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The nonlinear Schrödinger equation and the nonlinear heat equation reduction to linear form

The nonlinear Schrödinger equation and the nonlinear heat equation reduction to linear form This paper is dedicated to Natascha A. Brunswick. 1. Introduction Let X = f ( x ) be a nonlinear flow in n = 1, 2, 3, . . * < co dimensions, with resting point at the origin: f ( 0) = 0. Its reduction to the associated linear flow X = dF(O)x, by local change of coordinates, is the subject of a large literature: see notably POINCARE [ 18791, SIEGEL [ 19521, HARTMAN [ 19641, and NIKOLENKO [ 19861. McKEAN-SHATAH [ 199I] used familiar partial differential equations to illustrate how tricky it may be to extend the classical facts and/or their proofs to n = co dimensions. The present paper confirms the (local) reducibility of two co-dimensional nonlinear flows: ( 1 ) Schrodinger, and (2) heat, in d = 1,2, 3, - . - < 00 spatial dimensions. The nonlinear Schrodinger flow * : (1) Ilr-raulat = AU + 1U is reducible if ( 1 ' ) d q> 1 and ( 1") either q 2 [ d / 21 As to the nonlinear heat flow: + 5 , or else q is a whole number. it is reducible if (2') p is a whole number http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Communications on Pure & Applied Mathematics Wiley

The nonlinear Schrödinger equation and the nonlinear heat equation reduction to linear form

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References (7)

Publisher
Wiley
Copyright
Copyright © 1991 Wiley Periodicals, Inc., A Wiley Company
ISSN
0010-3640
eISSN
1097-0312
DOI
10.1002/cpa.3160440817
Publisher site
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Abstract

This paper is dedicated to Natascha A. Brunswick. 1. Introduction Let X = f ( x ) be a nonlinear flow in n = 1, 2, 3, . . * < co dimensions, with resting point at the origin: f ( 0) = 0. Its reduction to the associated linear flow X = dF(O)x, by local change of coordinates, is the subject of a large literature: see notably POINCARE [ 18791, SIEGEL [ 19521, HARTMAN [ 19641, and NIKOLENKO [ 19861. McKEAN-SHATAH [ 199I] used familiar partial differential equations to illustrate how tricky it may be to extend the classical facts and/or their proofs to n = co dimensions. The present paper confirms the (local) reducibility of two co-dimensional nonlinear flows: ( 1 ) Schrodinger, and (2) heat, in d = 1,2, 3, - . - < 00 spatial dimensions. The nonlinear Schrodinger flow * : (1) Ilr-raulat = AU + 1U is reducible if ( 1 ' ) d q> 1 and ( 1") either q 2 [ d / 21 As to the nonlinear heat flow: + 5 , or else q is a whole number. it is reducible if (2') p is a whole number

Journal

Communications on Pure & Applied Mathematics – Wiley

Published: Oct 1, 1991

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