Congruence solutions to the localized induction hierarchy in three-dimensional space forms (Q2435300)

The paper deals with the localized induction hierarchy in oriented three-dimensional Riemannian manifolds. The author studies the so-called ``congruence solutions'', moving without changing shape. They have the form \(\tilde \gamma(s, t) = \varphi^t(\gamma(s - ct))\), where \(\{ \varphi^t\}_{t \in \mathbb{R}}\) is a one-parameter group of isometries of the considered manifold, and \(\gamma\) is the so-called ``generating curve'', characterizing the shape of the congruence solution \(\tilde \gamma\). The author proves that in the spaces \(\mathbb{R}^3\), \(S^3\) and \(H^3\), the set of all the generating curves of congruence solutions to the \(n\)-th localized induction equation coincides with the set of all \((n+2)\)-th soliton curves (solutions of the \((n + 2)\)-th stationary equation). In the case when \(\{ \varphi^t\}_{t \in \mathbb{R}}\) is a group of translations in \(\mathbb{R}^3\), it is proved, that the set of all the generating curves, corresponding to the \(n\)-th localized induction equation, coincides with the set of all \((n+1)\)-th soliton curves. The author also investigates low-order soliton curves and obtain the following results. The sets of all first and second soliton curves in an arbitrary oriented three-dimensional Riemannian manifold coincide with the sets of all geodesics and helices, respectively. The set of all third soliton curves in \(\mathbb{R}^3\), \(S^3\) and \(H^3\) coincides with the set of all Kirchhoff rod centerlines.