Title: Conformal duality of the nonlinear Schrödinger equation: Theory and applications to parameter estimation Show Chinese title

Title: 非线性薛定谔方程的共形对偶性：理论及其在参数估计中的应用

Abstract: The nonlinear Schr\"odinger equation (NLSE) is a rich and versatile model, which in one spatial dimension has stationary solutions similar to those of the linear Schr\"odinger equation as well as more exotic solutions such as solitary waves and quantum droplets. Here we present the unified theory of the NLSE, showing that all stationary solutions of the local one-dimensional cubic-quintic NLSE can be classified according to a single number called the cross-ratio. Any two solutions with the same cross-ratio can be converted into one another using a conformal transformation, and the same also holds true for traveling wave solutions. Further, we introduce an optimization afterburner that relies on this conformal symmetry to substantially improve NLSE parameter estimation from noisy empirical data. The new method therefore should have far reaching practical applications for nonlinear physical systems. Show Chinese abstract

Abstract: 非线性薛定谔方程（NLSE）是一个丰富且多用途的模型，在一维空间中具有类似于线性薛定谔方程的定态解，以及孤波和量子液滴等更奇特的解。 在这里，我们提出了NLSE的统一理论，表明局部一维立方-五次NLSE的所有定态解都可以根据一个称为交叉比的单一数值进行分类。 任何两个具有相同交叉比的解都可以通过共形变换相互转换，旅行波解也是如此。 此外，我们引入了一个优化后燃器，它依赖于这种共形对称性，从而显著提高了从噪声经验数据中对NLSE参数的估计。 因此，新方法应在非线性物理系统中具有深远的实际应用。