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Traveling wave solutions of the Camassa-Holm equation. (English) Zbl 1082.35127

The Camassa-Holm equation (CHE) arises as a model for the unidirectional propagation of shallow water waves over a flat bottom, as well as a model for nonlinear waves in a cylindrical axially symmetric hyperelastic rod. CHE is a bi-Hamiltonian equation with infinitely many conservation laws. The author studies traveling wave solutions of the CHE using a natural weak formulation. It is shown that, in addition to smooth solutions, there are a multitude of traveling waves with singularities: peakons, cuspons, stumpons, and composite waves.

MSC:

35Q35 PDEs in connection with fluid mechanics
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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[1] Alber, M.; Camassa, R.; Holm, D.; Marsden, J., The geometry of peaked solitons and billiard solutions of a class of integrable PDE’s, Lett. Math. Phys., 32, 137-151 (1994) · Zbl 0808.35124
[2] Arnold, V., Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses application à l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16, 319-361 (1966) · Zbl 0148.45301
[3] Beals, R.; Sattinger, D.; Szmigielski, J., Acoustic scattering and the extended Korteweg-de Vries hierarchy, Adv. Math., 40, 190-206 (1998) · Zbl 0919.35118
[4] Beals, R.; Sattinger, D.; Szmigielski, J., Multi-peakons and a theorem of Stieltjes, Inverse Problems, 15, L1-L4 (1999) · Zbl 0923.35154
[5] Camassa, R.; Holm, D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71, 1661-1664 (1993) · Zbl 0972.35521
[6] Camassa, R.; Holm, D.; Hyman, J., A new integrable shallow water equation, Adv. Appl. Mech., 31, 1-33 (1994) · Zbl 0808.76011
[7] Constantin, A., On the Cauchy problem for the periodic Camassa-Holm equation, J. Differential Equations, 141, 218-235 (1997) · Zbl 0889.35022
[8] Constantin, A., On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155, 352-363 (1998) · Zbl 0907.35009
[9] Constantin, A., Existence of permanent and breaking waves for a shallow water equationa geometric approach, Ann. Inst. Fourier (Grenoble), 50, 321-362 (2000) · Zbl 0944.35062
[10] Constantin, A., On the blow-up of solutions of a periodic shallow water equation, J. Nonlinear Sci., 10, 391-399 (2000) · Zbl 0960.35083
[11] Constantin, A., On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London, 457, 953-970 (2001) · Zbl 0999.35065
[12] Constantin, A.; Escher, J., Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181, 229-243 (1998) · Zbl 0923.76025
[13] Constantin, A.; Escher, J., Global existence and blow-up for a shallow water equation, Ann. Sci. Norm. Sup. Pisa, 26, 303-328 (1998) · Zbl 0918.35005
[14] Constantin, A.; Escher, J., Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51, 475-504 (1998) · Zbl 0934.35153
[15] Constantin, A.; Escher, J., On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z., 233, 75-91 (2000) · Zbl 0954.35136
[16] Constantin, A.; Escher, J., Global weak solutions for a shallow water equation, Indiana Univ. Math. J., 47, 1527-1545 (1998) · Zbl 0930.35133
[17] Constantin, A.; Kolev, B., On the geometric approach to the motion of inertial mechanical systems, J. Phys. A, 35, R51-R79 (2002) · Zbl 1039.37068
[18] Constantin, A.; Kolev, B., Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78, 787-804 (2003) · Zbl 1037.37032
[19] Constantin, A.; Lenells, J., On the inverse scattering approach to the Camassa-Holm equation, J. Nonlinear Math. Phys., 10, 252-255 (2003) · Zbl 1038.35067
[20] Constantin, A.; McKean, H., A shallow water equation on the circle, Comm. Pure Appl. Math., 52, 949-982 (1999) · Zbl 0940.35177
[21] Constantin, A.; Molinet, L., Global weak solutions for a shallow water equation, Comm. Math. Phys., 211, 45-61 (2000) · Zbl 1002.35101
[22] Constantin, A.; Molinet, L., Orbital stability of solitary waves for a shallow water equation, Phys. D, 157, 75-89 (2001) · Zbl 0984.35139
[23] Constantin, A.; Strauss, W., Stability of peakons, Comm. Pure Appl. Math., 53, 603-610 (2000) · Zbl 1049.35149
