The nonlinear Schrödinger equation and the nonlinear heat equation reduction to linear form. (English) Zbl 0773.35075

Let \(\dot x=f(x)\) be a nonlinear flow in \(n=1,2,3,\dots<\infty\) dimensions, with resting point at the origin: \(f(0)=0\). Its reduction to the associated linear flow \(\dot x=dF(0)x\), by local change of coordinates, is the subject of a large literature. The authors [How real is resonance (to appear)] used familiar partial differential equations to illustrate how tricky it may be to extend the classical facts and/or their proofs to \(n=\infty\) dimensions.
The present paper confirms the (local) reducibility of two \(\infty\)- dimensional nonlinear flows: (1) Schrödinger, and (2) heat, in \(d=1,2,3,\dots<\infty\) spatial dimensions. The nonlinear Schrödinger flow: \[ \sqrt {-1} \partial u/\partial t=\Delta u+| u|^{2q} u\tag{1} \] is reducible if \((1') dq>1\) and \((1'')\) either \(q\geq [d/2]+{1\over 2}\), or else \(q\) is a whole number. As to the nonlinear heat flow: \[ \partial u/\partial t=\Delta u+u^ p,\tag{2} \] it is reducible if \((2')\) \(p\) is a whole number \(\geq 4\) for \(d=1\), \(\geq 3\) for \(d=2\) or 3, and \(\geq 2\) in higher dimensions.


35Q55 NLS equations (nonlinear Schrödinger equations)
35K55 Nonlinear parabolic equations
37C10 Dynamics induced by flows and semiflows
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[1] Ordinary Differential Equations, J. Wiley & Sons, New York, 1964.
[2] Klainerman, Comm. Pure Appl. Math. 40 pp 11– (1987)
[3] and , How real is resonance?, to appear.
[4] Morawetz, Comm. Pure Appl. Math. 25 pp 1– (1973)
[5] Moser, Ann. Scuola Normale Sup. di Pisa 20 pp 265– (1966)
[6] Nikolenko, Russian Math. Surveys 41 pp 63– (1986)
[7] Nirenberg, Ann. Scuola Normale Sup. di Pisa 13 pp 115– (1959)
[8] Thèse, 1879, Oeuvres 1, Paris, 1928.
[9] Simon, Schrödinger semigroups, BAMS 7 pp 447– (1982) · Zbl 0524.35002
[10] Über die Normalform analytischer Differentialgleichungen in der Nähe einer Gleich- gewichtlösung, Akad. Wiss. Göttingen, 1952, pp. 21–30.
[11] Nonlinear invariant wave equations, Lecture Notes in Physics 73, Springer-Verlag, 1978, pp. 197–247.
[12] Nonlinear wave equations, CBMS Regional Conf. Series in Math. 73, Amer. Math. Soc., 1989.
[13] Weissler, Israel J. Math. 38 pp 29– (1981)
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