Two-pulse solutions in the fifth-order KdV equation: Rigorous theory and numerical approximations
Two-pulse solutions in the fifth-order KdV equation: Rigorous theory and numerical approximations
复制标题
五阶 KdV 方程的两脉冲解：严格的理论和数值近似
DOI：
10.3934/dcdsb.2007.8.773
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发表时间：
2006
影响因子：
1.2
通讯作者：
D. Pelinovsky
中科院分区：
文献类型：
--
作者：
M. Chugunova;D. Pelinovsky
研究了五阶Korteweg-de弗里斯（KdV）方程双脉冲解的存在性和稳定性，得到了两个新的结果.首先，我们修改Petviashvili方法的连续迭代的脉冲的数值（谱）近似，并证明收敛的迭代在两个脉冲的解决方案。其次，我们证明了线性化KdV方程中负Krein签名嵌入特征值的结构稳定性。结合Pontryagin空间中的稳定性分析，完成了相应双脉冲解的谱稳定性的证明。线性化问题的特征值近似数值在指数加权空间，其中嵌入的特征值是孤立的连续谱。对五阶KdV方程的本征值的近似和全数值模拟证实了与有效相互作用势的极小值相关的双脉冲解的稳定性和与极大值点相关的双脉冲解的不稳定性。
We revisit existence and stability of two-pulse solutions in the fifth-order Korteweg–de Vries (KdV) equation with two new results. First, we modify the Petviashvili method of successive iterations for numerical (spectral) approximations of pulses and prove convergence of iterations in a neighborhood of two-pulse solutions. Second, we prove structural stability of embedded eigenvalues of negative Krein signature in a linearized KdV equation. Combined with stability analysis in Pontryagin spaces, this result completes the proof of spectral stability of the corresponding two-pulse solutions. Eigenvalues of the linearized problem are approximated numerically in exponentially weighted spaces where embedded eigenvalues are isolated from the continuous spectrum. Approximations of eigenvalues and full numerical simulations of the fifth-order KdV equation confirm stability of two-pulse solutions associated with the minima of the effective interaction potential and instability of two-pulse solutions associated with the maxima points.