Vera Mikyoung Hur
We prove that a Stokes' periodic wave of sufficiently small amplitude, traveling under gravity at the free surface of a two dimensional, infinitely deep, and irrotational flow, is spectrally unstable to slow modulation, rigorously justifying Benjamin and Feir's formal argument.
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Enhancement of the Benjamin-Feir instability with dissipation
结果表明，有一个被忽视的机制，即某些种类的耗散可以增强Benjamin-Feir不稳定性的水波。这一观察是新的，尽管它是违反直觉的，但这是由于Benjamin-Feir不稳定性涉及具有相反能量符号（相对于载波）的模式的碰撞，并且增强了负能量扰动。
Small-amplitude, traveling, space periodic solutions -- called Stokes waves -- of the 2 dimensional gravity water waves equations in deep water are linearly unstable with respect to long-wave perturbations, as predicted by Benjamin and Feir in 1967. We completely describe the behavior of the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent is turned on. We prove in particular the conjecture that a pair of non-purely imaginary eigenvalues depicts a closed figure eight, parameterized by the Floquet exponent, in full agreement with numerical s…
Spectral structure of the Benjamin-Feir instability in deep-water gravity-capillary Stokes waves
本文研究了深水重力-毛细 Stokes 波的 Benjamin-Feir 不稳定性，通过 Euler 方程的严格 Bloch-Floquet 谱分析，揭示了多特征值在原点的分裂，识别出非零实部的特征值对，从而获得表面张力参数下的稳定性区域。
The Benjamin-Feir instability in KdV-like equations with general dispersion and monomial nonlinearity
本文研究了具有通用色散和多项式非线性的KdV-like方程的Benjamin-Feir不稳定性，通过线性化周期行波，完整刻画了线性算子在原点附近的谱结构，证明了当模化不稳定性存在时，谱结构呈现闭合的八字形图案。
Benjamin-Feir spectrum of hydroelastic Stokes waves
针对二维有限深无旋Euler方程的小振幅水弹性Stokes波，在长波Floquet区域解析了从原点分岔的所有四个Bloch特征值，给出了Benjamin-Feir不稳定性的显式判据，并揭示了Wilton型共振消失的条件。
Whitham and Benjamin predicted in 1967 that small-amplitude periodic traveling Stokes waves of the 2d-gravity water waves equations are linearly unstable with respect to long-wave perturbations, if the depth $\mathtt h$ is larger than a critical threshold $\mathtt h_{WB} \approx 1.363$. In this paper we completely describe, for any value of $\mathtt h > 0$, the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent $μ$ is turned on. We prove in particular the existence of a unique depth $\mathtt h_{WB}$, which coincides with the one predicted by …
Full Benjamin-Feir instability of capillary-gravity Stokes waves in finite depth
本文利用Bloch-Floquet理论严格证明了有限深度毛细-重力Stokes波在长波扰动下零特征值的完全分裂，并证实其谱在原点附近呈现“8”字形，从而完整刻画了Benjamin-Feir不稳定性。
Full description of Benjamin-Feir instability for generalized Korteweg-de Vries equations
研究广义Korteweg-de Vries方程小振幅行波解的线性调制不稳定性，运用辛版本的Kato相似变换理论，将问题简化为确定矩阵特征值，通过块对角化过程，完整描述了不稳定谱，证实了“8”字形不稳定性。
Unstable Stokes waves
该研究针对小振幅周期Stokes波的谱不稳定性问题，发展周期Evans函数方法，证明了原点附近及远离原点处的不稳定性，结合数值验证提出所有足够小振幅的Stokes波均谱不稳定的猜想。
Sheets of Spectral Data of Stokes Waves in Weakly Nonlinear Models
研究弱非线性模型中斯托克斯波的小振幅频谱稳定性，通过扰动方法展开频谱数据，揭示不稳定岛的形成，并比较高频和本内特-费尔增长率。
Modulational spectrum of infinite-depth hydroelastic Stokes waves
该研究分析无限深度流体弹性Stokes波的调制谱，通过解耦特征值推导调制稳定性判据，识别弹性弯曲产生的稳定岛，揭示无限深度问题的奇异性。
McLean resonances and $3d$ spectral instability of Stokes waves
通过Kato扰动分析、KAM型解耦和解析延拓，首次严格描述了深水小振幅重力Stokes波在McLean共振曲线邻域内的三维不稳定谱，揭示了Benjamin-Feir不稳定性与高频孤立的共同根源。
It is proven that small-amplitude steady periodic water waves with infinite depth are unstable with respect to long-wave perturbations. This modulational instability was first observed more than half a century ago by Benjamin and Feir. It has been proven rigorously only in the case of finite depth. We provide a completely different and self-contained approach to prove the spectral modulational instability for water waves in both the finite and infinite depth cases.
