论文标题
临界长度的非线性KDV方程的衰减
Decay for the nonlinear KdV equations at critical lengths
论文作者
论文摘要
我们在右侧的界间隔内考虑非线性korteweg-de Vries（KDV）方程，并在右侧。众所周知，如果它们的初始数据属于有限的尺寸子空间$ \ m $，则有一组临界长度线性化系统的解决方案保存$ l^2 $ norm。在本文中，我们表明，当$ \ dim \ m = 1 $或$ \ dim \ m $均匀且满足特定条件时，非线性KDV系统的所有解决方案至少在$ 1/ t^{1/2} $中衰减至0，只要他们的初始数据足够小。我们的分析灵感来自功率系列扩展方法，并涉及准周期函数的理论。结果，我们重新发现了以前以$ \ dim \ m = 1 $或以$ \ dim \ m = 2 $为$ \ dim \ m = 1 $建立的已知结果，该结果使用中心歧管理论，并获得新的结果。我们还表明，对于所有关键长度，衰减率不超过$ \ ln（t + 2） / t $。
We consider the nonlinear Korteweg-de Vries (KdV) equation in a bounded interval equipped with the Dirichlet boundary condition and the Neumann boundary condition on the right. It is known that there is a set of critical lengths for which the solutions of the linearized system conserve the $L^2$-norm if their initial data belong to a finite dimensional subspace $\M$. In this paper, we show that all solutions of the nonlinear KdV system decay to 0 at least with the rate $1/ t^{1/2}$ when $\dim \M = 1$ or when $\dim \M$ is even and a specific condition is satisfied, provided that their initial data is sufficiently small. Our analysis is inspired by the power series expansion approach and involves the theory of quasi-periodic functions. As a consequence, we rediscover known results which were previously established for $\dim \M = 1$ or for the smallest critical length $L$ with $\dim \M = 2$ by a different approach using the center manifold theory, and obtain new results. We also show that the decay rate is not slower than $\ln (t + 2) / t$ for all critical lengths.
