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The Cauchy problem of a shallow water equation. (English) Zbl 1095.35041
The authors consider the initial value problem for a fifth order shallow water equation in the nonperiodic case, as opposed to the previously considered periodic case. The equation studied has the same nonlinear terms as the well-known Fuchssteiner-Fokas-Camassa-Holm equation, and in this sense can be considered to be a higher order analogue of this last.
35Q53KdV-like (Korteweg-de Vries) equations
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
Full Text: DOI
[1] Fokas, A. S., Fuchssteiner, B.: Symplectic structures, their Bäcklund transformation and hereditary symmetries. Phys. D., 4, 47--66 (1981--1982) · Zbl 1194.37114
[2] Camassa, R., Holm, D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Letters, 17, 1661--1664 (1993) · Zbl 0936.35153 · doi:10.1103/PhysRevLett.71.1661
[3] Constantin, A., Escher, J.: Well--posedness, global existence, and blowup phenomena for a periodic quasilinear hyperbolic equation. Comm. Pure Appl. Math., 51, 475--504 (1998) · Zbl 0934.35153 · doi:10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5
[4] Constantin, A., Escher, J.: Global weak solutions for a shallow water equation. Indiana Univ. Math. J., 47, 1525--1545 (1998) · Zbl 0930.35133 · doi:10.1512/iumj.1998.47.1466
[5] Constantin, A., Escher, J.: On the structure of a family of quasilinear equations arising in shallow water theory. Math. Ann., 312, 403--416 (1998) · Zbl 0923.76028 · doi:10.1007/s002080050228
[6] Himonas, A. A., Misiolek, G.: The Cauchy problem for a shallow water type equation. Comm. PDE, 23(2), 123--139 (1998) · Zbl 0895.35021 · doi:10.1080/03605309808821340
[7] Bourgain, J.: Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I, Schrödinger equations. GAFA, 3, 107--156 (1993) · Zbl 0787.35097 · doi:10.1007/BF01896020
[8] Bourgain, J.: Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations II, The Periodic KdV Equation. GAFA, 3, 209--262 (1993) · Zbl 0787.35098 · doi:10.1007/BF01895688
[9] Kenig, C. E., Ponce, G., Vega. L.: The Cauchy problem for the Korteweg--de Vries equation in Sobolev spaces of negative indices. Duke Math. J., 71, 1--21 (1993) · Zbl 0787.35090 · doi:10.1215/S0012-7094-93-07101-3
[10] Kenig, C. E., Ponce, G., Vega. L.: A bilinear estimate with applications to the KdV equation. J. Amer. Math., 9(2), 573--603 (1996) · Zbl 0848.35114 · doi:10.1090/S0894-0347-96-00200-7