Solitary waves in a Whitham equation with small surface tension
Solitary waves in a Whitham equation with small surface tension
复制标题
具有小表面张力的 Whitham 方程中的孤立波
DOI：
--
复制
发表时间：
2021
期刊：
影响因子：
--
通讯作者：
Miles H. Wheeler
中科院分区：
文献类型：
--
作者：
Mathew A. Johnson;Tien Truong;Miles H. Wheeler
利用非局部形式的中心流形定理和规范形约化，证明了具有弱表面张力的定常重力-毛细Whitham方程的小振幅广义孤立波解和调制孤立波解的存在性.通过中心流形定理的应用，孤立波剖面的非局部方程被简化为继承可逆性的四维常微分方程系统。沿着特定的参数曲线，直接与经典的重力毛细水波问题相关，相关的线性算子被认为经历可逆的02+（ik 0）$0^{2+}（ （2）如果一个分支（i）为2，则该分支（ ext{i}s）^2$分叉。通过正规形变换，沿着沿着每个相关参数曲线的常微分方程的简化系统被看作是由仅保留二阶或三阶项的截断系统很好地近似。这些截断系统直接与研究完整的重力-毛细水波动方程中得到的系统相关，因此，截断系统的广义和调制孤立波的存在性由经典著作保证，并且由于可逆性，它们很容易被视为重力-毛细惠特姆方程的解。因此，这项工作进一步阐明了重力-毛细管Whitham方程和全二维重力-毛细管水波问题之间的联系。
Using a nonlocal version of the center‐manifold theorem and a normal form reduction, we prove the existence of small‐amplitude generalized solitary‐wave solutions and modulated solitary‐wave solutions to the steady gravity‐capillary Whitham equation with weak surface tension. Through the application of the center‐manifold theorem, the nonlocal equation for the solitary wave profiles is reduced to a four‐dimensional system of ODEs inheriting reversibility. Along particular parameter curves, relating directly to the classical gravity‐capillary water wave problem, the associated linear operator is seen to undergo either a reversible 02+(ik0)$0^{2+}( ext{i}k_0)$ bifurcation or a reversible (is)2$( ext{i}s)^2$ bifurcation. Through a normal form transformation, the reduced system of ODEs along each relevant parameter curve is seen to be well approximated by a truncated system retaining only second‐order or third‐order terms. These truncated systems relate directly to systems obtained in the study of the full gravity‐capillary water wave equation and, as such, the existence of generalized and modulated solitary waves for the truncated systems is guaranteed by classical works, and they are readily seen to persist as solutions of the gravity‐capillary Whitham equation due to reversibility. Consequently, this work illuminates further connections between the gravity‐capillary Whitham equation and the full two‐dimensional gravity‐capillary water wave problem.
用于弹性、生物学和流体动力学中的拟线性问题的无相空间的中心流形
影响因子：
1.7
作者：
Chen, Robin Ming;Walsh, Samuel;Wheeler, Miles H
通讯作者：
Wheeler, Miles H