Bifurcation and Exact Traveling Wave Solutions for the Nonlinear Klein-Gordon Equation

This paper focuses on the construction of traveling wave solutions to the nonlinear Klein--Gordon equation by employing the qualitative theory of planar dynamical systems. Based on this theory, we analytically study the existence of periodic, kink (anti-kink), and solitary wave solutions. We then attempt to construct such solutions. For this purpose, we apply a well-known traveling wave solution to convert the nonlinear Klein--Gordon equation into an ordinary differential equation that can be written as a one-dimensional Hamiltonian system. The qualitative theory is applied to investigate and describe phase portraits of the Hamiltonian system. Based on the bifurcation constraints on the system parameters, we integrate the conserved quantities to build new wave solutions that can be classified into periodic, kink (anti-kink), and solitary wave solutions. Some of the obtained solutions are clarified graphically and their connection with the phase orbits is derived.