Solutions of mKdV in classes of functions unbounded at infinity

T. Kappeler; P. Perry; M. Shubin; P. Topalov

doi:10.1007/s12220-008-9013-3

Solutions of mKdV in classes of functions unbounded at infinity

T. Kappeler, P. Perry, M. Shubin, P. Topalov

Mathematics

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

Abstract

Using P. Lax's concept of a Lax pair we prove global existence and uniqueness for solutions of the initial value problem for mKdV in classes of smooth functions which can be unbounded at infinity, and in particular, may tend to infinity with respect to the space variable. Moreover, we establish the invariance of the spectrum and the unitary type of the Schrödinger operator under the KdV flow and the invariance of the spectrum and the unitary type of the impedance operator under the mKdV flow for potentials in these classes.

Original language	English
Pages (from-to)	443-477
Number of pages	35
Journal	Journal of Geometric Analysis
Volume	18
Issue number	2
DOIs	https://doi.org/10.1007/s12220-008-9013-3
State	Published - Apr 2008

Bibliographical note

Funding Information:
T. Kappeler supported in part by the Swiss National Science Foundation, the programme SPECT, and the European Community through the FP6 Marie Curie RTN ENIGMA (MRTN-CT-2004-5652). P. Perry partially supported by NSF-grant DMS-0408419. M. Shubin partially supported by NSF-grant DMS-0600196.

Keywords

KdV
Modified KdV
Spectra of Schrödinger operators

ASJC Scopus subject areas

Geometry and Topology

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10.1007/s12220-008-9013-3

Cite this

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title = "Solutions of mKdV in classes of functions unbounded at infinity",

abstract = "Using P. Lax's concept of a Lax pair we prove global existence and uniqueness for solutions of the initial value problem for mKdV in classes of smooth functions which can be unbounded at infinity, and in particular, may tend to infinity with respect to the space variable. Moreover, we establish the invariance of the spectrum and the unitary type of the Schr{\"o}dinger operator under the KdV flow and the invariance of the spectrum and the unitary type of the impedance operator under the mKdV flow for potentials in these classes.",

keywords = "KdV, Modified KdV, Spectra of Schr{\"o}dinger operators",

author = "T. Kappeler and P. Perry and M. Shubin and P. Topalov",

note = "Funding Information: T. Kappeler supported in part by the Swiss National Science Foundation, the programme SPECT, and the European Community through the FP6 Marie Curie RTN ENIGMA (MRTN-CT-2004-5652). P. Perry partially supported by NSF-grant DMS-0408419. M. Shubin partially supported by NSF-grant DMS-0600196.",

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language = "English",

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T1 - Solutions of mKdV in classes of functions unbounded at infinity

AU - Kappeler, T.

AU - Perry, P.

AU - Shubin, M.

AU - Topalov, P.

N1 - Funding Information: T. Kappeler supported in part by the Swiss National Science Foundation, the programme SPECT, and the European Community through the FP6 Marie Curie RTN ENIGMA (MRTN-CT-2004-5652). P. Perry partially supported by NSF-grant DMS-0408419. M. Shubin partially supported by NSF-grant DMS-0600196.

PY - 2008/4

Y1 - 2008/4

N2 - Using P. Lax's concept of a Lax pair we prove global existence and uniqueness for solutions of the initial value problem for mKdV in classes of smooth functions which can be unbounded at infinity, and in particular, may tend to infinity with respect to the space variable. Moreover, we establish the invariance of the spectrum and the unitary type of the Schrödinger operator under the KdV flow and the invariance of the spectrum and the unitary type of the impedance operator under the mKdV flow for potentials in these classes.

AB - Using P. Lax's concept of a Lax pair we prove global existence and uniqueness for solutions of the initial value problem for mKdV in classes of smooth functions which can be unbounded at infinity, and in particular, may tend to infinity with respect to the space variable. Moreover, we establish the invariance of the spectrum and the unitary type of the Schrödinger operator under the KdV flow and the invariance of the spectrum and the unitary type of the impedance operator under the mKdV flow for potentials in these classes.

KW - KdV

KW - Modified KdV

KW - Spectra of Schrödinger operators

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U2 - 10.1007/s12220-008-9013-3

DO - 10.1007/s12220-008-9013-3

M3 - Article

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VL - 18

SP - 443

EP - 477

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

IS - 2

ER -

Solutions of mKdV in classes of functions unbounded at infinity

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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