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The modulation of multiple phases leading to the modified Korteweg-de Vries equation
1Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU, United Kingdom.
Abstract:
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.
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