On Reducibility of 1d Wave Equation with Quasiperiodic in Time Potentials

In this paper we use a KAM theorem of Grébert and Thomann (Commun Math Phys 307:383–427, 2011) to prove the reducibility of the 1d wave equation with Dirichlet boundery conditions on \([0,\pi ]\) with a quasi-periodic in time potential under some symmetry assumptions. From Mathieu–Hill operator’s known results (Eastham in The spectral theory of periodic differential operators, Hafner, New York, 1974; Magnus and Winkler in Hill’s equation, Wiley-Interscience, London, 1969) and Bourgain’s techniques (Commun Math Phys 204:207–247, 1999), we prove that for any \(\epsilon \) small enough, there exist a \(0<m_{\epsilon }\le 1\) and one solution \(u_{\epsilon }(t,x)\) with

$$\begin{aligned} \Vert u_{\epsilon }(t_n,x)\Vert _{H^1({\mathbb {T}})}\rightarrow \infty , \qquad |t_n|\rightarrow \infty , \end{aligned}$$

where \(u_{\epsilon }(t,x)\) satisfies 1d wave equation

$$\begin{aligned} u_{tt}-u_{xx}+m_{\epsilon }u-\epsilon \cos 2t u=0, \end{aligned}$$

with Dirichlet boundery conditions on \([0,\pi ]\).

     

Chemodivergent Staudinger Reactions of Secondary Phosphine Oxides and Application to the Total Synthesis of LL–D05139β Potassium Salt

Unprecedented Staudinger reaction modes of secondary phosphine oxides (SPO) and organic azides are herein disclosed. By the application of various additives, selective nitrogen atom exclusion from the azide group has been achieved. Chlorotrimethylsilane mediates a stereoretentive Staudinger reaction with a 2-N exclusion which provides a valuable method for the synthesis of phosphinic amides and can be considered complementary to the stereoinvertive Atherton–Todd reaction. Alternatively, a 1-N exclusion pathway is promoted by acetic acid to provide the corresponding diazo compound. The effectiveness of this protocol has been further demonstrated by the total synthesis of the diazo-containing natural product LL-D05139β, which was prepared as a potassium salt for the first time in 6 steps and 26.5 % overall yield.