145– (1995) · Zbl 1194.35363 · doi:10.1016/0167-2789(95)00133-O
[29] Fuchssteiner B, Physica D 95 pp 229– (1996) · Zbl 0900.35345 · doi:10.1016/0167-2789(96)00048-6
[30] Fuchssteiner B, Physica D 4 pp 47– (1981) · Zbl 1194.37114 · doi:10.1016/0167-2789(81)90004-X
[31] Gesztesy F, Rev. Mat. Iberoamericana 19 pp 73– (2003) · Zbl 1029.37049 · doi:10.4171/RMI/339
[32] Holm D, Adv. Math. 137 pp 1– (1998) · Zbl 0951.37020 · doi:10.1006/aima.1998.1721
[33] Johnson RS, J. Fluid Mech. 455 pp 63– (2002) · Zbl 1037.76006 · doi:10.1017/S0022112001007224
[34] Johnson RS, Proc. Roy. Soc. London A 459 pp 1687– (2003) · doi:10.1098/rspa.2002.1078
[35] Johnson RS, J. Nonlinear Math. Phys. 10 (1) pp 72– (2003) · Zbl 1362.35264 · doi:10.2991/jnmp.2003.10.s1.6
[36] Lenells J, J. Nonlinear Math. Phys. 9 pp 389– (2002) · Zbl 1014.35082 · doi:10.2991/jnmp.2002.9.4.2
[37] Lenells J, Internat. Math. Res. Notices 10 pp 485– (2004) · Zbl 1075.35052 · doi:10.1155/S1073792804132431
[38] Li Y, J. Diff. Eq. 162 pp 27– (2000) · Zbl 0958.35119 · doi:10.1006/jdeq.1999.3683
[39] Misiolek G, J. Geom. Phys. 24 pp 203– (1998) · Zbl 0901.58022 · doi:10.1016/S0393-0440(97)00010-7
[40] McKean HP, Global Analysis, Springer Lecture Notes in Mathematics 755 pp 83– (1979) · doi:10.1007/BFb0069806
[41] McKean HP, Asian J. Math 2 pp 867– (1998) · Zbl 0959.35140 · doi:10.4310/AJM.1998.v2.n4.a10
[42] Rodriguez-Blanco G, Nonl. Anal. 46 pp 309– (2001) · Zbl 0980.35150 · doi:10.1016/S0362-546X(01)00791-X
[43] Xin Z, Comm. Pure Appl. Math. 53 pp 1411– (2000) · Zbl 1048.35092 · doi:10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5
[44] Yin Z, Dyn. Cont. Disc. Imp. Syst.
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.