Abstract
The Klein–Gordon equation is one of the highly studied partial differential equations in condensed matter physics. Typically, the nonlinear functions associated with the Klein–Gordon equation are in the form of sinusoidal functions or polynomials. However, when looking at phenomena where logarithmic quantum mechanics is involved, various forms of logarithmic nonlinearities can be incorporated within the Klein–Gordon equation. In this paper, we consider three different forms of logarithmic nonlinearity with the goal of obtaining exact traveling wave solutions. By employing the auxiliary equation method, we are able to obtain real-valued exact solutions that are bounded kinks, periodic and bell-shaped solutions or singular solutions.
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