### Abstract

The asymptotic behavior of the solution u(x, t) of the Korteweg-deVries equation u_{t} + uu_{x} + u_{xxx} = 0 is investigated for the class of problems where the initial data does not give rise to an associated discrete spectrum. It is shown that the behavior is different in the three regions (i) x ≫ t^{1/3}, (ii) x = 0(t^{1/3}), (iii) x ≪ - t^{1/3}. Asymptotic solutions in each of these regions are found which match at their respective boundaries. One of the Berezin-Karpman similarity solutions is the asymptotic state in (ii) and u(x, t) decays like 1/t^{2/3} in this region. For region (i) there is exponential decay, whereas in region (iii) the structure of u(x, t) is highly oscillatory and the amplitudes of the oscillations decay as 1/t^{1/2} when - x/t is of order unity. For x/t large and negative these amplitudes decay at least as 1/(- x/t)^{1/4}t^{1/2}.

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

Pages (from-to) | 1277-1284 |

Number of pages | 8 |

Journal | Journal of Mathematical Physics |

Volume | 14 |

Issue number | 9 |

State | Published - 1973 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Organic Chemistry

### Cite this

*Journal of Mathematical Physics*,

*14*(9), 1277-1284.

**The decay of the continuous spectrum for solutions of the Korteweg-deVries equation.** / Ablowitz, Mark J.; Newell, Alan C.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 14, no. 9, pp. 1277-1284.

}

TY - JOUR

T1 - The decay of the continuous spectrum for solutions of the Korteweg-deVries equation

AU - Ablowitz, Mark J.

AU - Newell, Alan C

PY - 1973

Y1 - 1973

N2 - The asymptotic behavior of the solution u(x, t) of the Korteweg-deVries equation ut + uux + uxxx = 0 is investigated for the class of problems where the initial data does not give rise to an associated discrete spectrum. It is shown that the behavior is different in the three regions (i) x ≫ t1/3, (ii) x = 0(t1/3), (iii) x ≪ - t1/3. Asymptotic solutions in each of these regions are found which match at their respective boundaries. One of the Berezin-Karpman similarity solutions is the asymptotic state in (ii) and u(x, t) decays like 1/t2/3 in this region. For region (i) there is exponential decay, whereas in region (iii) the structure of u(x, t) is highly oscillatory and the amplitudes of the oscillations decay as 1/t1/2 when - x/t is of order unity. For x/t large and negative these amplitudes decay at least as 1/(- x/t)1/4t1/2.

AB - The asymptotic behavior of the solution u(x, t) of the Korteweg-deVries equation ut + uux + uxxx = 0 is investigated for the class of problems where the initial data does not give rise to an associated discrete spectrum. It is shown that the behavior is different in the three regions (i) x ≫ t1/3, (ii) x = 0(t1/3), (iii) x ≪ - t1/3. Asymptotic solutions in each of these regions are found which match at their respective boundaries. One of the Berezin-Karpman similarity solutions is the asymptotic state in (ii) and u(x, t) decays like 1/t2/3 in this region. For region (i) there is exponential decay, whereas in region (iii) the structure of u(x, t) is highly oscillatory and the amplitudes of the oscillations decay as 1/t1/2 when - x/t is of order unity. For x/t large and negative these amplitudes decay at least as 1/(- x/t)1/4t1/2.

UR - http://www.scopus.com/inward/record.url?scp=0009065822&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0009065822&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0009065822

VL - 14

SP - 1277

EP - 1284

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 9

ER -