Dynamical analysis and closed-form soliton solutions for the doubly dispersive equation

Abstract The purpose of this work is to examine new exact soliton solutions for the doubly dispersive equation (DDE) by utilizing a finite series in terms of Jacobi elliptic functions (JEFs). For the exact analytical solution of nonlinear partial differential equations (NLPDEs), the JEF expansion approach is frequently employed. The suggested approach yields several solutions, including hyperbolic-type solutions, singular periodic wave solutions, JEF solutions, exponential solutions, bright, dark and bright-dark combo soliton solutions and Weierstrass elliptic double periodic solutions. Experts in engineering models will be interested in the findings, which will help them comprehend waves. To this end, we use the Galilean transformation to derive a dynamical system closely related to the equation. The bifurcation behaviours observed in this derived system were then investigated using concepts from the theory of planar dynamical systems. To examine the possible existence of chaotic behaviours, we carefully evaluate the DDE and add a perturbed term to the dynamical system. Comprehensive two- and three-dimensional (2D and 3D) phase portraits are presented to further enhance this inquiry. Some solutions are selected for the graphical behaviour, and these plots are explained according to real-life applications.