周期势场中的运动间隙孤子
Moving gap solitons in periodic potentials
摘要：
讨论了具有小周期势的Gross-Pitaevskii方程中运动间隙孤子（行波局域解）的存在性。运动的间隙孤子近似的耦合模系统的显式本地化的解决方案。然而，我们发现，指数衰减的Gross-Pitaevskii方程的行波解一般不存在于存在的周期性的潜在的有界振荡的尾巴，由于前面和后面的移动孤波。振荡的尾巴不占在耦合模式的形式主义，并估计通过使用空间动力学和本地中心稳定的流形减少技术。利用这些技巧证明了在空间尺度的有限大区间上具有振荡尾的单凸点的Gros-Pitaevskii方程的有界行波解的存在性。我们还展示了其他具有周期势的非线性方程的振荡尾的一般性。
英文摘要：
We address existence of moving gap solitons (traveling localized solutions) in the Gross-Pitaevskii equation with a small periodic potential. Moving gap solitons are approximated by the explicit localized solutions of the coupled-mode system. We show however that exponentially decaying traveling solutions of the Gross-Pitaevskii equation do not generally exist in the presence of a periodic potential due to bounded oscillatory tails ahead and behind the moving solitary waves. The oscillatory tails are not accounted in the coupled-mode formalism and are estimated by using techniques of spatial dynamics and local center-stable manifold reductions. Existence of bounded traveling solutions of the Gross--Pitaevskii equation with a single bump surrounded by oscillatory tails on a finite large interval of the spatial scale is proven by using these technique. We also show generality of oscillatory tails in other nonlinear equations with a periodic potential.
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