Virtual turning points and bifurcation of Stokes curves for higher order ordinary differential equations

由守恒能量导出的Stokes波不稳定特征值的分岔

Bifurcations of unstable eigenvalues for Stokes waves derived from conserved energy

针对无限深流体无旋重力波的Euler方程，严格证明Stokes波线性化运动方程的零特征值分岔出现在能量对陡度的极值点，且新不稳定特征值仅沿陡度增大方向产生。

Global bifurcation and highest waves on water of finite depth

We consider the two-dimensional problem for steady water waves with vorticity on water of finite depth. While neglecting the effects of surface tension we construct connected families of large amplitude periodic waves approaching the limiting wave, which is either a solitary wave, the highest solitary wave, the highest Stokes wave or a Stokes wave with a breaking profile. In particular, when the vorticity is nonnegative we prove the existence of highest Stokes waves with an included angle of 120 degrees. In contrast to previous studies we fix the Bernoulli constant and consider the wavelength …

Global bifurcation for the Whitham equation

We prove the existence of a global bifurcation branch of $2π$-periodic, smooth, traveling-wave solutions of the Whitham equation. It is shown that any subset of solutions in the global branch contains a sequence which converges uniformly to some solution of Hölder class $C^α$, $α< \frac{1}{2}$. Bifurcation formulas are given, as well as some properties along the global bifurcation branch. In addition, a spectral scheme for computing approximations to those waves is put forward, and several numerical results along the global bifurcation branch are presented, including the presence of a turning …

双峰斯托克斯波

Two-Crested Stokes Waves

本文研究无限深理想流体中的双峰第II类斯托克斯波，发现其分岔出现在速度-陡度图的前两次振荡中且互不连通，并追踪其次级与三级分支，揭示其速度振荡及120度角极限波特征。

Navier--Stokes 方程的渐近中心流形

Asymptotic Center--Manifold for the Navier--Stokes

本文通过扩展 Navier--Stokes 方程并应用中心流形定理，推导了图方程的渐近解，在分岔点附近有效，并成功应用于圆柱尾流和开腔流动案例，预测角频率与全系统吻合。

有限深度流体中的斯托克斯波

Stokes Waves in Finite Depth Fluids

针对有限深度理想流体表面行波问题，基于巴边科方程，采用牛顿-共轭梯度法计算不同共形深度的斯托克斯波，识别二次分岔点并发现浅水宽谷尖峰解，大陡度时波峰夹角为2π/3。

Dispersion equation for water waves with vorticity and Stokes waves on flows with counter-currents

The two-dimensional free-boundary problem of steady periodic waves with vorticity is considered for water of finite depth. We investigate how flows with small-amplitude Stokes waves on the free surface bifurcate from a horizontal parallel shear flow in which counter-currents may be present. Two bifurcation mechanisms are described: for waves with fixed Bernoulli's constant and fixed wavelength. In both cases the corresponding dispersion equations serve for defining wavelengths from which Stokes waves bifurcate. Sufficient conditions guaranteeing the existence of roots of these equations are ob…

Asymptotic expansion of the solution of the steady Stokes equation with variable viscosity in a two-dimensional tube structure

The Stokes equation with the varying viscosity is considered in a thin tube structure, i.e. in a connected union of thin rectangles with heights of order $\varepsilon<<1 $ and with bases of order 1 with smoothened boundary. An asymptotic expansion of the solution is constructed: it contains some Poiseuille type flows in the channels (rectangles) with some boundary layers correctors in the neighborhoods of the bifurcations of the channels. The estimates for the difference of the exact solution and its asymptotic approximation are proved.

Exact solutions to the Navier-Stokes equation for an incompressible flow from the interpretation of the Schroedinger wave function

The existence of the velocity potential is a direct consequence from the derivation of the continuity equation from the Schroedinger equation. This implies that the Cole-Hopf transformation is applicable to the Navier-Stokes equation for an incompressible flow and allows reducing the Navier-Stokes equation to the Einstein-Kolmogorov equation, in which the reaction term depends on the pressure. The solution to the resulting equation, and to the Navier-Stokes equation as well, can then be written in terms of the Green's function of the heat equation and is given in the form of an integral mappin…

Interior Structural Bifurcation of 2D Symmetric Incompressible Flows

The structural bifurcation of a 2D divergence free vector field $\mathbf{u}(\cdot, t)$ when $\mathbf{u}(\cdot, t_0)$ has an interior isolated singular point $\mathbf{x}_0$ of zero index has been studied by Ma and Wang. Although in the class of divergence free fields which undergo a local bifurcation around a singular point, the ones with index zero singular points are generic, this class excludes some important families of symmetric flows. In particular, when $\mathbf{u}(\cdot, t_0)$ is anti-symmetric with respect to $\mathbf{x}_0$, or symmetric with respect to the axis located on $\mathbf{x}_…