On the Existence and Conditional Energetic Stability of Solitary Gravity-Capillary Surface Waves on Deep Water
On the Existence and Conditional Energetic Stability of Solitary Gravity-Capillary Surface Waves on Deep Water
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深水中孤立重力毛细管面波的存在性及其条件能量稳定性
DOI：
10.1007/s00021-010-0034-x
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发表时间：
2011
影响因子：
1.3
通讯作者：
E. Wahlén
中科院分区：
文献类型：
--
作者：
Mark D. Groves;M. Groves;E. Wahlén;E. Wahlén
本文提出了无限深度水体表面重力-毛细孤立波的存在性和稳定性理论。利用经典的变分原理，我们证明了在约束$${{\mathcal I}=\sqrt{2}\mu}$$下波能极小值$${{\mathcal E}}$$的存在，其中$${{\mathcal I}}$$是波动量，$${0 < \mu \ll 1}$$。由于$${{\mathcal E}}$$和$${{\mathcal I}}$$都是守恒量，一个标准论证断言了最小值集合Dμ的稳定性：在适当定义的能量空间内，在它们的存在区间内，从Dμ附近开始的解保持在Dμ附近。在应用数学文献中，这种孤立水波被建模为具有三次聚焦非线性的非线性Schrödinger方程的解。我们表明，我们的变分方法检测到的波收敛（经过适当的重新缩放）到这个模型方程的解$${\mu \downarrow 0}$$。
This paper presents an existence and stability theory for gravity-capillary solitary waves on the surface of a body of water of infinite depth. Exploiting a classical variational principle, we prove the existence of a minimiser of the wave energy $${{\mathcal E}}$$ subject to the constraint $${{\mathcal I}=\sqrt{2}\mu}$$, where $${{\mathcal I}}$$ is the wave momentum and $${0 < \mu \ll 1}$$ . Since $${{\mathcal E}}$$ and $${{\mathcal I}}$$ are both conserved quantities a standard argument asserts the stability of the set Dμ of minimisers: solutions starting near Dμ remain close to Dμ in a suitably defined energy space over their interval of existence. In the applied mathematics literature solitary water waves of the present kind are modelled as solutions of the nonlinear Schrödinger equation with cubic focussing nonlinearity. We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of this model equation as $${\mu \downarrow 0}$$ .