一类广义耦合的非线性波动方程组时间周期解的存在性

研究了一类广义耦合的非线性波动方程组关于时间周期解的问题.首先利用Galerkin方法构造近似时间周期解序列,然后利用先验估计和Laray-Schauder不动点原理,证明近似时间周期解序列的收敛性,从而得到该问题时间周期解的存在性.     

On the Existence of Solitary Traveling Waves for Generalized Hertzian Chains

We consider the question of existence of ??bell-shaped?? (i.e., nonincreasing for x>0 and nondecreasing for x<0) traveling waves for the strain variable of the generalized Hertzian model describing, in the special case of a p=3/2 exponent, the dynamics of a granular chain. The proof of existence of such waves is based on the English and Pego (Proc. Am. Math. Soc. 133:1763, 2005) formulation of the problem. More specifically, we construct an appropriate energy functional, for which we show that the constrained minimization problem over bell-shaped entries has a solution. We also provide an alternative proof of the Friesecke?CWattis result (Commun. Math. Phys. 161:391, 1994) by using the same approach (but where the minimization is not constrained over bell-shaped curves). We briefly discuss and illustrate numerically the implications on the doubly exponential decay properties of the waves, as well as touch upon the modifications of these properties in the presence of a finite precompression force in the model.     

Traveling Waves in Degenerate Diffusion Equations

In this survey, we present a literature review on the study of traveling waves in degenerate diffusion equations by illustrating the interesting and singular wave behavior caused by degeneracy. The main results on wave existence and stability are presented for the typical degenerate equations, including porous medium equations, flux limited diffusion equations, delayed degenerate diffusion equations, and other strong degenerate diffusion equations.     

Periodic Traveling Waves in Diatomic Granular Chains

We study bifurcations of periodic traveling waves in diatomic granular chains from the anti-continuum limit, when the mass ratio between the light and heavy beads is zero. We show that every limiting periodic wave is uniquely continued with respect to the mass ratio parameter, and the periodic waves with a wavelength larger than a certain critical value are spectrally stable. Numerical computations are developed to study how this solution family is continued to the limit of equal mass ratio between the beads, where periodic traveling waves of homogeneous granular chains exist.     

一类广义BBM方程周期行波解的存在性

本文引入和讨论一类新的广义BBM方程,得到了关于这类广义BBM方程周期行波解的一些存在性定理。     

Transverse Orbital Stability of Periodic Traveling Waves for Nonlinear Klein–Gordon Equations

In this paper, we establish the orbital stability of a class of spatially periodic wave train solutions to multidimensional nonlinear Klein–Gordon equations with periodic potential. We show that the orbit generated by the one‐dimensional wave train is stable under the flow of the multidimensional equation under perturbations which are, on one hand, coperiodic with respect to the translation or Galilean variable of propagation, and, on the other hand, periodic (but not necessarily coperiodic) with respect to the transverse directions. That is, we show their transverse orbital stability. The class of periodic wave trains under consideration is the family of subluminal rotational waves, which are periodic in the momentum but unbounded in their position.     

Birkhoff Normal Form and Long Time Existence for Periodic Gravity Water Waves

We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth and give a rigorous proof of a conjecture of Dyachenko-Zakharov [16] concerning the approximate integrability of these equations. More precisely, we prove a rigorous reduction of the water waves equations to its integrable Birkhoff normal form up to order 4. As a consequence, we also obtain a long-time stability result: periodic perturbations of a flat interface that are initially of size ε remain regular and small up to times of order ${\epsilon }^{-3}$. This time scale is expected to be optimal. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.     

Envelope Solitary Waves and Periodic Waves in the AB Equations

This article deals with the envelope solitary waves and periodic waves in the AB equations that serve as model equations describing marginally unstable baroclinic wave packets in geophysical fluids and also ultra‐short pulses in nonlinear optics. An envelope solitary wave has a width proportional to its velocity and inversely proportional to its amplitude. The velocity of the envelope solitary wave is partially dependent on its amplitude in the sense that the amplitude determines the upper or lower limit of the velocity. When two envelope solitary waves collide, they survive the collision and retain their identities except for a shift in the positions of both the envelopes and the carrier waves. The periodic wave solutions in sine wave form may be stable or unstable depending upon the wave parameters. When the sine wave is destabilized by small perturbations, its long‐time evolution shows a Fermi–Pasta–Ulam‐type oscillation.     

