On the (in)validity of the NLS-KdV system in the study of water waves

TL;DR

This paper critically examines the NLS-KdV system's applicability to water wave modeling, demonstrating it cannot be derived from the full Euler equations, thus questioning its validity.

Contribution

The paper provides a rigorous analysis proving that the NLS-KdV system is not a valid reduction of the full water wave equations, clarifying longstanding assumptions.

Findings

NLS-KdV system cannot be derived from Euler equations for water waves

Linear Schrödinger-KdV system also invalid as a water wave model

Clarifies limitations of coupled wave models in water wave theory

Abstract

It is universally accepted that the cubic, nonlinear Schrodinger equation (NLS) models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves, while the Kortewegde Vries equation (KdV) models the propagation of long waves in dispersive media. A system that couples the two equations seems attractive to model the interaction of long and short waves and such a system has been studied over the last few decades. However, questions about the validity of the system in the study of water waves were raised in our previous work where we presented our analysis using the fifth-order KdV as the starting point. In this paper, these questions are settled unequivocally as we show that the NLS-KdV system or even the linear Schrodinger-KdV system cannot be resulted from the full Euler equations formulated in the study of water waves.