Bifurcation analysis of generalized damped forced KDV equation and its analytical solitary wave solutions

In the present study, we investigate a generalized form of the Korteweg–de Vries (KdV) equation which is given by ut + Punux + Quxxx = 0. The analytical solution is generated using the sine-cosine method, which produces multi-solitons for different values of the parameters involved. The effects of the non-linear coefficient (P), dispersion parameter (Q), and the exponent ’n ’ on to the solitons have been studied. Furthermore, the generalized KdV equation is modified to generalized damped forced KdV (DFKdV equation given by ut + Punux + Quxxx + Su = f (t), where additional damping term Su and forcing term f have been introduced. For this DFKdV, a few standard components such as phase portraits and Lyapunov exponents from the bifurcation theory of planar dynamical systems [4] have been employed to discuss various conditions on the parameters for the existence of the various types of solutions including chaos.