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            <title xml:lang="en">Semiclassical microlocal normal forms and global solutions of modified one-dimensional KG equations</title>
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            <note type="commentary">68 pages. This is the final version to be published in Annales de l'Institut Fourier. Some misprints have been corrected and the bibliography has been updated.</note>
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                <title xml:lang="en">Semiclassical microlocal normal forms and global solutions of modified one-dimensional KG equations</title>
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                    <forename type="first">Jean-Marc</forename>
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                <title level="j">Annales de l'Institut Fourier</title>
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                <term xml:lang="en">Semiclassical analysis</term>
                <term xml:lang="en">Microlocal normal forms</term>
                <term xml:lang="en">Klainerman vector fields</term>
                <term xml:lang="en">Global solution of Klein-Gordon equations</term>
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              <p>The method of Klainerman vector fields plays an essential role in the study of global existence of solutions of nonlinear hyperbolic PDEs, with small, smooth, decaying Cauchy data. Nevertheless, it turns out that some equations of physics, like the one dimensional water waves equation with finite depth, do not possess any Klainerman vector field. The goal of this paper is to design, on a model equation, a substitute to the Klainerman vector fields method, that allows one to get global existence results, even in the critical case for which linear scattering does not hold at infinity. The main idea is to use semiclassical pseudodifferential operators instead of vector fields, combined with microlocal normal forms, to reduce the nonlinearity to expressions for which a Leibniz rule holds for these operators.</p>
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