We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth, and prove a rigorous reduction of these equations to Birkhoff normal form up to degree four. This proves a conjecture of Zakharov-Dyachenko [55] suggested by the formal Birkhoff integrability of the water waves Hamiltonian truncated at order four. As a consequence, we also obtain a long-time stability result: periodic perturbations of a flat interface that are of size $\varepsilon$ in a sufficiently smooth Sobolev space lead to solutions that remain regular and small up to times of order $\v…
Modulational instability in the full-dispersion Camassa-Holm equation
我们确定的稳定性和不稳定性的一个足够小的周期性行波长波长的扰动，一个非线性色散方程，其中扩展了Camassa-Holm方程，包括所有的色散水波和Whitham方程，包括非线性介质振幅波。在没有表面张力的影响下，结果定性地与Stokes波的Benjamin-Feir不稳定性一致。在表面张力的影响下，它与物理问题的形式渐近展开定性地一致，并改进了Whitham方程，正确地预测了强表面张力的极限。讨论了Camassa-Holm方程及相关模型的调制稳定性和不稳定性。
Instability of the solitary waves for the Generalized Benjamin-Bona-Mahony Equation
本文针对广义Benjamin-Bona-Mahony方程，在临界频率c=c_0(p)下证明了孤立波的轨道不稳定性，填补了已知稳定与不稳定区域间的空白。
Modulational Instability in the Whitham Equation for Water Waves
我们表明，周期性的Whitham方程，其中包括表面水波的色散关系和浅水方程的非线性振幅足够小的行波，光谱不稳定的长波长扰动，如果波数大于一个临界值，轴承Benjamin-Feir Stokes波的不稳定性;他们是光谱稳定的平方可积〔的〕扰动，否则。的证明涉及到相关的线性化运营商的Floquet指数和小振幅参数的谱扰动。我们将结果扩展到相关的非线性色散方程。
Comparison of modulational instabilities in full-dispersion shallow water models
我们研究了浅水模型的调制不稳定性，有和没有表面张力，其中概括的Whitham方程，包括双向传播。在没有表面张力的情况下，如果小振幅周期行波的波数大于临界波数，则小振幅周期行波是调制不稳定的，预测Benjamin-Feir型不稳定性，并且结果定性地与[HP 16]中的浅水模型一致。对于表面张力，除了[HP 16]中浅水模型准确预测的大表面张力极限外，结果定性上与物理问题一致。我们还比较了结果与Whitham和全色散Camassa-Holm方程。
We generalize the periodic Evans function approach recently used to study the spectral stability of Stokes wave and gravity-capillary (including Wilton ripples) in water of finite depth to study spectral stability of Stokes waves in water of infinite depth. We prove waves of sufficiently small amplitude are always low-frequency unstable regardless of the wave number and gravity, giving an alternative proof for the Benjamin-Feir modulational instability in the infinite depth case. Here, the first proof for the infinite depth case is recently obtained by Nguyen and Strauss. We also study the spe…
Modulational instability in a full-dispersion shallow water model
我们提出了一个浅水模型相结合的色散关系的水波和Boussinesq方程，并扩展了Whitham方程允许双向传播。我们确定，只要波数大于临界值，其足够小的周期波训练对于长波长扰动在光谱上是不稳定的，就像斯托克斯波的本杰明-菲尔不稳定性一样。我们验证了关联的线性算子具有无穷多个纯虚本征值的碰撞，但它们不会导致从谱平面原点到振幅参数首阶的不稳定性。讨论了表面张力对调制不稳定性的影响。所得结果与物理问题的形式渐近展开和数值计算的结果一致。
Linear asymptotic stability of small-amplitude periodic waves of the generalized Korteweg--de Vries equations
