The skyrmion core, percolating the volume of the magnet, forms a skyrmion string—a topological Dirac-string-like object. Here we analyze the nonlinear dynamics of a skyrmion string in a low-energy regime by means of the collective variables approach, which we generalized for the case of strings. Using the perturbative method of multiple scales (both in space and time), we show that the weakly nonlinear dynamics of the translational mode propagating along the string is captured by the focusing-type nonlinear Schrödinger equation. As a result, the basic “planar-wave” solution, which has the form of a helix-shaped wave, experiences modulational instability. The latter leads to the formation of cnoidal waves. Both types of cnoidal waves, dn- and cn-waves, as well as the separatrix soliton solution, are confirmed by micromagnetic simulations. Beyond the class of traveling-wave solutions, we found Ma-breather propagating along the string. Finally, we proposed a generalized approach that enables one to describe the nonlinear dynamics of the modes of different symmetries, e.g., radially symmetrical or elliptical.

• Received 27 June 2023
• Revised 20 August 2023
• Accepted 27 September 2023

DOI:https://doi.org/10.1103/PhysRevB.108.144412