Analytical properties and numerical solutions of the derivative nonlinear Schroedinger equation

From the analysis of the symmetries of the derivative nonlinear Schroedinger (DNLS) equation, a new constant of motion is obtained, 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, conclusions are drawn 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. The relationship between the stability with respect to form and the symmetries that are broken by the soliton solutions is examined. The analysis is completed with some numerical simulations: the DNLS equation is solved taking a slightly perturbed soliton as an initial condition and study its temporal evolution is studied and it is found that, as expected, they are stable with respect to form. (author)