本文将KdV方程椭圆余弦波的线性化动力学研究推广到广义KdV方程中不受Benjamin–Feir不稳定性影响的小振幅周期行波，并证明在色散谱稳定性条件下，任意振幅波也具有Sobolev空间全局有界稳定性和色散型渐近稳定性。
On the Stability of Periodic Solutions of the Generalized Benjamin-Bona-Mahony Equation
本文研究了广义Benjamin-Bona-Mahony方程的四参数空间周期行波解族对两类扰动的稳定性：一类是与基本波具有相同周期结构的周期扰动，另一类是长波长局域扰动.特别是，我们推导出谱不稳定的必要条件的扰动，这两类扰动通过推导适当的渐近展开的周期埃文斯函数，我们概述了一个非线性稳定性理论的周期扰动的基础上变分方法，有效地扩展了我们的周期谱稳定性的结果。
Motivated by the ongoing study of dispersive shock waves in non integrable systems, we propose and analyze a set of wave parameters for periodic waves of a large class of Hamiltonian partial differential systems -- including the generalized Korteweg de Vries equations and the Euler-Korteweg systems -- that are well-behaved in both the small amplitude and small wavelength limits. We use this parametrization to determine fine asymptotic properties of the associated modulation systems, including detailed descriptions of eigenmodes. As a consequence, in the solitary wave limit we prove that modula…
Modulational instability of small amplitude periodic traveling waves in the $b$-family of Novikov equation
本文研究了$b$-族Novikov方程中周期行波的调制不稳定性，通过谱扰动理论推导出稳定性指数，证明了长波扰动下的谱不稳定性。
We examine the spectral stability and instability of periodic traveling waves for regularized long-wave models. Examples include the regularized Boussinesq, Benney--Luke, and Benjamin--Bona--Mahony equations. Of particular interest is a striking new instability phenomenon -- spectrum off the imaginary axis extending into infinity. The spectrum of the linearized operator of the generalized Korteweg--de Vries equation, for instance, lies along the imaginary axis outside a bounded set. The spectrum for a regularized long-wave model, by contrast, can vary markedly with the parameters of the period…
Modulational Instability of Small Amplitude Periodic Traveling Waves in the Novikov Equation
研究Novikov方程小振幅周期行波的谱稳定性，发现波数大于临界值时出现Benjamin-Feir不稳定性，小于临界值时谱稳定，方法基于谱扰动理论。
Modulational Instability in the Whitham Equation with Surface Tension and Vorticity
我们研究调制的稳定性和不稳定性的Whitham方程，结合水波的色散关系和浅水方程的非线性，并修改允许表面张力和常涡的影响。当表面张力系数是大的，我们表明，一个周期性的行波足够小的振幅是不稳定的长波长的扰动，如果波数大于临界值，否则稳定，类似于本杰明-Feir不稳定的重力波。在弱表面张力的情况下，我们发现稳定和不稳定的波数，其边界与群速度的极值，第一和第二谐波之间的共振，长波和短波之间的共振，色散和非线性之间的共振的间隔。对于每一个恒定的涡度，我们表明，一个周期性的行波足够小的振幅是不稳定的，如果波数大于临界值，否则稳定。而且，对于足够大的涡量，它可以是稳定的。结果与物理问题的数值计算或形式多尺度展开的结果一致。
Modulational instability in nonlinear nonlocal equations of regularized long wave type
我们研究了Benjamin- Bona-Mahony（BBM）方程和正则化Boussinesq型方程在谱平面原点附近的稳定性和不稳定性。我们最近的Korteweg-de Vries型方程的结果进行了扩展，并推导出调制不稳定性指数作为下伏波的波数的函数。我们证明了BBM方程的一个足够小的周期行波对于长波长扰动是谱不稳定的，如果波数大于一个临界值，正则化Boussinesq方程的一个足够小的周期行波对于平方可积〔的〕是稳定的.
