Generalized Solitary Wave Approximate Analytical and Numerical Solutions for Generalized Damped Forced KdV and Generalized Damped Forced MKdV Equations

In this article, we study the non-linear partial differential equation given by $$u_t+Pu^k u_x+Qu_{xxx}+Su=f(t)$$ , where P, Q, S denote non-linear coefficient, dispersion coefficient, and damping coefficient, respectively; f(t) denotes external hyperbolic forcing term, $$f_0 cosh(\omega t)$$ . The parameter ‘k’ denotes the non-linear exponent. For $$k= n$$ , where $$n \in N$$ , the equation represents the Generalized Damped Forced KdV (GDFKdV) equation, and for $$k= n/2$$ , it can be referred to as the Generalized Modified Damped Forced KdV (GMDFKdV) equation. Initially, analytical solution of the Generalized KdV (GKdV) equation and the Generalized modified KdV (GMKdV) equation are derived employing sine-cosine method. Further, we obtain the solitary wave analytical solutions to the GDFKdV and GMDFKdV equations by using the direct assumption technique. We construct the generalized forms of the solutions, which involve two new parameters, ‘a’ and ‘b’. In the first instance, the solutions to GDFKdV, and GMDFKdV may look very similar. However, in this article, it has been shown that the nature of solitons and their topological structures emerging from these two equations are very different. Using the method of dynamical systems, we analyse the bifurcation and nature of the solutions. Finally, the pseudo-spectral method, which we employed to approximate the solutions, is proven to be ineffective concerning time and the increasing value of exponent power n. Our theoretical results are supported by our numerical experiments.