Oscillatory instability of solitary waves in a continuum model of lattice vibrations
Oscillatory instability of solitary waves in a continuum model of lattice vibrations
复制标题
晶格振动连续模型中孤立波的振荡不稳定性
DOI：
10.1088/0951-7715/8/6/003
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发表时间：
1995
期刊：
影响因子：
1.7
通讯作者：
M. Weinstein
中科院分区：
文献类型：
--
作者：
R. Pego;P. Smereka;M. Weinstein
我们研究了两个耦合布辛涅斯克方程的孤立波稳定性，该方程模拟了立方晶格中的弱非线性振动。提出了哈密顿公式。据观察，已知的变分方法无法建立稳定性或检测不稳定性。相反，该问题被线性化并使用埃文斯函数进行研究，埃文斯函数是一种解析函数，其零点在右半平面时对应于离散的不稳定特征值。证明了如果存在线性指数不稳定性，则在转变时右半平面中出现一对复共轭特征值。埃文斯函数是通过数值计算的，我们观察到交叉到右半平面的复共轭零对。第一对交叉点与 Christiansen、Lomdahl 和 Muto (1990 Nonlinearity 4 477) 的结论非常一致，这些结论是根据初值问题的数值计算解得出的。该系统中的不稳定机制与有限维哈密顿系统中的典型不稳定机制不同，在有限维哈密顿系统中，通过虚部特征值的碰撞发生向不稳定的转变。在这里，转变涉及共振极点，即左半平面中埃文斯函数的零点，它们穿过连续谱并以不稳定的特征值出现。
We study the stability of solitary waves of two coupled Boussinesq equations which model weakly nonlinear vibrations in a cubic lattice. A Hamiltonian formulation is presented. Known variational methods are observed to be incapable of establishing stability or detecting instability. Instead, the problem is linearized and studied using the Evans function, an analytic function whose zeros, when in the right half plane, correspond to discrete unstable eigenvalues. It is proved that if there is a linear exponential instability, then at transition a pair of complex conjugate eigenvalues emerges into the right half plane. The Evans function is computed numerically and we observe complex conjugate pairs of zeros crossing into the right half plane. The first pair that crosses does so in close agreement with the conclusions of Christiansen, Lomdahl and Muto (1990 Nonlinearity 4 477), which were drawn from numerically computed solutions of the initial-value problem. The instability mechanism in this system differs from that typical in finite-dimensional Hamiltonian systems, where transition to instability occurs via collisions of imaginary eigenvalues. Here, the transition involves resonance poles, which are the zeros of the Evans function in the left half plane, that cross the continuous spectrum and emerge as unstable eigenvalues.