Quadratic Klein-Gordon equations with a potential in one dimension
Quadratic Klein-Gordon equations with a potential in one dimension
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DOI：
10.1017/fmp.2022.9
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发表时间：
2020-06
期刊：
影响因子：
--
通讯作者：
P. Germain;F. Pusateri
中科院分区：
文献类型：
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作者：
P. Germain;F. Pusateri
本文对（拓扑）孤子的渐近稳定性问题提出了一个相当普遍的新观点。我们的方法是基于使用的扭曲傅立叶变换在非线性水平，它不依赖于仅在Hohartz或维里估计，因此能够处理低功率非线性（因此也非局域孤子）和捕获的全球（在空间和时间）的解决方案的行为。更具体地说，我们考虑二次非线性Klein-Gordon方程的定期和衰减的潜力在一个空间维。附加的假设，使失真的傅里叶变换的解决方案在零频率消失。还假设相关联的薛定谔算子没有负特征值，我们获得了全局的时间界，包括尖锐的逐点衰减和修改的渐近性，小的解决方案。这些结果有一些直接的应用（拓扑）孤子的渐近稳定性，以及其他几个潜在的应用到各种相关的问题。例如，我们获得了双sine-Gordon问题（在适当的变形参数范围内）关于奇摄动扭结的完全渐近稳定性。对于$\phi ^4$问题，只要非线性项投影到连续谱上，我们得到了小奇解的渐近稳定性.我们的结果也超越了这些例子，因为我们的框架允许在二次相互作用的水平上存在完全相干的现象（时空共振），这在扭曲的傅立叶空间中产生了简并。我们设计了一个合适的框架，结合这一点，并使用多线性谐波分析的扭曲设置，以控制所有的非线性相互作用。
Abstract This paper proposes a fairly general new point of view on the question of asymptotic stability of (topological) solitons. Our approach is based on the use of the distorted Fourier transform at the nonlinear level; it does not rely only on Strichartz or virial estimates and is therefore able to treat low-power nonlinearities (hence also nonlocalised solitons) and capture the global (in space and time) behaviour of solutions. More specifically, we consider quadratic nonlinear Klein-Gordon equations with a regular and decaying potential in one space dimension. Additional assumptions are made so that the distorted Fourier transform of the solution vanishes at zero frequency. Assuming also that the associated Schrödinger operator has no negative eigenvalues, we obtain global-in-time bounds, including sharp pointwise decay and modified asymptotics, for small solutions. These results have some direct applications to the asymptotic stability of (topological) solitons, as well as several other potential applications to a variety of related problems. For instance, we obtain full asymptotic stability of kinks with respect to odd perturbations for the double sine-Gordon problem (in an appropriate range of the deformation parameter). For the $\phi ^4$ problem, we obtain asymptotic stability for small odd solutions, provided the nonlinearity is projected on the continuous spectrum. Our results also go beyond these examples since our framework allows for the presence of a fully coherent phenomenon (a space-time resonance) at the level of quadratic interactions, which creates a degeneracy in distorted Fourier space. We devise a suitable framework that incorporates this and use multilinear harmonic analysis in the distorted setting to control all nonlinear interactions.