论文标题
修改后的Korteweg-de Vries方程的多层的稳定性
Stability of the multi-solitons of the modified Korteweg-de Vries equation
论文作者
论文摘要
我们建立了修改后的Korteweg-De Vries（MKDV）方程的$ n $ -soliton解决方案的非线性稳定性。 $ n $ -soliton解决方案是MKDV的全球解决方案，表现为（正和负）时间无穷大，为$ 1 $ -solitons的总和，$ 0 <c_1 <c_1 <\ cdots <c_n $。证明依赖于$ n $ solitons的变化表征。 We show that the $N$-solitons realize the local minimum of the $(N+1)$-th mKdV conserved quantity subject to fixed constraints on the $N$ first conserved quantities.To this aim, we construct a functional for which $N$-solitons are critical points, we prove that the spectral properties of the linearization of this functional around a $N$-soliton are preserved on the extended timeline, and we analyze线性化运算符无穷大的频谱左右$ 1 $ - 苏里顿。我们分析中的主要新成分是基于MKDV方程的惯性和递归算子的广义西尔维斯特定律的新操作员身份。
We establish the nonlinear stability of $N$-soliton solutions of the modified Korteweg-de Vries (mKdV) equation. The $N$-soliton solutions are global solutions of mKdV behaving at (positive and negative) time infinity as sums of $1$-solitons with speeds $0<c_1<\cdots< c_N$.The proof relies on the variational characterization of $N$-solitons. We show that the $N$-solitons realize the local minimum of the $(N+1)$-th mKdV conserved quantity subject to fixed constraints on the $N$ first conserved quantities.To this aim, we construct a functional for which $N$-solitons are critical points, we prove that the spectral properties of the linearization of this functional around a $N$-soliton are preserved on the extended timeline, and we analyze the spectrum at infinity of linearized operators around $1$-solitons. The main new ingredients in our analysis are a new operator identity based on a generalized Sylvester law of inertia and recursion operators for the mKdV equation.
