Nonlinear dynamics of skyrmion strings

The skyrmion core, percolating the volume of the magnet, forms a skyrmion string -- the topological Dirac-string-like object. Here we analyze the nonlinear dynamics of 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 nonlinear Schroedinger equation of the focusing type. As a result, the basic "planar-wave" solution, which has a 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 the micromagnetic simulations. Beyond the class of the traveling-wave solutions, we found Ma-breather propagating along the string. Finally, we proposed a generalized approach, which enables one to describe nonlinear dynamics of the modes of different symmetries, e.g. radially symmetrical or elliptical.