标题： 变分法逆问题的高阶常微分方程通解族扩展——六阶和八阶方程 显示英文标题

标题： Extensions of the General Solution Families for the Inverse Problem of the Calculus of Variations for Sixth- and Eighth-order Ordinary Differential Equations

摘要： 本文推导了新的三阶和四阶拉格朗日层次结构。 leading项中的自由系数满足目前关于变分公式存在性的最一般的微分几何标准，这些标准是由求解六阶和八阶标量常微分方程（ODE）的逆变分问题得出的。 这里得到的拉格朗日函数具有更大的自由度，因为它们只需要对单个系数施加条件。 特别是，它们包含四个任意函数，因此基于现有的一般变分表示准则的一些研究是我们模型族的特例。 由我们广义拉格朗日函数得到的变分方程也可以表示各种非线性演化方程的行波，其中一些恢复了已知的物理模型。 对于广义变分ODEs的典型成员，在适当的参数范围内，还推导出了正则孤波和嵌入孤波的族。 与通常一样，嵌入孤波仅存在于其存在的参数空间的一部分上的孤立曲线上。 显示英文摘要

摘要： New third- and fourth-order Lagrangian hierarchies are derived in this paper. The free coefficients in the leading terms satisfy the most general differential geometric criteria currently known for the existence of a variational formulation, as derived by solution of the full inverse problem of the Calculus of Variations for scalar sixth- and eighth-order ordinary differential equations (ODEs). The Lagrangians obtained here have greater freedom since they require conditions only on individual coefficients. In particular, they contain four arbitrary functions, so that some investigations based on the existing general criteria for a variational representation are particular cases of our families of models. The variational equations resulting from our generalized Lagrangians may also represent traveling waves of various nonlinear evolution equations, some of which recover known physical models. For a typical member of our generalized variational ODEs, families of regular and embedded solitary waves are also derived in appropriate parameter regimes. As usual, the embedded solitons are found to occur only on isolated curves in the part of parameter space where they exist.