We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth and we establish the existence and the linear stability of small amplitude, quasi-periodic in time, traveling waves. This provides the first existence result of quasi-periodic water waves solutions bifurcating from a \emph{completely resonant} elliptic fixed point. The proof is based on a Nash-Moser scheme, Birkhoff normal form methods and pseudo-differential calculus techniques. We deal with the combined problems of \emph{small divisors} and the \emph{fully-nonlinear} nature of the equations…
The Transverse Instability of Periodic Waves in Zakharov-Kuznetsov Type Equations
本文研究了二维空间中广义Korteweg-de Vries方程的一维稳定周期行波解对广义Zakharov-Kuznetsov方程的长波长横向扰动的不稳定性.通过导出周期Evans函数的适当的渐近展开，我们导出了一个指数，它产生横向不稳定性发生的充分条件。这个指数是几何性质的，适用于任何周期性行波轮廓下的一些轻微的平滑假设的非线性。我们还描述了广义Benjamin-Bona-Mahony方程的周期行波与gBBM-Zakharov-Kuznetsov方程的长波长横向扰动的类似理论。
Nonlinear Modulational Instability of Dispersive PDE Models
我们证明了几种色散偏微分方程的周期行波的周期扰动和局部扰动的非线性调制不稳定性，包括KDV型方程（如Whitham方程，广义KDV方程，Benjamin-Ono方程），非线性薛定谔方程和BBM方程.首先，利用线性化偏微分方程的Hamilton结构得到了证明非线性所需的半群估计：其次，对于KDV型方程，在两种互补的情况下克服了非线性项中导数的损失：（1）对于光滑非线性项和一般色散算子，我们构造了高阶近似解，然后利用能量型估计;（2）对于低正则性的非线性项，在对色散算子作了一些附加假设的情况下，我们利用自举方法克服了导数的损失。
Spectral stability in the modified Camassa-Holm equation
研究修正Camassa-Holm方程小振幅周期行波解的谱稳定性，利用Kato扰动理论完整描述线性化算子原点附近谱，证明波数k²≤3时谱稳定，k²>3时出现不稳定性。
Stability and instability of solitary waves in fractional generalized KdV equation in all dimensions
该研究针对任意维分数阶广义KdV方程，基于基态衰减与正则性新估计，证明了L²次临界情形孤立波的轨道稳定性与超临界情形的不稳定性，并拓展了一维Benjamin-Ono类方程的不稳定性结论。
Stability properties of solitary waves for fractional KdV and BBM equations
本文对Korteweg-de Vries型和Benjamin\&Bona\&Mahoney型模型的孤波解在色散很小时的稳定性作了新的研究.利用紧性、解析性和渐近扰动理论，我们建立了线性化问题指数增长解存在的充分条件，从而得到了两种模型的孤波线性不稳定性判据.此外，对于某些特定的参数方案，这两个模型的基态解的非线性稳定性和线性不稳定性。通过李雅普诺夫策略和变分分析，我们得到了临界分数阶KdV方程孤波爆破的稳定性。本文的研究结果对其他非线性发展方程行波解的不稳定性研究具有一定的参考价值。
A Rational Weakly Non-Linear Theory: Taylor-Couette Instability With a Continuous Spectrum
本文研究了圆形Couette流第一分叉点附近不稳定波的连续谱的非线性演化。扰动由所有可能的轴向波数的傅里叶积分表示，并导出了连续谱的振幅密度函数的积分微分方程。描述经典弱非线性不稳定性理论的单色波和慢变波包演化的方程是特殊的极限情况。积分微分方程的数值积分表明，最终的平衡态取决于初始扰动，如实验所观察到的，它不是唯一的。稳定的超临界泰勒涡的波数的预测范围被发现是窄于从线性理论的中性曲线的跨度。泰勒涡流与波数以外的这个范围被发现是不稳定的，衰减，但激发另一个波内的窄带。这个结果与Eckhaus和Benjamin-Feir边带不稳定性一致。自1965年Coles的重大贡献以来，人们就知道在给定几何形状的Taylor-Couette流动中，在固定雷诺数下存在多个解。值得注意的是，Coles发现的多重解的存在性不同于当前流行的分歧理论。除Taylor-Couette流动外，混合对流也有Coles意义下的多解（Yao & Ghosh Moulic 1993，1994）。我们认为，Coles意义的非唯一性，它补充了分歧理论，是所有流体流动的一般属性。
