### Abstract

The moment formulas that globally characterize the zero-dispersion limit of the Korteweg-deVries (KdV) equation are known to be expressed in terms of the solution of a maximization problem. Here we establish a direct relation between this maximizer and the zero-dispersion limit of the logarithm of the Jost functions associated with the inverse spectral transform. All the KdV conserved densities are encoded in the spatial derivative of these functions, known as Weyl functions. We show the Weyl functions are densities of measures that converge in the weak sense to a limiting measure. This limiting measure encodes all of the weak limits of the KdV conserved densities. Moreover, we establish the weak limit of spectral measures associated with the Dirichlet problem.

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

Pages (from-to) | 119-143 |

Number of pages | 25 |

Journal | Communications in Mathematical Physics |

Volume | 183 |

Issue number | 1 |

State | Published - 1997 |

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### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics

### Cite this

*Communications in Mathematical Physics*,

*183*(1), 119-143.

**The behavior of the Weyl function in the zero-dispersion KdV limit.** / Ercolani, Nicholas M; Levermore, C. David; Zhang, Taiyan.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 183, no. 1, pp. 119-143.

}

TY - JOUR

T1 - The behavior of the Weyl function in the zero-dispersion KdV limit

AU - Ercolani, Nicholas M

AU - Levermore, C. David

AU - Zhang, Taiyan

PY - 1997

Y1 - 1997

N2 - The moment formulas that globally characterize the zero-dispersion limit of the Korteweg-deVries (KdV) equation are known to be expressed in terms of the solution of a maximization problem. Here we establish a direct relation between this maximizer and the zero-dispersion limit of the logarithm of the Jost functions associated with the inverse spectral transform. All the KdV conserved densities are encoded in the spatial derivative of these functions, known as Weyl functions. We show the Weyl functions are densities of measures that converge in the weak sense to a limiting measure. This limiting measure encodes all of the weak limits of the KdV conserved densities. Moreover, we establish the weak limit of spectral measures associated with the Dirichlet problem.

AB - The moment formulas that globally characterize the zero-dispersion limit of the Korteweg-deVries (KdV) equation are known to be expressed in terms of the solution of a maximization problem. Here we establish a direct relation between this maximizer and the zero-dispersion limit of the logarithm of the Jost functions associated with the inverse spectral transform. All the KdV conserved densities are encoded in the spatial derivative of these functions, known as Weyl functions. We show the Weyl functions are densities of measures that converge in the weak sense to a limiting measure. This limiting measure encodes all of the weak limits of the KdV conserved densities. Moreover, we establish the weak limit of spectral measures associated with the Dirichlet problem.

UR - http://www.scopus.com/inward/record.url?scp=0031540140&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031540140&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0031540140

VL - 183

SP - 119

EP - 143

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -