Theory of bichromatic wave groups amplitude amplification using implicit variational method

Abstract

This research aims to study the maximal amplification factor for the evolution of nonlinear wave groups, particularly the evolution of bichromatic wave groups governed by temporal nonlinear Schr¨odinger equation. A new numerical method: implicit variational method has been proposed to simulate the nonlinear wave groups’ evolution. The scheme combines the implicit differences and variational techniques. When the results are compared to the exact solutions of one soliton and bisoliton wave groups, the implicit variational method proved to be better in terms of accuracy and conserving the energy property than the existing known method, the explicit forward difference method and the Crank- Nicolson implicit method. Given an initial condition, the simulation based on the implicit variational method can be used to predict the maximal amplification factor for various form of wave groups at any location. Two analytical approximate models, namely the low-dimensional model and the optimization model with conserved properties, are developed to further exploit the amplification factor. The low-dimensional model takes into account the primary mode and the third order mode which are most relevant for bichromatic waves with small frequency differences. Given an initial condition, an analytical expression for the maximal amplitude of the evolution of bichromatic wave groups within this truncated model can be readily obtained. Good agreement is observed between the analytical and numerical solutions: when the initial amplitude is not too large, or when the difference of frequencies is not too small. The optimization model with conserved properties can predict the maximal amplification factor for all forms of wave groups’ evolutions. Given an initial condition and a prescribed location, the analytical expression of a function which not only achieves the maximal amplitude at that location but is also consistent with the initial values of the conservation of energy, linear momentum and hamiltonian of the nonlinear Schr¨odinger equation, can be obtained. When tested with bichromatic wave groups, the model gives rather accurate prediction of the maximal amplification factor compared to the numerical simulations. The results obtained from numerical simulations and the two analytical approximate models are motivated and relevant in the generation of waves in hydrodynamic laboratories.