一类双色散波方程的Cauchy问题

This paper concerns with the Cauchy problem for the nonlinear double dispersive wave equation.By the priori estimates and the method in [9],It proves that the Cauchy problem admits a unique global classical solution.And by the concave method,we give sufficient conditions on the blowup of the global solution for the Cauchy problem.     

一类非线性色散型发展方程的反问题

具有主部u_t-u_(xxt)的伪抛物型方程近些年来已引起广泛注意,因为这类方程有着明显的物理背景。众所周知,形如u_t u_x uu_x-u_(xxt)=0的方程是研究小振幅长波的数学模型,并由T.B.Benjamin,J.L.Bona和J.J.Maho     

一类非线性发展方程的精确孤波解

本文首先求出了非线性常微分方程ｕ″（ξ）＋ｍｕ２（ξ）＋ｎｕ３（ξ）＋ｐｕ（ξ）＝ｃ（Ⅰ）和ｕ″（ξ）＋ｒｕ′（ξ）＋ｍｕ２（ξ）＋ｎｕ３（ξ）＋ｐｕ（ξ）＝ｃ（Ⅱ）的显式精确解．进而求出了组合ＢＢＭ方程、Ｂｕｒｇｅｒｓ方程与组合ＢＢＭ方程混合型的钟状孤波解和扭状孤波解，同时还求出了广义Ｂｏｕｓｓｉｎｅｓｑ方程和广义ＫＰ方程的钟状和扭状孤波解．文中指出了其行波解可化为（Ⅰ）的发展方程既有钟状又有扭状孤波解，而其行波解可化为（Ⅱ）的发展方程没有钟状孤波解．     

一类耦合非线性波方程的行波解分支

利用动力系统的Hopf分支理论来研究耦合非线性波方程周期行波解的存在性和稳定性．应用行波法把一类耦合非线性波方程转换为三维动力系统来研究,从而给在不同的参数条件下给出了周期解存在和稳定性的充分条件．     

一类高阶非线性波方程的子方程与精确行波解

结合子方程和动力系统分析的方法研究了一类五阶非线性波方程的精确行波解.得到了这类方程所蕴含的子方程, 并利用子方程在不同参数条件下的精确解, 给出了研究这类高阶非线性波方程行波解的方法, 并以Sawada Kotera方程为例, 给出了该方程的两组精确谷状孤波解和两组光滑周期波解.该研究方法适用于形如对应行波系统可以约化为只含有偶数阶导数、一阶导数平方和未知函数的多项式形式的高阶非线性波方程行波解的研究.     

非线性波方程准确孤立波解的符号计算

该文将机械化数学方法应用于偏微分方程领域，建立了构造一类非线性发展方程孤立波解的一种统一算法，并在计算机数学系统上加以实现，推导出了一批非线性发展方程的精确孤立波解．算法的基本原理是利用非线性发展方程孤立波解的局部性特点，将孤立波表示为双曲正切函数的多项式．从而将非线性发展方程（组）的求解问题转化为非线性代数方程组的求解问题．利用吴文俊消元法在计算机代数系统上求解非线性代数方程组，最终获得非线性发展方程（组）的准确孤立波解.     

一类非线性色散型发展方程反问题的整体解

将一类非线性色散型发展方程反问题转化为抽象空间非线性发展方程Cauchy问题。利用半群方法和赋等价范数技巧,建立了该类抽象发展方程整体解的存在唯一性定理,并应用于所论反问题,得到了该类非线性色散型发展方程反问题整体解的存在唯一性定理,本质地改进了袁忠信得出的解的局部存在唯一性结果。     

一类非线性波方程的光滑与非光滑行波解

讨论了一类偏微分方程的行波解。该方程的行波方程对应于一个平面三次多项式系统,因而可将行波解的研究化为对平面系统所定义的相轨线的拓扑分类研究。应用平面动力系统理论在三参数空间内作定性分析,首先获得三次多项式系统的完整拓扑分类,再将相平面分析的结果返回到非线性波解u(ξ) 。考虑到解关于变量ξ=x-ct在“奇线”近旁的不连续性,可得到各种光滑与非光滑行波的存在条件。     

一类非线性波方程初边值问题解的爆破

该文研究如下具有非线性阻尼项和非线性源项的波方程的初边值问题 u_tt -u_xxt -u_xx -(σ(u²_x)u_x)_x+δ|u_t|^p-1u_t=μ|u|^q-1u, 0 < x <1, 0≤ t ≤T, (0.1) u(0, t)=u(1, t)=0, 0≤t≤ T, (0.2) u(x, 0)=u₀(x), u_t(x, 0)=u₁(x),0≤x≤1.(0.3) 文章将给出问题(0.1)--(0.3)的解在有限时刻爆破的充分条件, 同时将证明问题的局部广义解和局部古典解的存在性和唯一性.     

