journal article Jan 01, 1998

Wave breaking for nonlinear nonlocal shallow water equations

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References
35
[1]
Alber, M. S., Camassa, R., Holm, D. &Marsden, J. E., The geometry of peaked solitons and billiard solutions of a class of integrable PDE's.Lett. Math. Phys., 32 (1994), 137–151. 10.1007/bf00739423
[2]
Alinhac, S.,Blow-up for Nonlinear Hyperbolic Equations. Progr. Nonlinear Differential Equations Appl., 17. Birkhäuser Boston, Boston, MA, 1995. 10.1007/978-1-4612-2578-2
[3]
—, Blow-up of classical solutions of nonlinear hyperbolic equations: a survey of recent results, inPartial Differential Equations and Mathematical Physics (Copenhagen, 1995; Lund, 1995), pp. 15–24. Progr. Nonlinear Differential Equations Appl., 21, Birkhäuser Boston, Boston, MA, 1996. 10.1007/978-1-4612-0775-7_2
[4]
Boyd, J. P., Peakons and coshoidal waves: traveling wave solutions of the Camassa-Holm equation,Appl. Math. Comput., 81 (1997), 173–187. 10.1016/0096-3003(95)00326-6
[5]
Caffarelli, L. &Friedman, A., Differentiability of the blow-up curve for one-dimensional nonlinear wave equations.Arch. Rational Mech. Anal., 91 (1985), 83–98. 10.1007/bf00280224
[6]
Calogero, F. &Françoise, J.-P., A completely integrable Hamiltonian system.J. Math. Phys., 37 (1996), 2863–2871. 10.1063/1.531536
[7]
Camassa, R. &Holm, D., An integrable shallow water equation with peaked solitions.Phys. Rev. Lett., 71 (1993), 1661–1664. 10.1103/physrevlett.71.1661
[8]
Camassa, R., Holm, D. &Hyman, J., A new integrable shallow water equation.Adv. in Appl. Mech., 31 (1994), 1–33. 10.1016/s0065-2156(08)70254-0
[9]
Constantin, A. &Escher, J., Global existence and blow-up for a shallow water equation.Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 303–328.
[10]
Constantin, A. & McKean, H. P., A shallow water equation on the circle. To appear inComm. Pure Appl. Math.
[11]
Cooper, F. &Shepard, H., Solitons in the Camassa-Holm shallow water equation.Phys. Lett. A, 194 (1994), 246–250. 10.1016/0375-9601(94)91246-7
[12]
Dieudonné, J.,Foundations of Modern Analysis. Academic Press, New York, 1969.
[13]
Dodd, R. K., Eilbeck, J. C., Gibbon, J. D. &Morris, H. C.,Solitons and Nonlinear Wave Equations. Academic Press, New York, 1984.
[14]
Evans, L. &Gariepy, R.,Measure Theory and Fine Properties of Functions. Stud. Adv. Math. CRC Press, Boca Raton, FL, 1992.
[15]
Symplectic structures, their Bäcklund transformations and hereditary symmetries

B. Fuchssteiner, A.S. Fokas

Physica D: Nonlinear Phenomena 1981 10.1016/0167-2789(81)90004-x
[16]
Fuchssteiner, B., Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation.Phys. D, 95 (1996), 229–243. 10.1016/0167-2789(96)00048-6
[17]
Glassey, R., Finite-time blow-up for solutions of nonlinear wave equations,Math. Z., 177 (1981), 323–340. 10.1007/bf01162066
[18]
Hörmander L.,Lectures on Nonlinear Hyperbolic Differential Equations. Mathématiques & Applications, 26. Springer-Verlag, Berlin, 1997.
[19]
—, The lifespan of classical solutions of nonlinear hyperbolic equations, inPseudo-Differential Operators (Oberwolfach, 1986), pp. 214–280. Lecture Notes in Math., 1256. Springer-Verlag, Berlin-New York, 1987. 10.1007/bfb0077745
[20]
John, F., Formation of singularities in one-dimensional nonlinear wave propagation.Comm. Pure Appl. Math., 27 (1974), 337–405. 10.1002/cpa.3160270307
[21]
—,Nonlinear Wave Equations—Formation of Singularities. Univ. Lecture Ser., Amer. Math. Soc., Providence, RI, 1990. 10.1090/ulect/002
[22]
Kato, T., Blow-up solutions of some nonlinear hyperbolic equations.Comm. Pure Appl. Math., 33 (1980), 501–505. 10.1002/cpa.3160330403
[23]
—, Quasi-linear equations of evolution, with applications to partial differential equations, inSpectral Theory and Differential Equations (Dundee, 1974), pp. 25–70. Lecture Notes in Math., 448. Springer-Verlag, Berlin-New York, 1975. 10.1007/bfb0067080
[24]
Kenig, C., Ponce, G. &Vega, L., Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle.Comm. Pure Appl. Math., 46 (1993), 527–620. 10.1002/cpa.3160460405
[25]
Klainerman, S. &Majda, A., Formation of singularities for wave equations including the nonlinear vibrating string.Comm. Pure Appl. Math., 33 (1980), 241–263. 10.1002/cpa.3160330304
[26]
Lax, P., Development of singularities of solutions of nonlinear hyperbolic partial differential equations.J. Math. Phys., 5 (1964), 611–613. 10.1063/1.1704154
[27]
Liu, P., Development of singularities in the nonlinear waves for quasilinear hyperbolic partial differential equations.J. Differential Equations, 33 (1979), 92–111. 10.1016/0022-0396(79)90082-2
[28]
McKean, H. P., Integrable systems and algebraic curves, inGlobal Analysis (Calgary, 1978), pp. 83–200. Lecture Notes in Math., 755. Springer-Verlag, Berlin, 1979. 10.1007/bfb0069806
[29]
Naumkin, P. &Shishmarev, I.,Nonlinear Nonlocal Equations in the Theory of Waves. Transl. Math. Monographs, 133. Amer. Math. Soc., Providence, RI, 1994. 10.1090/mmono/133
[30]
Ostrowski, A.,Vorlesungen über Differential- und Integralrechnung, Vol. III. Birkhäuser, Basel-Stuttgart, 1954.
[31]
Schiff, J., Zero curvature formulations of dual hierarchies.J. Math. Phys., 37 (1996), 1928–1938. 10.1063/1.531486
[32]
Seliger, R., A note on the breaking of waves.Proc. Roy. Soc. Ser. A, 303 (1968), 493–496.
[33]
Sideris, T. C., Formation of singularities in solutions to nonlinear hyperbolic equations.Arch. Rational Mech. Anal., 86 (1984), 369–381. 10.1007/bf00280033
[34]
Strauss, W.,Nonlinear Wave Equations. CBMS Regional Conf. Ser. in Math., 73. Amer. Math. Soc., Providence, RI, 1989.
[35]
Whitham, G. B.,Linear and Nonlinear Waves. J. Wiley & Sons, New York, 1980.
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1,200
Citations
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References
Details
Published
Jan 01, 1998
Vol/Issue
181(2)
Pages
229-243
Cite This Article
Adrian Constantin, Joachim Escher (1998). Wave breaking for nonlinear nonlocal shallow water equations. Acta Mathematica, 181(2), 229-243. https://doi.org/10.1007/bf02392586
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