×
Clarkson, Peter A.; Kruskal, Martin D.

New similarity reductions of the Boussinesq equation. (English) Zbl 0698.35137

J. Math. Phys. 30, No. 10, 2201-2213 (1989).
Summary: Some new similarity reductions of the Boussinesq equation, which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering, are presented. These new similarity reductions, including some new reductions to the first, second, and fourth Painlevé equations, cannot be obtained using the standard Lie group method for finding group-invariant solutions of partial differential equations; they are determined using a new and direct method that involves no group theoretical techniques.

Cited in 20 Reviews
Cited in 319 Documents

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
58J72 Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)

Keywords:

Boussinesq equation; similarity reductions; Painlevé equations
PDF BibTeX XML Cite
\textit{P. A. Clarkson} and \textit{M. D. Kruskal}, J. Math. Phys. 30, No. 10, 2201--2213 (1989; Zbl 0698.35137)
Full Text: DOI

References:

[1] Boussinesq J., Comptes Rendus 72 pp 755– (1871)
[2] Boussinesq J., J. Math. Pures Appl. 7 pp 55– (1872)
[3] DOI: 10.1017/S0305004100028887
[4] DOI: 10.1016/0370-1573(75)90018-6
[5] Zakharov V. E., Sov. Phys. JETP 38 pp 108– (1974)
[6] DOI: 10.1063/1.522460
[7] DOI: 10.1063/1.522460
[8] DOI: 10.1063/1.522460
[9] DOI: 10.1063/1.522460
[10] DOI: 10.1103/PhysRevLett.19.1095
[11] DOI: 10.1007/BF02259838 · Zbl 0555.65076
[12] DOI: 10.1137/1030094 · Zbl 0664.35004
[13] Bluman G. W., J. Math. Mech. 18 pp 1025– (1969)
[14] DOI: 10.1137/0147018 · Zbl 0621.35007
[15] DOI: 10.1137/0147018 · Zbl 0621.35007
[16] DOI: 10.1080/00411457108231446 · Zbl 0292.49008
[17] DOI: 10.1080/00411457108231446 · Zbl 0292.49008
[18] DOI: 10.1063/1.527974 · Zbl 0669.58037
[19] DOI: 10.1088/0305-4470/22/15/010 · Zbl 0694.35159
[20] DOI: 10.1016/0375-9601(82)90326-7
[21] DOI: 10.1016/0375-9601(86)90026-5
[22] DOI: 10.1103/PhysRevLett.38.1103
[23] DOI: 10.1063/1.524491 · Zbl 0445.35056
[24] DOI: 10.1063/1.524491 · Zbl 0445.35056
[25] DOI: 10.1063/1.524491 · Zbl 0445.35056
[26] DOI: 10.1016/0375-9601(82)90817-9
[27] DOI: 10.1063/1.525260 · Zbl 0504.34022
[28] DOI: 10.1063/1.525260 · Zbl 0504.34022
[29] DOI: 10.1143/JPSJ.47.1698 · Zbl 1334.35256
[30] DOI: 10.1137/0149071 · Zbl 0694.35005
[31] DOI: 10.1137/0149071 · Zbl 0694.35005
[32] DOI: 10.1137/0149071 · Zbl 0694.35005
[33] DOI: 10.1016/0022-1694(82)90016-6
[34] DOI: 10.1016/0370-1573(89)90090-2
[35] DOI: 10.1016/0370-1573(89)90090-2
[36] DOI: 10.1016/0370-1573(89)90090-2
[37] DOI: 10.1016/0370-1573(89)90090-2
[38] DOI: 10.1016/0370-1573(89)90090-2
[39] DOI: 10.1016/0370-1573(89)90090-2
[40] DOI: 10.1016/0370-1573(89)90090-2
[41] DOI: 10.1002/cpa.3160030302 · Zbl 0039.10403
[42] DOI: 10.1002/cpa.3160030302 · Zbl 0039.10403
[43] DOI: 10.1143/JPSJ.32.1681
[44] DOI: 10.1143/JPSJ.32.1681
[45] DOI: 10.1063/1.527526 · Zbl 0663.76001
[46] DOI: 10.1063/1.527526 · Zbl 0663.76001
[47] DOI: 10.1063/1.527526 · Zbl 0663.76001
[48] DOI: 10.1063/1.527526 · Zbl 0663.76001
[49] DOI: 10.1063/1.527526 · Zbl 0663.76001
[50] DOI: 10.1063/1.527526 · Zbl 0663.76001
[51] DOI: 10.1063/1.527526 · Zbl 0663.76001
[52] DOI: 10.1063/1.527526 · Zbl 0663.76001
[53] DOI: 10.1063/1.527526 · Zbl 0663.76001
[54] DOI: 10.1063/1.527526 · Zbl 0663.76001
[55] DOI: 10.1063/1.527526 · Zbl 0663.76001
[56] DOI: 10.1063/1.527526 · Zbl 0663.76001
[57] DOI: 10.1063/1.527526 · Zbl 0663.76001
[58] DOI: 10.1063/1.527526 · Zbl 0663.76001
[59] DOI: 10.1063/1.527526 · Zbl 0663.76001
[60] DOI: 10.1063/1.527526 · Zbl 0663.76001
[61] DOI: 10.1063/1.527526 · Zbl 0663.76001
[62] DOI: 10.1063/1.527526 · Zbl 0663.76001
[63] DOI: 10.1063/1.527526 · Zbl 0663.76001
[64] DOI: 10.1063/1.527526 · Zbl 0663.76001
[65] DOI: 10.1063/1.527526 · Zbl 0663.76001
[66] DOI: 10.1143/JPSJ.52.4059
[67] DOI: 10.1063/1.525752 · Zbl 0525.35022
[68] DOI: 10.1143/JPSJ.51.1678
[69] DOI: 10.1063/1.1664700 · Zbl 0283.35018
[70] DOI: 10.1143/PTP.57.797 · Zbl 1098.81547
[71] Gromak V. I., Differ. Eqs. 23 pp 506– (1987)
[72] Gromak V. I., Diff. Urav. 23 pp 760– (1988)
[73] DOI: 10.1007/BF02393131 · JFM 42.0340.03
[74] DOI: 10.1016/0378-4371(83)90022-5 · Zbl 0515.35020
[75] Tajiri M., Math. Japon. 28 pp 125– (1983)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.