The Cauchy problem of a shallow water equation. (English) Zbl 1095.35041

Liu, Xiaofeng; Jin, Yongyang

Acta Math. Sin., Engl. Ser. 21, No. 2, 393-408 (2005).

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.

Reviewer: Andrew Pickering (Madrid)

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
76B15	Water waves, gravity waves; dispersion and scattering, nonlinear interaction

Keywords:

wellposedness; nonperiodic shallow water equation; Camassa-Holm equation

References:

[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
[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
[4]	Constantin, A., Escher, J.: Global weak solutions for a shallow water equation. Indiana Univ. Math. J., 47, 1525–1545 (1998) · Zbl 0930.35133
[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
[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
[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
[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
[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
[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