Analytical properties and numerical solutions of the derivative nonlinear Schrödinger equation

Abstract

From the analysis of the symmetries of the derivative nonlinear Schrödinger (DNLS) equation, we obtain a new constant of motion, which may be formally considered as a charge and which is related to the helicity of the physical System. From comparison of these symmetries and those of the soliton solutions, we draw conclusions about the number of constraints that must be imposed and the way a Liapunov functional must be constructed in order to study the solitons' stability. We also examine the relationship between the stability with respect to form and the symmetries that are broken by the soliton solutions. We complete the analysis with some numerical simulations: we solve the DNLS equation taking a slightly perturbed soliton as an initial condition and study its temporal evolution, finding that, as expected, they are stable with respect to form.