Emergence of Coupled Korteweg–de-Vries Equations in m $m$ 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 (mcKdV) 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 $m$ coupled KdV-type equations that is parameterized by $m$ non-zero real numbers and two symmetric functions of those $m$ numbers. Second, we show that physically relevant setups such as $N\ge m+1$ multi-component nonlinear Schrödinger equations (mcNLS), under scaling and perturbative treatment, reduce to such a mcKdV equation for a specific choice of the symmetric functions. The reduction from mcNLS to mcKdV requires one to be in a suitable parameter regime where the associated sound speeds of mcNLS are repeated. Hence, we connect the assumptions made in the derivation of the mcKdV system to physically interpretable assumptions for the mcNLS 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.