AbstractWe used the sine-Gordon expansion method to find kink solutions of the Oskolkov equation. A solution can be found by matching coefficients and choosing some parameters of the series. We found two possible solutions—one is kink and the other is a hybrid of kink and pulse solitons. These solutions can be used for further studies, such as their stability or their interaction. Specific parameters from the solution could be useful for controlling the physical behavior of a system governed by the Oskolkov equation.