This paper proposes and analyzes an ultra-weak local discontinuous Galerkin scheme for one-dimensional nonlinear biharmonic Schr\"{o}dinger equations. We develop the paradigm of the local discontinuous Galerkin method by introducing the second-order spatial derivative as an auxiliary variable instead of the conventional first-order derivative. The proposed semi-discrete scheme preserves a few physically relevant properties such as the conservation of mass and the conservation of Hamiltonian accompanied by its stability for the targeted nonlinear biharmonic Schr\"{o}dinger equations. We also derive optimal $L^2$-error estimates of the scheme that measure both the solution and the auxiliary variable. Several numerical studies demonstrate and support our theoretical findings.

中文翻译：

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非线性双调和薛定谔方程的一种不连续伽辽金方法

本文针对一维非线性双调和 Schr\"{o}dinger 方程提出并分析了一种超弱局部不连续 Galerkin 格式。我们通过引入二阶空间导数作为辅助变量开发了局部不连续 Galerkin 方法的范式而不是传统的一阶导数。所提出的半离散方案保留了一些物理相关的特性，例如质量守恒和哈密顿量守恒，并伴随着目标非线性双调和 Schr\"{o}dinger 方程的稳定性。我们还推导出了方案的最优 $L^2$ 误差估计，该方案测量了解决方案和辅助变量。一些数值研究证明并支持我们的理论发现。