Wave operators for the coupled Klein-Gordon-Schrodinger equations in two space dimensions
Wave operators for the coupled Klein-Gordon-Schrodinger equations in two space dimensions
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DOI：
10.1619/fesi.47.63
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发表时间：
2004
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影响因子：
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通讯作者：
A. Shimomura
中科院分区：
文献类型：
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作者：
A. Shimomura
本文研究了二维二次型Yukawa相互作用耦合KleinGordon-Schrodinger方程(KGS方程)的散射理论。众所周知，对于具有|u|p−1U形式的典型非线性的二维解耦的非线性薛定谔方程和克莱因-戈登方程，当p>；2时存在波算符，否则它们不存在。也就是说，二次非线性相互作用是两个空间维度中的临界功率。然而，对于小散射态，我们证明了具有二次相互作用的KGS方程的波算符的存在性。证明是基于KGS方程解的适当的二次近似的构造，这意味着相互作用项的改进的时间衰减估计，因此Cook-Kuroda方法是适用的。
In this paper, we study the scattering theory for the coupled KleinGordon-Schrodinger equation (the KGS equation) with the Yukawa type interaction, which is the certain quadratic interaction, in two space dimensions. It is well-known that for the two dimensional decoupled nonlinear Schrodinger and Klein-Gordon equations with the typical nonlinearity of the form |u|p−1u, there exist wave operators if p > 2 and they do not exist otherwise. Namely, quadratic nonlinear interaction is the critical power in two space dimensions. We, however, prove the existence of wave operators to the KGS equation with that quadratic interaction for small scattered states. The proof is based on the construction of suitable second approximations of the solution to the KGS equation which imply the improved time decay estimates of the interaction terms so that the Cook-Kuroda method is applicable.