Nonlinear Dirac Equation
Nonlinear Dirac Equation
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非线性狄拉克方程
DOI：
10.1090/surv/244
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
A. Comech
中科院分区：
文献类型：
--
作者：
N. Boussaid;A. Comech
这本专著给出了一个全面的处理谱（线性）稳定性弱相对论孤立波的非线性狄拉克方程。结果表明，不稳定性不是狄拉克方程的固有性质，狄拉克方程只能在狄拉克海假设的第二量子化框架中解决。尽管狄拉克-麦克斯韦方程和类似方程的一般结果还没有，我们可以考虑带有标量自相互作用的狄拉克方程，该模型于1938年首次引入。在这本书中，我们表明，在特殊情况下，孤立波在这个模型中可能是谱稳定的（没有线性不稳定性）。这一结果是证明孤立波渐近稳定性的第一步。这本书提出了必要的概述的功能分析，谱理论，和存在和线性稳定性孤波的非线性薛定谔方程。给出了必要的工具，如极限吸收原理和适用于Dirac算子的Carleman估计，并证明了Dirac-Pauli定理的一般形式。所有这些结果都被用来证明非线性Dirac方程弱相对论孤波解的谱稳定性。
This monograph gives a comprehensive treatment of spectral (linear) stability of weakly relativistic solitary waves in the nonlinear Dirac equation. It turns out that the instability is not an intrinsic property of the Dirac equation that is only resolved in the framework of the second quantization with the Dirac sea hypothesis. Whereas general results about the Dirac-Maxwell and similar equations are not yet available, we can consider the Dirac equation with scalar self-interaction, the model first introduced in 1938. In this book we show that in particular cases solitary waves in this model may be spectrally stable (no linear instability). This result is the first step towards proving asymptotic stability of solitary waves. The book presents the necessary overview of the functional analysis, spectral theory, and the existence and linear stability of solitary waves of the nonlinear Schrödinger equation. It also presents the necessary tools such as the limiting absorption principle and the Carleman estimates in the form applicable to the Dirac operator, and proves the general form of the Dirac-Pauli theorem. All of these results are used to prove the spectral stability of weakly relativistic solitary wave solutions of the nonlinear Dirac equation.