We solve the one-dimensional time-independent Klein–Gordon equation in the presence of a smooth potential well. The bound state solutions are given in terms of the Whittaker [Formula: see text] function, and the antiparticle bound state is discussed in terms of potential parameters.
A linear stability analysis is performed for a two-phase flow in a channel to demonstrate the feasibility of using momentum flux parameters to improve the one-dimensional two-fluid model. It is shown that the proposed model is stable within a practical range of pressure and void fraction for a bubbly and a slug flow.
We present a study of the one-dimensional Klein–Gordon equation by a smooth barrier. The scattering solutions are given in terms of the Whittaker Mκ,μ(x) function. The reflection and transmission coefficients are calculated in terms of the energy, the height, and the smoothness of the potential barrier. For any value of the smoothness parameter we observed transmission resonances.
We study the linear stability of equilibrium points of a semilinear phase-field model, giving criteria for stability and instability. In the one-dimensional case, we study the distribution of equilibria and also prove the existence of metastable solutions that evolve very slowly in time.
Bound states of a Klein–Gordon particle in the presence of a smooth potential well