We obtain real-valued, time-periodic and radially symmetric solutions of the cubic Klein–Gordon equation which are weakly localized in space. Various families of such 'breather' solutions are shown to bifurcate from any given nontrivial stationary solution. The construction of weakly localized breathers in three space dimensions is, to the author's knowledge, a new concept and based on the reformulation of the cubic Klein–Gordon equation as a system of coupled nonlinear Helmholtz equations involving suitable conditions on the far field behavior.

中文翻译：

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三次 Klein-Gordon 方程的 Breather 解

我们获得了三次 Klein-Gordon 方程的实值、时间周期和径向对称解，它们在空间中弱局部化。这种“呼吸”解决方案的各种家族被证明可以从任何给定的非平凡固定解决方案中分叉出来。在三个空间维度上构建弱局部呼吸器，据作者所知，是一个新概念，它基于三次 Klein-Gordon 方程的重新表述，作为耦合非线性亥姆霍兹方程的系统，涉及远场行为的合适条件。