Perturbation Methods

Abstract As we have seen in Chapters 3 and 4, the soliton is a solution of an integrable system (the NLS equation) which is obtained by the leading order approximation of the original equation (the Maxwell equations in a dielectric medium). The most important property of solitons is that even during the process of mutual interaction, each of their identities, such as the amplitude (or width) and velocity (or frequency), remains constant (but not phase shift). This remarkable property of the soliton leads to the concept of the fundamental mode of the nonlinear system. However, in a real situation of physical applications of solitons, we have to consider effects of higher order terms and external perturbations to study the stability of the soliton under the influence of these effects. Here we present several perturbation methods to deal with this issue. In doing so, we first derive in Section 5.1 an infinite number of conservation laws of the NLS equation, the fact of which is a consequence of the integrability and is that which leads to the remarkable property of the soliton solution. In this section, we also discuss the Hamiltonian structure of the NLS equation, and the stability of one soliton solutions. Then we observe how these conserved quantities deteriorate due to perturbations, and in the case of a small perturbation, we see in Section 5.2 that this observation actually determines the behaviour of a perturbed soliton solution. This approach gives a simple method to study a perturbed problem. However, it should be noted that the method gives only limited information on the soliton parameters, and not the structure of solitons including their phase and position. These are essential in considering a multi-soliton perturbation problem.