Applicability of the hodograph method for the problem of long-scale nonlinear dynamics of a thin vortex filament near a flat boundary

Hamiltonian dynamics of a thin vortex filament in an ideal incompressible fluid near a flat fixed boundary is considered under the conditions that at any point of the curve, determining the shape of the filament, the angle between tangent vector and the boundary plane is small. Also the distance from a point on the curve to the plane is small in comparison with the curvature radius. The dynamics is shown to be effectively described by a nonlinear system of two (1+1)-dimensional partial differential equations. The hodograph transformation reduces this system to a single linear differential equation of the second order with separable variables. Simple solutions of the linear equation are investigated for real values of spectral parameter λ, when the filament projection on the boundary plane has shape of a two-branch spiral or a smoothed angle, depending on the sign of λ.