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

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.