The Alber equation is a phase-averaged second-moment model for the statistics of a sea state, which recently has been attracting renewed attention. We extend it in two ways: firstly, we derive a generalized Alber system starting from a system of nonlinear Schrödinger equations, which contains the classical Alber equation as a special case but can also describe crossing seas, i.e. two wavesystems with different wavenumbers crossing. (These can be two completely independent wavenumbers, i.e. in general different directions and different moduli.) We also derive the associated 2-dimensional scalar…
Stable and unstable capillary-gravity waves
本文通过周期性Evans函数方法研究非共振毛细重力波和小振幅Wilton涟漪的稳定性，提出新的分析框架，证明了非共振频率附近稳定性及共振频率附近稳定性与不稳定性特征。
Orbital instability of solitary waves for the generalized BBM equation at the negative critical endpoint
该研究针对广义Benjamin–Bona–Mahony方程，证明在负临界端点速度下，满足特定参数条件的孤立波具有轨道不稳定性，证明结合了余维二coercivity估计、两参数调制与修正局部位势泛函。
Stability of Small Periodic Waves in Fractional KdV Type Equations
考虑了色散和非线性对KdV型非线性偏微分方程周期行波解稳定性的影响，包括广义KdV方程和Benjamin-Ono方程.在这项研究中，我们考虑的谱稳定性的解决方案，出现的小扰动的平衡态。我们的分析的一个关键特点是一个非局部Floquet理论，适合于分析相关的线性化运营商的$L^2(\RM)$频谱的发展。利用谱扰动理论，我们推导出了非线性项的幂次与分数阶色散算子的符号之间的关系，该关系决定了对任意小的局部微扰的谱稳定性和不稳定性。
This paper investigates the stability of traveling wave solutions to the free boundary Euler equations with a submerged point vortex. We prove that sufficiently small-amplitude waves with small enough vortex strength are conditionally orbitally stable. In the process of obtaining this result, we develop a quite general stability/instability theory for bound state solutions of a large class of infinite-dimensional Hamiltonian systems in the presence of symmetry. This is in the spirit of the seminal work of Grillakis, Shatah, and Strauss, but with hypotheses that are relaxed in a number of ways …
Various approaches to studying the stability of solutions of nonlinear PDEs lead to explicit formulae determining the stability or instability of the wave for a wide range of classes of equations. However, these are typically specialized to a particular equation and checking the stability conditions may not be not straightforward. We present results for a large class of problems that reduce the determination of spectral stability of a wave to a simple task of locating zeros of explicitly constructed polynomials. We study spectral stability of small-amplitude periodic waves in scalar Hamiltonia…
Stokes waves at the critical depth are modulational unstable
本文证明临界深度处的斯托克斯波在长波扰动下线性不稳定，通过高阶泰勒展开解决了退化问题，并揭示了特征值过渡的“8”字形结构。
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