### Abstract

The Bäcklund transformation provides a mathematical tool which displays the interaction of solitons. Here a simple, systematic Bäcklund formalism is introduced which permits the explicit construction of these transformations for a restricted class of nonlinear wave equations. Traditionally a Bäcklund transformation has been viewed as a transformation of a solution surface of a partial differential equation into another surface which may not satisfy the same equation. In the present paper the term "restricted Bäcklund transformation" (hereafter abbreviated R-B) is used to refer to the case in which the transformed surface does satisfy the same equation. This formalism clarifies the nature of those transformations which have already been used to study nonlinear interactions in many physical problems. The formalism is introduced through a form of the linear Klein-Gordon equation. For this linear example a complete set of Fourier components is generated by a sequence of R-B transformations. This concrete example also indicates the type of results one can expect in the nonlinear case. For the nonlinear equation φ_{xy} = F(φ), a theorem is established which states that R-B transformations exist if and only if the nonlinearity F(·) satisfies F″ = κF, where κ is a constant. For such nonlinearities, the R-B transformations are explicitly constructed and are used to display exact nonlinear interactions. A relationship between the condition F″ = κF, the existence of an infinite number of conservation laws, and the transformation theory is briefly discussed.

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

Pages (from-to) | 1817-1828 |

Number of pages | 12 |

Journal | Journal of Mathematical Physics |

Volume | 14 |

Issue number | 12 |

State | Published - 1973 |

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### ASJC Scopus subject areas

- Organic Chemistry

### Cite this

*Journal of Mathematical Physics*,

*14*(12), 1817-1828.

**A restricted Bäcklund transformation.** / McLaughlin, David W.; Scott, Alwyn C.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 14, no. 12, pp. 1817-1828.

}

TY - JOUR

T1 - A restricted Bäcklund transformation

AU - McLaughlin, David W.

AU - Scott, Alwyn C.

PY - 1973

Y1 - 1973

N2 - The Bäcklund transformation provides a mathematical tool which displays the interaction of solitons. Here a simple, systematic Bäcklund formalism is introduced which permits the explicit construction of these transformations for a restricted class of nonlinear wave equations. Traditionally a Bäcklund transformation has been viewed as a transformation of a solution surface of a partial differential equation into another surface which may not satisfy the same equation. In the present paper the term "restricted Bäcklund transformation" (hereafter abbreviated R-B) is used to refer to the case in which the transformed surface does satisfy the same equation. This formalism clarifies the nature of those transformations which have already been used to study nonlinear interactions in many physical problems. The formalism is introduced through a form of the linear Klein-Gordon equation. For this linear example a complete set of Fourier components is generated by a sequence of R-B transformations. This concrete example also indicates the type of results one can expect in the nonlinear case. For the nonlinear equation φxy = F(φ), a theorem is established which states that R-B transformations exist if and only if the nonlinearity F(·) satisfies F″ = κF, where κ is a constant. For such nonlinearities, the R-B transformations are explicitly constructed and are used to display exact nonlinear interactions. A relationship between the condition F″ = κF, the existence of an infinite number of conservation laws, and the transformation theory is briefly discussed.

AB - The Bäcklund transformation provides a mathematical tool which displays the interaction of solitons. Here a simple, systematic Bäcklund formalism is introduced which permits the explicit construction of these transformations for a restricted class of nonlinear wave equations. Traditionally a Bäcklund transformation has been viewed as a transformation of a solution surface of a partial differential equation into another surface which may not satisfy the same equation. In the present paper the term "restricted Bäcklund transformation" (hereafter abbreviated R-B) is used to refer to the case in which the transformed surface does satisfy the same equation. This formalism clarifies the nature of those transformations which have already been used to study nonlinear interactions in many physical problems. The formalism is introduced through a form of the linear Klein-Gordon equation. For this linear example a complete set of Fourier components is generated by a sequence of R-B transformations. This concrete example also indicates the type of results one can expect in the nonlinear case. For the nonlinear equation φxy = F(φ), a theorem is established which states that R-B transformations exist if and only if the nonlinearity F(·) satisfies F″ = κF, where κ is a constant. For such nonlinearities, the R-B transformations are explicitly constructed and are used to display exact nonlinear interactions. A relationship between the condition F″ = κF, the existence of an infinite number of conservation laws, and the transformation theory is briefly discussed.

UR - http://www.scopus.com/inward/record.url?scp=0002909909&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0002909909&partnerID=8YFLogxK

M3 - Article

VL - 14

SP - 1817

EP - 1828

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 12

ER -