【作者】 王兆娟；

【导师】 唐生强；

【作者基本信息】 桂林电子科技大学 ， 应用数学， 2009， 硕士

【摘要】 随着非线性科学的发展,许多物理、化学和生命科学模型都可以转化为非线性方程,如非线性常微分方程、偏微分方程和差分方程等.非线性方程的求解已经成为非线性科学领域的一个重要研究课题.非线性Zakharov-Kuznetsov方程(简称ZK方程)于1974年由Zakharov和Kuznetsov提出.由于该方程是KdV方程在二维空间的典型推广形式之一,因此研究该方程具有广泛的理论意义和实践意义.2005年, Wazwaz结合ZK方程和BBM方程构造了ZK-BBM方程,并运用tanh方法和sine-cosine方法获得了ZK-BBM方程的一些精确解.本文利用动力系统分支理论研究了广义ZK方程和广义ZK-BBM方程,由于它们的行波系统具有奇性,因此本文借助微分方程定性理论研究了对应的正则系统,获得了正则系统有界轨道的定性性质,指出了奇异直线的存在是导致正则系统出现非光滑的周期尖波、孤立尖波、compacton和破缺波的原因,进而分析了广义ZK方程和广义ZK-BBM方程的光滑行波解和非光滑行波解产生的分支参数条件,获得了各种有界行波解存在的充分条件,并求出了上述部分解的精确参数表示.更多还原

【Abstract】 In the view of nonlinear science, a lot of physics, chemistry, and life sciences modelscan be changed into nonlinear equations, such as nonlinear ODE, PDE and di?erenceequation. Solving nonlinear equations has become an important research topic in thefield of nonlinear science.In 1974, Zakharov and Kuznetsov posed the nonlinear Zakharov-Kuznetsov equa-tion(ZK equations in short). This equation is one of the best known two-dimensionalgeneralizations of the KdV equation. Studying this equation is important not only intheory but also in practice.In 2005, Wazwaz constructed the ZK-BBM equation by combining the BBM equa-tion with the ZK equation, and obtained some exact solutions for the ZK-BBM equationby means of the tanh and the sine-cosine methods.In this paper, by using the bifurcation theory of planar dynamical systems to thegeneralized ZK equations and the generalized ZK-BBM equations, since their travellingwave systems have singularity, after applying qualitative theory of di?erential equationsto the corresponding regular systems, the qualitative properties of bounded orbits ofthe regular system are obtained. It is emphasized that the existence of singular straightline is the reason for the appearance of non-smooth periodic cusp wave solutions,solitary cusp wave solutions and breaking wave solutions. Various su?cient conditionsto guarantee the existence of smooth and non-smooth travelling wave solutions aregiven. Therefore the bifurcation parameter conditions which lead to smooth and non-smooth travelling wave solutions to the generalized ZK equations and the generalizedZK-BBM equations are analyzed and the various su?cient conditions to guarantee theexistence of the bounded travelling wave solutions are obtained. Some exact explicitparametric representations of the above waves are determined.更多还原