Stationary Harry-Dym's equation and its relation with geodesics on ellipsoid

Abstract

Harry-Dym's equation (HD) $$r_\tau = (r^{ - \tfrac{1}{2}} )_{xxx} $$ is well-known for its cusp soliton solutions. In this paper, relations are revealed between HD and a completely integrable Hamiltonian system in Liouville sense given by $$H = \frac{1}{2}\left\langle {p,p} \right\rangle - \left\langle {q,q} \right\rangle ^{ - 1} $$ in the symplectic manifold (ℝ2N ,dp/∩). Here 〈ξ,η〉 is the standard inner product in ℝ N .