The modulation of multiple phases leading to the modified Korteweg–de Vries equation

This paper seeks to derive the modified Korteweg–de Vries (mKdV) equation using a novel approach from systems generated from abstract Lagrangians possessing a two-parameter symmetry group. The method utilises a modified modulation approach, which results in the mKdV emerging with coefficients related to the conservation laws possessed by the original Lagrangian system. Alongside this, an adaptation of the method of Kuramoto is developed, providing a simpler mechanism to determine the coefficients of the nonlinear term. The theory is illustrated using two examples of physical interest, one in stratified hydrodynamics and another using a coupled Nonlinear Schrödinger model, to illustrate how the criterion for the mKdV equation to emerge may be assessed and its coefficients generated. Interacting nonlinear waves of two or more phases are a rich source of instability, which lead to the development of defects which then evolve over time to form further coherent structures, such as solitary pulses or nonlinear periodic forms. We present here one way in which the evolution of these defects can be modelled, by using the method of modulation to derive nonlinear partial differential equations which govern their evolution. In particular, we extend previous studies in this context to show that one may obtain a modified Korteweg–de Vries (mKdV) equation, whose coefficients come from derivatives of the conservation of wave action associated with the original wavetrain. To help illustrate how this approach can be applied in practice, we study two physically relevant systems, a stratified shallow water system and a coupled Nonlinear Schrödinger model, in order to show how the conditions for the mKdV equation to emerge can be found and the relevant coefficients calculated.