Qualitative behavior and cusped solitons for a partial differential equation

This paper deals with a partial differential equation by making use of the qualitative theory of differential equation. The main results are twofold. First, this work improves the previous result by analyzing the qualitative behavior of degenerate singular points. Second, new types of cusped solitons are obtained by setting the partial differential equation under inhomogeneous boundary condition. Asymptotic analysis and numerical simulations are provided for the cusped solitons of the equation.