[24] Constantin, A.; Strauss, W., Stability of the Camassa-Holm solitons, J. Nonlinear Sci., 12, 415-422 (2002) · Zbl 1022.35053
[25] Dai, H., Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 127, 193-207 (1998) · Zbl 0910.73036
[26] Danchin, R., A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14, 953-988 (2001) · Zbl 1161.35329
[27] Fokas, A., On a class of physically important integrable equations, Physica D, 87, 145-150 (1995) · Zbl 1194.35363
[28] Fokas, A.; Olver, P.; Rosenau, P., A plethora of integrable bi-Hamiltonian equations, Progr. Nonlinear Differential Equations Appl., 26, 93-101 (1997) · Zbl 0865.35121
[29] Fuchssteiner, B., Some tricks from the symmetry-toolbox for nonlinear equationsgeneralizations of the Camassa-Holm equation, Physica D, 95, 229-243 (1996) · Zbl 0900.35345
[30] Fuchssteiner, B.; Fokas, A., Symplectic structures their Bäcklund transformation and hereditary symmetries, Physica D, 4, 47-66 (1981) · Zbl 1194.37114
[31] Ferreira, M.; Kraenkel, R.; Zenchuk, A., Soliton-cuspon interaction for the Camassa-Holm equation, J. Phys. AMath. Gen., 32, 8665-8670 (1999) · Zbl 0946.35088
[32] Gesztesy, F.; Holden, H., Algebro-geometric solutions of the Camassa-Holm hierarchy, Rev. Mat. Iberoamericana, 19, 73-142 (2003) · Zbl 1029.37049
[33] Hewitt, E.; Stromberg, K., Real and Abstract Analysis (1965), Springer: Springer Berlin · Zbl 0137.03202
[34] Hörmander, L., The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis (2003), Springer: Springer Berlin · Zbl 1028.35001
[35] Johnson, R., Camassa-Holm Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455, 63-82 (2002) · Zbl 1037.76006
[36] Johnson, R., The Camassa-Holm equation for water waves moving over a shear flow, Fluid Dynam. Res., 33, 97-111 (2003) · Zbl 1032.76519
[37] Johnson, R., The classical problem of water wavesa reservoir of integrable and nearly-integrable equations, J. Nonlinear Math. Phys., 10, Suppl. 1, 72-92 (2003) · Zbl 1362.35264
[38] Johnson, R., On solutions of the Camassa-Holm equation, Proc. Roy. Soc. London A, 459, 1687-1708 (2003) · Zbl 1039.76006
[39] Kraenkel, R.; Zenchuk, A., Camassa-Holm equationtransformation to deformed sinh-Gordon equations, cuspon and soliton solutions, J. Phys. AMath. Gen., 32, 4733-4747 (1999) · Zbl 0941.35094
[40] Lenells, J., The scattering approach for the Camassa-Holm equation, J. Nonlinear Math. Phys., 9, 389-393 (2002) · Zbl 1014.35082
[41] Lenells, J., Stability of periodic peakons, Internat. Math. Res. Notices, 10, 485-499 (2004) · Zbl 1075.35052
[42] Lenells, J., A variational approach to the stability of periodic peakons, J. Nonlinear Math. Phys., 11, 2, 151-163 (2004) · Zbl 1067.35076
[43] Li, Y.; Olver, P., Convergence of solitary-wave solutions in a perturbed bi-Hamiltonian dynamical system. I. Compactons and peakons, Discrete Continus Dynamics Systems, 3, 419-432 (1997) · Zbl 0949.35118
[44] Li, Y.; Olver, P., Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162, 27-63 (2000) · Zbl 0958.35119
[45] Misiolek, G., A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24, 203-208 (1998) · Zbl 0901.58022
[46] H. McKean, Integrable systems and algebraic curves, Global Analysis, Springer Lecture Notes in Mathematics, vol. 755, 1979, pp. 83-200.; H. McKean, Integrable systems and algebraic curves, Global Analysis, Springer Lecture Notes in Mathematics, vol. 755, 1979, pp. 83-200. · Zbl 0449.35080
[47] McKean, H., Breakdown of a shallow water equation, Asian J. Math., 2, 867-874 (1998) · Zbl 0959.35140
[48] Phillips, E., An Introduction to Analysis and Integration Theory (1984), Dover: Dover New York
[49] Rudin, W., Real and Complex Analysis (1987), McGraw-Hill: McGraw-Hill New York · Zbl 0925.00005
[50] Xin, Z.; Zhang, P., On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53, 1411-1433 (2000) · Zbl 1048.35092
[51] Z. Yin, On the blow-up of solutions of the periodic Camassa-Holm equation, Dynamics of Continuous Discrete Impulsive System, to appear.; Z. Yin, On the blow-up of solutions of the periodic Camassa-Holm equation, Dynamics of Continuous Discrete Impulsive System, to appear.
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