Creation of solitons for sine-Gordon and Korteweg–de Vries equations by means of connections defining the representations of zero curvature

It is possible to create traveling wave type solutions (and in particular soliton solutions) of partial differential equations by means of connections defining the representations of zero curvature. In this paper we create the solitons of the sine-Gordon equation and the Korteweg–de Vries equation. In the final section we compare the proposed method for soliton creation with the inverse scattering method. We systematically use the Cartan–Laptev invariant analytic method in the work.