Exact solutions of the Hirota equation and vortex filaments motion

•We construct an explicit soliton solution formula for the Hirota equation.•We get our solutions via the Inverse Scattering Method and the matrix triplet method.•Our formula contains a new class of solutions called multipole soliton solutions.•Every solution corresponds to a vortex filament motion with nonzero axial velocity.

By using the Inverse Scattering Transform we construct an explicit soliton solution formula for the Hirota equation. The formula obtained allows one to get, as a particular case, the NN-soliton solution, the breather solution and, most relevantly, a new class of solutions called multipole soliton solutions. We use these exact solutions to study the motion of a vortex filament in an incompressible Euler fluid with nonzero axial velocity.