### Abstract

A new approach for finding the class of integrable evolution equations associated with a given eigenvalue problem is developed. The key point to note is that the squares of the eigenfunctions form a natural basis in which to expand the solutions of the evolution equation. Once this step is taken, the class of integrable equations may usually be read off by inspection. Of particular interest are those equations for which the bound state eigenvalues are not invariant but move in a way prescribed by the coefficients of the evolution equation. The corresponding solitons have the property that they retain their identity on collision with other solution components, but this identity is no longer a constant one. The Hamiltonian structure and the causality properties of these systems are also explored.

Original language | English (US) |
---|---|

Pages (from-to) | 67-100 |

Number of pages | 34 |

Journal | Advances in Mathematics |

Volume | 31 |

Issue number | 1 |

DOIs | |

State | Published - 1979 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Evolution equations, singular dispersion relations, and moving eigenvalues.** / Kaup, David J.; Newell, Alan C.

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 31, no. 1, pp. 67-100. https://doi.org/10.1016/0001-8708(79)90021-5

}

TY - JOUR

T1 - Evolution equations, singular dispersion relations, and moving eigenvalues

AU - Kaup, David J.

AU - Newell, Alan C

PY - 1979

Y1 - 1979

N2 - A new approach for finding the class of integrable evolution equations associated with a given eigenvalue problem is developed. The key point to note is that the squares of the eigenfunctions form a natural basis in which to expand the solutions of the evolution equation. Once this step is taken, the class of integrable equations may usually be read off by inspection. Of particular interest are those equations for which the bound state eigenvalues are not invariant but move in a way prescribed by the coefficients of the evolution equation. The corresponding solitons have the property that they retain their identity on collision with other solution components, but this identity is no longer a constant one. The Hamiltonian structure and the causality properties of these systems are also explored.

AB - A new approach for finding the class of integrable evolution equations associated with a given eigenvalue problem is developed. The key point to note is that the squares of the eigenfunctions form a natural basis in which to expand the solutions of the evolution equation. Once this step is taken, the class of integrable equations may usually be read off by inspection. Of particular interest are those equations for which the bound state eigenvalues are not invariant but move in a way prescribed by the coefficients of the evolution equation. The corresponding solitons have the property that they retain their identity on collision with other solution components, but this identity is no longer a constant one. The Hamiltonian structure and the causality properties of these systems are also explored.

UR - http://www.scopus.com/inward/record.url?scp=0002592333&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0002592333&partnerID=8YFLogxK

U2 - 10.1016/0001-8708(79)90021-5

DO - 10.1016/0001-8708(79)90021-5

M3 - Article

AN - SCOPUS:0002592333

VL - 31

SP - 67

EP - 100

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 1

ER -