## 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.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35K55 Nonlinear parabolic equations 37C10 Dynamics induced by flows and semiflows
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### References:

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