Multi solitary waves for a fourth order nonlinear Schrödinger type equation

In this article we prove the existence of multi solitary waves of a fourth order Schrödinger equation (4NLS) which describes the motion of the vortex filament. These solutions behave at large time as sum of stable Hasimoto solitons. It is obtained by solving the system backward in time around a sequence of approximate multi solitary waves and showing convergence to a solution with the desired property. The new ingredients of the proof are modulation theory, virial identity adapted to 4NLS and energy estimates. Compare to NLS, 4NLS does not preserve Galilean transform which contributes the main difficulty in spectral analysis of the corresponding linearized operator around the Hasimoto solitons.