标题： $m$场中耦合Korteweg-de Vries方程的出现 显示英文标题

标题： Emergence of coupled Korteweg-de Vries equations in $m$ fields

摘要： Korteweg-de Vries（KdV）方程在广泛的学科中具有基础重要性，其推广到多组分系统对于多组分流体和冷原子混合物相关。 我们提出一个通用框架，在合理的非线性耦合性质假设下，从更广泛的数学结构中自然出现一组多组分KdV（mKdV）方程。 特别是，我们推导出一种适用于$m$个KdV方程的系统通用形式，该形式由$m$个非零实数参数化，并且由这些$m$个数的两个对称函数决定。 其次，我们表明物理上有意义的情况，如$N\geq m+1$个多组分非线性薛定谔方程（MNLS），在进行尺度变换和微扰处理后，会简化为特定对称函数选择下的mKdV方程。 从MNLS到mKdV的约化要求处于一个合适的参数区域，其中相关的声速是重复的。 因此，我们将mKdV系统推导中所做的假设与MNLS方程的物理可解释假设联系起来。 最后，我们的方法提供了一个系统的基础，以促进从一般数学结构出发，自然地出现多组分偏微分方程。 显示英文摘要

摘要： The Korteweg-de Vries (KdV) equation is of fundamental importance in a wide range of subjects with generalization to multi-component systems relevant for multi-species fluids and cold atomic mixtures. We present a general framework in which a family of multi-component KdV (mKdV) equations naturally arises from a broader mathematical structure under reasonable assumptions on the nature of the nonlinear couplings. In particular, we derive a universal form for such a system of $m$ KdV equations that is parameterized by $m$ non-zero real numbers and two symmetric functions of those $m$ numbers. Secondly, we show that physically relevant setups such as $N\geq m+1$ multi-component nonlinear Schr\"odinger equations (MNLS), under scaling and perturbative treatment, reduce to such a mKdV equation for a specific choice of the symmetric functions. The reduction from MNLS to mKdV requires one to be in a suitable parameter regime where the associated sound speeds are repeated. Hence, we connect the assumptions made in the derivation of mKdV system to physically interpretable assumptions for the MNLS equation. Lastly, our approach provides a systematic foundation for facilitating a natural emergence of multi-component partial differential equations starting from a general mathematical structure.