Global existence and blow-up for a shallow water equation
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 4, Volume 26 (1998) no. 2, p. 303-328
@article{ASNSP_1998_4_26_2_303_0,
author = {Constantin, Adrian and Escher, Joachim},
title = {Global existence and blow-up for a shallow water equation},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola normale superiore},
volume = {Ser. 4, 26},
number = {2},
year = {1998},
pages = {303-328},
zbl = {0918.35005},
mrnumber = {1631589},
language = {en},
url = {http://www.numdam.org/item/ASNSP_1998_4_26_2_303_0}
}

Constantin, Adrian; Escher, Joachim. Global existence and blow-up for a shallow water equation. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 4, Volume 26 (1998) no. 2, pp. 303-328. http://www.numdam.org/item/ASNSP_1998_4_26_2_303_0/

[1] M.S. Alber - R. Camassa - D. Holm - J.E. Marsden, On the link between umbilic geodesics and soliton solutions of nonlinear PDE's, Proc. Roy. Soc. London, Ser. A 450 (1995), 677-692. | MR 1356178 | Zbl 0835.35125

[2] M.S. Alber - R. Camassa - D. Holm - J.E. Marsden, The geometry of peaked solitons and billiard solutions of a class of integrable PDE's, Lett. Math. Phys. 32 (1994), 137-151. | MR 1296383 | Zbl 0808.35124

[3] B. Benjamin - J.L. Bona - J.J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London, Ser. A 272 (1972), 47-78. | MR 427868 | Zbl 0229.35013

[4] J.L. Bona - W.G. Pritchard - L.R. Scott, Solitary wave interaction, Phys. Fluids 23 (1980), 438-441. | Zbl 0425.76019

[5] J. Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations II, Geom. Funct. Anal. 3 (1993), 209-262. | MR 1215780 | Zbl 0787.35098

[6] F. Calogero, An integrable Hamiltonian system, Phys. Lett. A 201 (1995), 306-310. | MR 1331829 | Zbl 1020.37524

[7] F. Calogero - J.P. Francoise, A completely integrable Hamiltonian system, J. Math. Phys. 37 (1996), 2863-2871. | MR 1390240 | Zbl 0864.58025

[8] F. Calogero - J.F. Vandiejen, Solvable quantum version of an integrable Hamiltonian system, J. Math. Phys. 37 (1996), 4243-4251. | MR 1408090 | Zbl 0863.58022

[9] R. Camassa - D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993), 1661-1664. | MR 1234453 | Zbl 0972.35521

[10] R. Camassa - D. Holm - J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech. 31 (1994), 1-33. | Zbl 0808.76011

[11] A. Constantin, The Hamiltonian structure of the Camassa-Holm equation, Expositiones Math. 15 (1997), 53-85. | MR 1438436 | Zbl 0881.35094

[12] A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation, J. Differential Equations 141 (1997), 218-235. | MR 1488351 | Zbl 0889.35022

[13] A. Constantin - J. Escher, Well-posedness and existence of global solutions for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math. 51 (1998), 475-504. | MR 1604278 | Zbl 0934.35153

[14] F. Cooper - H. Shepard, Solitons in the Camassa-Holm shallow water equation, Phys. Lett. A 194 (1994), 246-250. | MR 1301972 | Zbl 0961.76512

[15] R.K. Dodd - J.C. Eilbeck - J.D. Gibbon - H.C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, New York, 1984. | MR 696935 | Zbl 0496.35001

[16] A. Fokas - B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Phys. D 4 (1981), 47-66. | MR 636470

[17] B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation, Phys. D 95 (1996), 296-343. | MR 1406283 | Zbl 0900.35345

[18] D. Gilbarg - N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Verlag, Berlin, 1977. | MR 473443 | Zbl 0361.35003

[19] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, In: "Spectral Theory and Differential Equations ", 448, Springer Lecture Notes in Mathematics, 1975, pp. 25-70. | MR 407477 | Zbl 0315.35077

[20] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Stud. Appl. Math. 8 (1983), 93-126. | MR 759907 | Zbl 0549.34001

[21] P. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467-490. | MR 235310 | Zbl 0162.41103

[22] H.P. Mckean, Integrable systems and algebraic curves, In: "Global Analysis ", 755, Springer Lecture Notes in Mathematics, 1979, pp. 83-200. | MR 564904 | Zbl 0449.35080

[23] P.I. Naumkin - I. Shishmarev, Nonlinear Nonlocal Equations in the Theory of Waves, vol. 133, Transl. Math. Monographs, Providence, Rhode Island, 1994. | MR 1261868 | Zbl 0802.35002

[24] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York, 1983. | MR 710486 | Zbl 0516.47023

[25] J. Schiff, Zero curvature formulations of dual hierarchies, J. Math. Phys. 37 (1996), 1928-1938. | MR 1380881 | Zbl 0863.35093

[26] E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, 1993. | MR 1232192 | Zbl 0821.42001

[27] G.B. Whitham, Linear and Nonlinear Waves, J. Wiley & Sons, New York, 1980. | MR 1699025