摘要
在Ortega等人关于Sine-Gordon方程研究结论的基础上,为扩大研究对象并进一步完善概周期解理论,研究了一类在物理中常用的Sine-Gordon方程广义形式的概周期解.该广义形式是把一般形式中变量的正弦函数变换为关于正弦函数的奇数次多项式.在研究概周期解的过程中利用了上下解的方法、紧性准则和概周期函数点态定义的知识.首先介绍了所用到的定义和结论;接下来研究了在一定条件下,广义Sine-Gordon方程解的存在性,并结合文献中已有的结论证得了该弱解的唯一性;最后,证明了当受迫项为概周期函数时,之前所得到的唯一弱解即为Sine-Gordon方程广义形式的概周期解,即得到了概周期解的存在性,并得到了该解在一定范围内的唯一性,取得了较好的结果.所得结果覆盖了已有的结论,具有一定的理论和实际意义.
Based on study conclusions of researchers, including Ortega, the paper researches the almost periodic solution of the Sine-Gordon equation in physics to extend the research objects and then perfect the almost periodic solution theory. In the generalized Sine-Gordon equation the "sinu" is changed by its odd number polynomial. The method of upper and lower soiutions, compactness criterion and definition of the almost periodic function are applied in the research. First, the definition and conclusion usedin this paper are introduced;then the solution of generalized Sine-Gordon equation is found under special condition, and its uniqueness is also got with the conclusion in literature; finally, the almost periodic solution is found with the almost periodic forced, proving that the only weak salution obtained previously is the almost periodic solution to generalized Sine-Gordon equation, and we also obtain its existance and uniqueness within certain limits. The above results cover the accepted conclusions and also have some significance in theory and reality.
出处
《河北北方学院学报(自然科学版)》
2009年第3期11-13,18,共4页
Journal of Hebei North University:Natural Science Edition
关键词
广义Sine-Gordon方程
概周期解
上下解证法
generalized Sine-Gordon equation
almost periodic solution
the method of upper and lower solutions