论文标题
正弦方程的长期渐进渐变和稳定性
Long-time asymptotics and stability for the sine-Gordon equation
论文作者
论文摘要
在本文中，我们研究了正弦 - 戈登方程的长期动态和稳定性$$ f_ {tt} -f_ {xx}+\ sin f = 0。$ $$首先，我们使用非线性陡峭下降来实现Riemann-hilbert的最陡峭下降，以使长期的求解方程式属于Sine-gord-Gord-Gord-Gord-Gord-Gord-Gord-Gord-Gord-Gord-Gord-eque exterip。其次，我们研究正弦方程的渐近稳定性。众所周知，能量空间中正弦仪方程的渐近稳定性的障碍物是存在小呼吸器的存在，这也与扭曲扭结的出现密切相关。结合了长期渐近学和精致的近似参数，我们分析了加权能量空间中正弦戈登方程的渐近稳定性。我们的稳定性分析给出了重量的标准，该标准是尖锐到终点，以使渐近稳定性保持。
In this paper, we study the long-time dynamics and stability properties of the sine-Gordon equation $$f_{tt}-f_{xx}+\sin f=0.$$ Firstly, we use the nonlinear steepest descent for Riemann-Hilbert problems to compute the long-time asymptotics of the solutions to the sine-Gordon equation whose initial condition belongs to some weighted Sobolev spaces. Secondly, we study the asymptotic stability of the sine-Gordon equation. It is known that the obstruction to the asymptotic stability of the sine-Gordon equation in the energy space is the existence of small breathers which is also closely related to the emergence of wobbling kinks. Combining the long-time asymptotics and a refined approximation argument, we analyze the asymptotic stability properties of the sine-Gordon equation in weighted energy spaces. Our stability analysis gives a criterion for the weight which is sharp up to the endpoint so that the asymptotic stability holds.
