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含扰动非线性波方程的Jacobi椭圆函数展开法及其在KdV类方程中的应用
【作者】 贾秀娟；
【导师】 化存才；
【作者基本信息】 云南师范大学 ， 应用数学， 2008， 硕士
【摘要】 非线性波方程是非线性数学物理,特别是孤立子理论中最前沿的研究课题之一。对非线性波方程的求解有助于我们弄清系统在非线性作用下的运动变化规律。以实际物理问题作为背景的含有参数扰动因素的非线性波方程的研究是当代非线性科学的一个重要研究方向。通过对受扰动因素下非线性波方程的求解和定性研究,更能合理地解释复杂的自然现象,更深刻地描述系统的本质特征,极大推动相关学科的发展。本文在许多专家学者研究工作的基础上,对现有求解传统非线性波方程的方法进行分析研究,吸取求解变系数偏微分方程行波解的构造方法,基于Jacobi椭圆函数展开法、齐次平衡法和转化的思想,改进了Jacobi椭圆函数展开法,并将方法应用于受扰情形下的变系数KdV方程,变系数组合KdV—mKdV方程和两类非线性耦合KdV方程,获得了一些有意义的新结果,其中包括变速类孤立波解、变速类三角双曲型周期解,以及变速类三角函数型周期解。本文的方法在求解受扰情形下的非线性波方程中具有普遍性,可推广到其它的非线性波方程及高阶非线性耦合方程组。
【Abstract】 Nonlinear wave equations are one of the forefront topics in the studies of nonlinear mathematical physics. The research on finding solutions of nonlinear wave equations can help us understand the motion laws of the nonlinear systems under the nonlinear interactions. The nonlinear wave equations with a spatiotemporal perturbation based on actual physical problems is one of the significant subjects in contemporary study of nonlinear science. The research on finding solutions and analyzing the qualitative behavior of solutions of nonlinear wave equations with a spatiotemporal perturbation can explain the corresponding natural phenomena reasonably, describe the essential properties of the systems more deeply and promote the development of related subjects greatly.In this dissertation, based on the method of Jacobian elliptic function expansion, homogeneous balance method and transformed thought, the method of Jacobian elliptic function expansion is improved and is applied to KdV equation with variable coefficient, the combined KdV-mKdV equation with variable coefficients and two kinds of the coupled KdV equations. As a result, many helpful conclusions are obtained. In fact, speed-changed similar solitary wave solutions, similar periodic solutions with trigonometric function and periodic solutions with hyperbolic function.It is worthwhile to mention that the above method can also be applied to some other nonlinear wave equations and higher-order coupled equations.