Qualitative Analysis and Periodic Cusp Waves to a Class of Generalized Short Pulse Equations

In this paper, we qualitatively study periodic cusp waves to a class of generalized short pulse equations, which are of the general form of three special generalized short pulse equations, from the perspective of dynamical systems. We show the existence of smooth periodic waves, periodic cusp wave and compactons, obtain exact expression of periodic cusp wave and illustrate the limiting process of periodic cusp wave from smooth periodic waves.     

格微分方程的行波解研究简介(英文)

本文介绍了格微分方程关于行波解及其相关渐近性态方面的最新研究进展,如行波解的存在性、唯一性和稳定性,格微分方程的传播失败现象、最小波速和传播速度.本文力求展示相关问题的研究背景和线索脉络,阐述相关结论和研究方法,以期待相关主题研究的进一步深入.     

Traveling Waves for Nonlocal and Non-monotone Delayed Reaction-difusion Equations

We study the existence of traveling wave solutions for a nonlocal and non-monotone delayed reaction-difusion equation.Based on the construction of two associated auxiliary reaction difusion equations with monotonicity and by using the traveling wavefronts of the auxiliary equations,the existence of the positive traveling wave solutions for c≥c is obtained.Also,the exponential asymptotic behavior in the negative infnity was established.Moreover,we apply our results to some reactiondifusion equations with spatio-temporal delay to obtain the existence of traveling waves.These results cover,complement and/or improve some existing ones in the literature.     

具有时滞的非线性微分方程的概周期解的存在性及唯一性

本文研究了一类具有时滞的非线性微分方程的有界解及概周期解的存在性及唯一性问题。利用变量替换和不动点方法及逼近法,我们得到了此类方程有界解和概周期解的存在性及唯一性的一些有趣的新结果。我们的结果大大改进和推广了文［２］的主要结果。     

具有分布时滞的KdV方程的周期行波解的存在性

将研究具有分布时滞的K dV方程Ut+(f*U)Ux+τUxx+Uxxx=0,得出当时滞核函数为弱一般核时,时滞方程周期行波解的存在性.     

Existence of Periodic Solutions in Impulsive Differential Equations

In this paper, we are concerned with the problem of existence of periodic solutions for a class of second order impulsive differential equations. By Poincaré-Bohl theorem, we give several criteria to guarantee that the impulsive differential equation has periodic solutions under assumptions that the nonlinear term satisfies the linear growth conditions. Two specific examples are presented to illustrate the obtained results.     

时间映射与二阶拟线性微分方程周期解的存在性

本文研究二阶拟线性微分方程(φp(x'))'+g(z)=p(t)周期解的存在性.借助于推广的时间映射,在两组不同的条件下证明了该方程至少存在一个周期解.     

广义Hammerstein型方程的存在定理

本文利用单调型强制映射满射性和逼近方法建立了广义Ｈａｍｍｅｒｓｔｅｉｎ型方程ｕ＋Ｋ（ｕ）Ｆ（ｕ）＝０解的存在性定理，这里对于实自反Ｂａｎａｃｈ空间Ｘ的共轭空间Ｘ^*的每个ｕ，Ｋ（ｕ）：Ｘ→Ｘ^*是线性映射，Ｆ：Ｘ^*→Ｘ是任一映射．所得结果推广了Ｓｃｈｉｌｉｎｇ，Ｓｒｉｋａｎｔｈ ａｎｄ Ｊｏｓｈｉ等相应的结果     

About the Existence Time of Solutions for Field Equations in the One Space Dimension

In this paper, we study the lower bounds problem for the existence time or solutions to the different massive Dinac-Kiein-Gordon equations and with different massive Klein-Gordon equations, in one space dimension, for weakly decaying Cauchy data, of size ε. The results assert that the existence time is (almost) larger than ε^{-4}.     

含奇异线的广义KdV方程的行波解

研究了一个广义KdV方程的行波解,在行波变换下,该方程转化成含奇异线的平面系统,通过平衡点分析定性地得到不同参数条件下系统解的特性.特别的,由于相平面上的奇异线的存在,系统具有一些特殊结构的解,例如compactons、kink-compactons、anti-kink-compactons,给出了这些解的积分表达式,并且由椭圆函数积分求出了精确解.     

Existence of Eulerian Solutions to the Semigeostrophic Equations in Physical Space: The 2-Dimensional Periodic Case

In this article we use new regularity and stability estimates for Alexandrov solutions to Monge-Ampère equations, recently established by De Philippis and Figalli [14 De Philippis , G. , Figalli , A. W^{2, 1} regularity for solutions of the Monge-Ampère equation. Preprint.  [Google Scholar]], to provide global in time existence of distributional solutions to the semigeostrophic equations on the 2-dimensional torus, under very mild assumptions on the initial data. A link with Lagrangian solutions is also discussed.