Symmetries and similarity reductions of the Dym equation

Abstract

The eigenfunctions of the recursion operator Φ with eigenvalue λi and the inverse of the recursion operator Φi ≡ Φ − λi for the Dym equation (DE) are obtained by means of the pseudopotentials or the isospectral functions of the isospectral problem. The symmetries and algebras of the DE are also given. Using some symmetry subalgebra of the DE, thirteen types of the significant similarity reductions are obtained by virtue of the classical Lie approach. For six types of reductions, the general solutions can be obtained by means of the Weierstrass elliptic function, Riemann's zeta function, Jacobi elliptic functions and the solutions of a Riccati equation implicitly. Three types of reductions can all be solved by means of the Painlevé II equation but with different independent arguments. Some types of nontravelling singular solitary waves exist for the Dym equation there are two types of nonsingular nontravelling solitary wave solutions for some suitable potentials.