On the Cauchy problem for higher-order nonlinear dispersive equations

We study the higher-order nonlinear dispersive equation

     

Cauchy problem for a class of nonlinear dispersive wave equations arising in elasto-plastic flow

The paper studies the existence, both locally and globally in time, stability, decay estimates and blowup of solutions to the Cauchy problem for a class of nonlinear dispersive wave equations arising in elasto-plastic flow. Under the assumption that the nonlinear term of the equations is of polynomial growth order, say α, it proves that when α>1, the Cauchy problem admits a unique local solution, which is stable and can be continued to a global solution under rather mild conditions; when α?5 and the initial data is small enough, the Cauchy problem admits a unique global solution and its norm in L1,p(R) decays at the rate for 2<p?10. And if the initial energy is negative, then under a suitable condition on the nonlinear term, the local solutions of the Cauchy problem blow up in finite time.     

Analyticity for a class of nonlocal Kuramoto‐Sivashinsky equations arising in interfacial electrohydrodynamics

In this article, we study the analyticity properties of solutions of the nonlocal Kuramoto‐Sivashinsky equations, defined on 2π‐periodic intervals, where ν is a positive constant; μ is a nonnegative constant; p is an arbitrary but fixed real number in the interval [3,4); and is an operator defined by its symbol in Fourier space, with be the Hilbert transform. We establish spatial analyticity in a strip around the real axis for the solutions of such equations, which possess universal attractors. Also, a lower bound for the width of the strip of analyticity is obtained.     

Inverse problem for a class of one-dimensional wave equations with piecewise constant coefficients

We consider the problem of reconstructing the piecewise constant coefficient of a one-dimensional wave equation on the halfline from the knowledge of the displacement on the boundary caused by an impulse at time zero. This problem is formulated as a nonlinear optimization problem. The objective function of this optimization problem has several special features that have been exploited in building an ad hoc optimization method. The optimization method is based on the solution of a nonlinear system of equations by an algorithm consisting of the evaluation of the unknowns one by one.The research of the third author has been made possible through the support and sponsorship of the Italian Government through the Ministero Pubblica Istruzione under Contract M.P.I. 60% 1987 at the Università di Roma—La Sapienza.     

On the Radius of Spatial Analyticity for the Inviscid Boussinesq Equations

In this paper, we study the problem of analyticity of smooth solutions of the inviscid Boussinesq equations. If the initial datum is real-analytic, the solution remains real-analytic on the existence interval. By an inductive method we can obtain a lower bound on the radius of spatial analyticity of the smooth solution.     

The limit behavior of the solutions to a class of nonlinear dispersive wave equations

In this paper, we study the limit behavior of the solutions to a class of nonlinear dispersive wave equations. We also demonstrate that the solutions to Eq. (1.1) converge to the solution to the corresponding BBM equation as the parameter γ converges to zero. And the convergence of solutions to Eq. (1.1) as α²→0 is studied in Hs(R), . Given the discussion of the parameters in the nonlinear dispersive wave equation (1.1), one will obtain some conditions in which compacton and peakon occur.     

Decay estimates for a class of wave equations

In this paper we use a unified way studying the decay estimate for a class of dispersive semigroup given by , where is smooth away from the origin. Especially, the decay estimates for the solutions of the Klein-Gordon equation and the beam equation are simplified and slightly improved.     

New multisoliton solutions of the (2+1)-dimensional dispersive long wave equations

IntroductionSoliton is a complicated mathematical structure based on the nonlinear evolution equation.(1+ 1)-dimensional soliton and solitary wave solutions have been studied we1l and widely appliedto many physics fields like the condense matter physics, fluid mechanics, plasma physics, optics,etc. However, to find some exact physically significant soliton solutions in (2+l)-dimensions ismuch more difficult than in (1+1)-dimensions. Recently, by using some different approashes,one special type…     

The Cauchy problem for a class of quasilinear parabolic equations

The present paper is concerned with the Cauchy problem for the parabolic equation u_t+H(t,x,u,u)=u. New conditions guaranteeing the global classical solvability are formulated. Moreover, it is shown that the same conditions guarantee the global existence of the Lipschitz continuous viscosity solution for the related Hamilton–Jacobi equation. Mathematics Subject Classification (2000) 35K15, 35F25     

Unique solvability of the Cauchy problem for certain quasilinear pseudoparabolic equations

We study the Cauchy problem in the layer Π _T =ℝ ⁿ ×[0,T] for the equationu _t =cGΔu _t +Δϕ(u), wherec is a positive constant and the functionϕ(p) belongs toC ¹(ℝ₊) and has a nonnegative monotone non-decreasing derivative. The unique solvability of this Cauchy problem is established for the class of nonnegative functionsu(x,t) ∈C _x,t ^2,1 (Π _T ) with the properties: , . Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 356–362, September, 1996. This research was partially supported by the International Science Foundation under grant No. MX6000.