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

TL;DR

This work develops a general Hamiltonian-based framework to derive coupled multi-component KdV (mKdV) equations for $m$ fields, revealing a universal all-to-all, equal-coupling structure parameterized by $m$ real numbers and two symmetric functions. It then connects this framework to physics by showing that, for an $N$-component MNLS with $N \ge m+1$, small-amplitude perturbations reduce to the mKdV system when sound speeds are degenerate (repeated), with explicit coupling coefficients expressed through weights $w_j$ and functions $s_1, s_2$. The analysis uses a Madelung transform and multiple-scales perturbation to derive a coupled KdV system, culminating in a rescaled set of equations where the nonlinear coupling is captured by a cubic quadratic form in the $u_j$ and governed by $s(\bar{w})=(1+\sum_j w_j)^{-1}$. This provides a systematic foundation for the emergence of multi-component PDEs from general structures and offers pathways to extend the approach to other multi-component nonlinear wave systems and symmetry-preserving reductions.

Abstract

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 Schrödinger 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.