Bäcklund transformation, conservation laws, and inverse scattering transform of a model integrodifferential equation for water waves. (English) Zbl 0729.45013
The author studies a nonlinear integro-partial differential equation in one space dimension, and time. The equation which has been introduced by the author in an earlier paper is supposed to interpolate between models describing deep and shallow water waves (keyword: solitons).
The purpose of the paper is to construct a Bäcklund transformation, an infinite number of conservation laws, and the scattering inverse of the equation. This formidable task is accomplished within Hirota’s bilinear differential calculus.
The author further shows that his model passes the litmus test for integrability, namely the Painlevé test. On passing to the limit, he obtains the conservation laws for the deep and the shallow water limits of his model. Open questions are presented.
The purpose of the paper is to construct a Bäcklund transformation, an infinite number of conservation laws, and the scattering inverse of the equation. This formidable task is accomplished within Hirota’s bilinear differential calculus.
The author further shows that his model passes the litmus test for integrability, namely the Painlevé test. On passing to the limit, he obtains the conservation laws for the deep and the shallow water limits of his model. Open questions are presented.
Reviewer: G.Leugering (Washington)
MSC:
| 45K05 | Integro-partial differential equations |
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
Keywords:
inverse scattering transform; nonlinear integro-partial differential equation; shallow water waves; solitons; Bäcklund transformation; conservation laws; Hirota’s bilinear differential calculus; Painlevé testReferences:
| [1] | DOI: 10.1063/1.528575 · Zbl 0709.76573 · doi:10.1063/1.528575 |
| [2] | DOI: 10.1063/1.528575 · Zbl 0709.76573 · doi:10.1063/1.528575 |
| [3] | DOI: 10.1143/JPSJ.40.611 · Zbl 1334.76016 · doi:10.1143/JPSJ.40.611 |
| [4] | DOI: 10.1088/0305-4470/10/12/002 · doi:10.1088/0305-4470/10/12/002 |
| [5] | DOI: 10.2514/3.63127 · doi:10.2514/3.63127 |
| [6] | DOI: 10.1017/S002211206700103X · Zbl 0147.46502 · doi:10.1017/S002211206700103X |
| [7] | DOI: 10.1143/JPSJ.39.1082 · Zbl 1334.76027 · doi:10.1143/JPSJ.39.1082 |
| [8] | DOI: 10.1016/0375-9601(79)90779-5 · doi:10.1016/0375-9601(79)90779-5 |
| [9] | DOI: 10.1016/0375-9601(79)90534-6 · doi:10.1016/0375-9601(79)90534-6 |
| [10] | DOI: 10.1063/1.525393 · Zbl 0489.35004 · doi:10.1063/1.525393 |
| [11] | DOI: 10.1103/PhysRevLett.27.1192 · Zbl 1168.35423 · doi:10.1103/PhysRevLett.27.1192 |
| [12] | DOI: 10.1143/JPSJ.57.1577 · doi:10.1143/JPSJ.57.1577 |
| [13] | DOI: 10.1143/JPSJ.58.1948 · doi:10.1143/JPSJ.58.1948 |
| [14] | DOI: 10.1063/1.524548 · Zbl 0445.35057 · doi:10.1063/1.524548 |
| [15] | DOI: 10.1063/1.524548 · Zbl 0445.35057 · doi:10.1063/1.524548 |
| [16] | DOI: 10.1063/1.525721 · Zbl 0514.35083 · doi:10.1063/1.525721 |
| [17] | DOI: 10.1063/1.526841 · Zbl 0588.35020 · doi:10.1063/1.526841 |
| [18] | DOI: 10.1063/1.526841 · Zbl 0588.35020 · doi:10.1063/1.526841 |
| [19] | DOI: 10.1063/1.526841 · Zbl 0588.35020 · doi:10.1063/1.526841 |
| [20] | DOI: 10.1063/1.526841 · Zbl 0588.35020 · doi:10.1063/1.526841 |
| [21] | DOI: 10.1063/1.526841 · Zbl 0588.35020 · doi:10.1063/1.526841 |
| [22] | DOI: 10.1063/1.526841 · Zbl 0588.35020 · doi:10.1063/1.526841 |
| [23] | DOI: 10.1016/0370-1573(89)90024-0 · doi:10.1016/0370-1573(89)90024-0 |
| [24] | DOI: 10.1103/PhysRevLett.53.1 · doi:10.1103/PhysRevLett.53.1 |
| [25] | DOI: 10.1143/JPSJ.51.667 · doi:10.1143/JPSJ.51.667 |
| [26] | DOI: 10.1143/JPSJ.51.3375 · doi:10.1143/JPSJ.51.3375 |
| [27] | DOI: 10.1088/0305-4470/12/4/019 · Zbl 0399.45014 · doi:10.1088/0305-4470/12/4/019 |
| [28] | DOI: 10.1098/rspa.1976.0149 · Zbl 0346.35024 · doi:10.1098/rspa.1976.0149 |
| [29] | DOI: 10.1143/JPSJ.43.692 · Zbl 1334.81041 · doi:10.1143/JPSJ.43.692 |
| [30] | DOI: 10.1002/sapm1980623189 · Zbl 0431.35073 · doi:10.1002/sapm1980623189 |
| [31] | DOI: 10.1143/PTP.51.1355 · Zbl 1125.35400 · doi:10.1143/PTP.51.1355 |
| [32] | DOI: 10.1016/0375-9601(80)90343-6 · doi:10.1016/0375-9601(80)90343-6 |
| [33] | DOI: 10.1143/JPSJ.58.2285 · doi:10.1143/JPSJ.58.2285 |
| [34] | DOI: 10.1143/JPSJ.58.2705 · doi:10.1143/JPSJ.58.2705 |
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