论文标题
具有可变系数二次非线性的1D klein-gordon方程的渐近学
Asymptotics for 1D Klein-Gordon equations with variable coefficient quadratic nonlinearities
论文作者
论文摘要
我们启动对具有可变系数二次非线性的一维klein-gordon方程的小解决方案的渐近行为的研究。这项工作的主要发现是可变系数的特定空间频率与溶液的时间振荡之间的引人注目的谐振相互作用。在共振的情况下，发生了一种新型的改良散射行为，它表现出对对数沿某些射线的衰减速率的对数。在非共鸣的情况下，我们引入了一种新的变量系数二次正常形式，并在存在恒定的恒定系数立方非线性的情况下建立尖锐的衰减估计值和渐近性。本文考虑的Klein-Gordon模型是由对实际线上经典非线性标量场方程的扭结解决方案的渐近稳定性的研究进行的。
We initiate the study of the asymptotic behavior of small solutions to one-dimensional Klein-Gordon equations with variable coefficient quadratic nonlinearities. The main discovery in this work is a striking resonant interaction between specific spatial frequencies of the variable coefficient and the temporal oscillations of the solutions. In the resonant case a novel type of modified scattering behavior occurs that exhibits a logarithmic slow-down of the decay rate along certain rays. In the non-resonant case we introduce a new variable coefficient quadratic normal form and establish sharp decay estimates and asymptotics in the presence of a critically dispersing constant coefficient cubic nonlinearity. The Klein-Gordon models considered in this paper are motivated by the study of the asymptotic stability of kink solutions to classical nonlinear scalar field equations on the real line.
