### Abstract

Solutions of the "magma equation", f_{t}= [f ^{n}"{(f^{-m}f_{t})_{x}-1}]_{x}, are investigated for compact support initial conditions. It is shown analytically that traveling wave solutions with compact support do not exist for this equation. Numerical integration demonstrates that although initial conditions with compact support produce solitary waves, these solitary waves do not interact like solitons. Moreover, a relationship between the magma equation and the KdV equation is established, and this relationship is used to explain previously observed soliton-like behavior. Both equations are shown to be related to the modified magma equation, F_{t}+ [F^{3}(3F _{XX}+1)]_{x}=0, which is proven to have N-soliton solutions with compact support and an infinite number of conserved densities.

Original language | English |
---|---|

Pages (from-to) | 1941-1953 |

Number of pages | 13 |

Journal | Journal of the Physical Society of Japan |

Volume | 59 |

Issue number | 6 |

Publication status | Published - 1990 Jun |

Externally published | Yes |

### Fingerprint

### Keywords

- Compact support solution
- Integrable
- KdV equation
- Magma equation
- Soliton

### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Journal of the Physical Society of Japan*,

*59*(6), 1941-1953.

**Properties of the magma and modified magma equations.** / Takahashi, Daisuke; Sachs, Jeffrey R.; Satsuma, Junkichi.

Research output: Contribution to journal › Article

*Journal of the Physical Society of Japan*, vol. 59, no. 6, pp. 1941-1953.

}

TY - JOUR

T1 - Properties of the magma and modified magma equations

AU - Takahashi, Daisuke

AU - Sachs, Jeffrey R.

AU - Satsuma, Junkichi

PY - 1990/6

Y1 - 1990/6

N2 - Solutions of the "magma equation", ft= [f n"{(f-mft)x-1}]x, are investigated for compact support initial conditions. It is shown analytically that traveling wave solutions with compact support do not exist for this equation. Numerical integration demonstrates that although initial conditions with compact support produce solitary waves, these solitary waves do not interact like solitons. Moreover, a relationship between the magma equation and the KdV equation is established, and this relationship is used to explain previously observed soliton-like behavior. Both equations are shown to be related to the modified magma equation, Ft+ [F3(3F XX+1)]x=0, which is proven to have N-soliton solutions with compact support and an infinite number of conserved densities.

AB - Solutions of the "magma equation", ft= [f n"{(f-mft)x-1}]x, are investigated for compact support initial conditions. It is shown analytically that traveling wave solutions with compact support do not exist for this equation. Numerical integration demonstrates that although initial conditions with compact support produce solitary waves, these solitary waves do not interact like solitons. Moreover, a relationship between the magma equation and the KdV equation is established, and this relationship is used to explain previously observed soliton-like behavior. Both equations are shown to be related to the modified magma equation, Ft+ [F3(3F XX+1)]x=0, which is proven to have N-soliton solutions with compact support and an infinite number of conserved densities.

KW - Compact support solution

KW - Integrable

KW - KdV equation

KW - Magma equation

KW - Soliton

UR - http://www.scopus.com/inward/record.url?scp=0039066125&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0039066125&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0039066125

VL - 59

SP - 1941

EP - 1953

JO - Journal of the Physical Society of Japan

JF - Journal of the Physical Society of Japan

SN - 0031-9015

IS - 6

ER -