论文标题
第六阶和八阶变异计算的逆问题的通用溶液家族的扩展
Extensions of the General Solution Families for the Inverse Problem of the Calculus of Variations for Sixth- and Eighth-order Ordinary Differential Equations
论文作者
论文摘要
本文得出了新的第三阶和四阶层次结构。主术语中的自由系数满足了目前以变异公式的存在而闻名的最通用的差异几何标准，这是通过解决标量六阶和八阶和八阶普通微分方程（ODES）变化的完整反向问题的解决方案得出的。这里获得的拉格朗日人具有更大的自由，因为它们仅需要个人系数的条件。特别是，它们包含四个任意功能，因此基于现有的一般标准的一些调查是我们模型家族的特定情况。由我们的广义拉格朗日人产生的变分方程也可能代表各种非线性进化方程的行进波，其中一些方程恢复已知的物理模型。对于我们广义变异ODE的典型成员，也以适当的参数制度得出了常规和嵌入式孤立波的家族。像往常一样，发现嵌入的孤子仅发生在存在参数空间的部分的孤立曲线上。
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.
