Construction of the soliton solutions and modulation instability analysis for the Mel'nikov system. (November 2020) - Kumar, Vineesh; Patel, Arvind
Highlights: This paper discusses three methods for solving the Mel'nikov system. Complex amplitude ansatz method gives bright, dark, kink and breather solitons. He's variational method gives bright, homoclinic orbits and breather solitons. Modulation instability of the system has been investigated. The results have applications in the validation of numerical solutions. Abstract: This paper addresses the soliton solutions of the Mel'nikov system by the complex amplitude ansatz method and the He's variational principle. The bright, anti-bright, dark, anti-dark, kink, anti-kink, bell-shaped, breather and homoclinic orbits are obtained as different possible solutions of the Mel'nikov system by the two methods. The solutions and their dynamical behaviour captured by the two methods are discussed. The solutions contain enough number of free physical parameters. These solutions give a better explanation of the associated physical phenomena. The modulation instability analysis of the steady-state solutions of the Mel'nikov system is investigated. This investigation has revealed that the Mel'nikov system is stable against small perturbation for a certain range of perturbation wave numbers. Apart from this, these solutions also help in validating the amount of accuracy in the numerical solutions and to perform the stability analysis further.