Abstract
The qualitative theory of differential equations is applied to the generalized Korteweg–de Vries (KdV) equation. Smooth, peaked and cusped solitary wave solutions of the generalized KdV equation under the boundary condition \(\lim \nolimits _{x \rightarrow \pm \infty }{u}=A \) (\(A\) is a constant) are obtained. The parametric conditions of existence of the smooth, peaked and cusped solitary wave solutions are given using the phase portrait analytical technique. Asymptotic analysis is provided for smooth, peaked and cusped solitary wave solutions of the generalized KdV equation.
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