两类非线性Schrdinger型方程的局部临界周期分支

Local bifurcation of critical periods for two nonlinear Schrdinger type equations

针对非线性Schrdinger型方程的局部临界周期分支问题,通过行波变换将两类Schrdinger型方程转换为同一等价Hamiltonian系统,应用Mathematica计算Hamiltonian系统的周期常数,得到原点为一阶细中心的充要条件,并证明了该系统在原点邻域存在一个局部临界周期分支。分析结果表明,两类非线性Schrdinger型方程均恰有一个局部临界周期分支,即在原点附近邻域内闭轨周期的单调性变换一次。

To solve the local bifurcation of critical periods of nonlinear Schrodinger type equation, the generalized nonlinear de- rivative Schr6dinger equation and the high-order dispersive nonlinear Schrodinger equation are converted to a equivalence Hamiltonian system by using traveling wave transformation. The period constants of Hamiltonian system are calculated and the necessary and sufficient condition of the origin with one order weak center is obtained by using Mathematica software. It is proved that there is one local bifurcation of critical periods in the origin of the system. The result shows that the two non- linear Schrodinger type equations have only one local bifurcation of critical periods in the neighborhood of near origin, which means that the monotonicity of the periodic for two nonlinear Schrodinger type equations changes once.