Breather solutions of the discrete p-Schr"odinger equation
Breather solutions of the discrete p-Schr"odinger equation
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离散p-Schr"odinger方程的呼吸解
DOI：
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发表时间：
2013
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影响因子：
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通讯作者：
Y. Starosvetsky
中科院分区：
文献类型：
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作者：
G. James;Y. Starosvetsky
考虑离散的p-Schr“odinger（DpS）方程，它近似了具有α = p-1 >1阶完全非线性近邻相互作用的振子链中的小振幅振荡.利用映射的方法，我们证明了具有偶宇称或奇宇称反射对称的DpS方程呼吸子解的存在性。此外，我们推导出解析近似的呼吸配置文件和相应的交叉稳定和不稳定的流形，有效的整个范围内的非线性订单α。在弱非线性极限下（α-> 1^+），我们引入了一个连接定常DpS和对数非线性Schr“odinger方程的连续极限.在这个极限下，呼吸子渐近对应于高斯同宿解。我们数值分析呼吸解决方案的稳定性取决于他们的奇偶对称性。当α接近于1时，不稳定呼吸子的扰动通常导致平移运动（行进呼吸子），而对于较大的α值，钉扎成为主导。
We consider the discrete p-Schr"odinger (DpS) equation, which approximates small amplitude oscillations in chains of oscillators with fully-nonlinear nearest-neighbors interactions of order alpha = p-1 >1. Using a mapping approach, we prove the existence of breather solutions of the DpS equation with even- or odd-parity reflectional symmetries. We derive in addition analytical approximations for the breather profiles and the corresponding intersecting stable and unstable manifolds, valid on a whole range of nonlinearity orders alpha. In the limit of weak nonlinearity (alpha --> 1^+), we introduce a continuum limit connecting the stationary DpS and logarithmic nonlinear Schr"odinger equations. In this limit, breathers correspond asymptotically to Gaussian homoclinic solutions. We numerically analyze the stability properties of breather solutions depending on their even- or odd-parity symmetry. A perturbation of an unstable breather generally results in a translational motion (traveling breather) when alpha is close to unity, whereas pinning becomes predominant for larger values of alpha.