一类三维二阶可积拉格朗日量及其色散变形
On a class of 3D second-order integrable Lagrangians and their dispersive deformations
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中文总结 AI 辅助
该研究确定了一类三维二阶拉格朗日量的四种类型，推导了其欧拉-拉格朗日方程的色散变形，为达布系统的连续及离散版本提供了拉格朗日表述，揭示了其与三角学公式的关联。
中文摘要 AI 辅助
我们研究形如∫f(u_{xy},u_{xt},u_{yt})dxdydt的三维二阶拉格朗日量对应的欧拉-拉格朗日方程的可积性。研究表明，这类拉格朗日密度f恰好有四种不同类型：第一种为f=√(u_{xy}u_{xt}u_{yt})；第二种和第三种更复杂，但仍可由初等函数表示；最一般的第四种可通过罗巴切夫斯基函数表达，揭示出与球面/双曲三角学及施莱夫利型公式的意外联系。我们还研究了对应欧拉-拉格朗日方程的无色散Lax对及可色散变形。第一种拉格朗日密度的色散变形与经典达布系统的拉格朗日量一致，该系统源于ℝ³中三重正交坐标系理论；其余三种情况的色散变形则分别提供了达布系统的半离散、全离散版本的拉格朗日公式，对应1、2、3个离散变量。
英文摘要
We investigate integrability of Euler-Lagrange equations associated with 3D second-order Lagrangians of the form \begin{equation*} \int f(u_{xy},u_{xt},u_{yt})\ \text{d}x\text{d}y\text{d}t. \end{equation*} It is demonstrated that there are exactly four different types of such Lagrangian densities $f$: the first one is given by the formula $f=\sqrt{u_{xy}u_{xt}u_{yt}}$, the second and the third are more complicated (although still representable in elementary functions), whereas the most generic fourth one is expressible in terms of the Lobachevsky function, revealing unexpected links to spherical/hyperbolic trigonometry and Schläfly-type formulas. Dispersionless Lax pairs and integrable dispersive deformations of the corresponding Euler-Lagrange equations are also constructed.Remarkably, dispersive deformation of the first Lagrangian density coincides with the Lagrangian of the classical Darboux system arising in the theory of triply-orthogonal coordinate systems in $\mathbb{R}^3$. Dispersive deformations of the three other cases provide Lagrangian formulation of semi-discrete and fully discrete versions of the Darboux system, with one, two and three discrete variables, respectively.
