×

Resonance and long time existence for the quadratic semilinear Schrödinger equation. (English) Zbl 0774.34063

The paper starts from a hint by J. Shatah, who adapted in [Commun. Pure Appl. Math. 38, 685-696 (1985; Zbl 0597.35101)] Poincaré’s theory of normal forms, which deals with ordinary differential equations of the type \(\dot x=f(x)\), to the case of infinitely many dimensions, to reduce the quadratically nonlinear Klein-Gordon equation to a cubically nonlinear one. This higher degree of nonlinearity can later be exploited to prove global existence of solutions of the initial value problem for the original quadratic equation. Here the semilinear quadratic Schrödinger equation \(v_ t=iv_{xx}+Q(v,v_ x)\) is considered, showing that the sample nonlinearity is nonresonant (i.e., the nonlinearity \(Q\) can be removed by Poincaré’s procedure) for some specific \(Q\)’s, such as \(Q(v,v_ x)=\bar v^ 2_ x\). However, the class of acceptable nonlinearities appears to be rather small. Then, a smooth change of variable \(w=w(v)\) reduces the equation to one of the form \(w_ t=iw_{xx}+C(w,w_ x)+\cdots\), where \(C\) is cubic. From this point of view, the method of normal forms is in fact extended, by developing techniques with which one can recover, from this last equation for \(w\), estimates for \(v\) that are significantly stronger than those that can be derived directly from the original equation for \(v\). Using these estimates, the author manages to prove a long time existence theorem for the solution of the initial value problem: \(v_ t=iv_{xx}+\bar v^ 2_ x\), \(v(x,0)=g(x)\). The techniques are built around an interesting result from nonlinear harmonic analysis due to R. R. Coifman and Y. Meyer [Ann. Math. Stud. 112, 3-45 (1986; Zbl 0623.47052)]. A weak point of the whole procedure is the fact that, in the lack of a comprehensive theory of normal forms for the case of infinitely many dimensions, the criteria for nonresonance has to be based on heuristic arguments and formal calculations.

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, New York, 1983.
[2] and , Non-Linear Harmonic Analysis, Operator Theory, and P. D. E., Beijing Lectures in Harmonic Analysis, ed., Princeton University Press, 1986.
[3] Resonance and Long Time Existence for the Quadratically Nonlinear Schrödinger Equation, Doctoral Dissertation, Department of Mathematics, New York University, 1990.
[4] Klainerman, Comm. Pure Appl. Math. 36 pp 133– (1983)
[5] Shatah, Comm. Pure Appl. Math. 38 pp 685– (1985)
[6] Shock Waves and Reaction Diffusion Equations, Springer-Verlag, New York, 1984.
[7] Linear and Nonlinear Waves, Wiley Interscience, New York, 1974.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.