Two model equations with a second degree logarithmic nonlinearity and their Gaussian solutions

In the paper, we try to study the mechanism of the existence of Gaussian waves in high degree logarithmic nonlinear wave motions. We first construct two model equations which include the high order dispersion and a second degree logarithmic nonlinearity. And then we prove that the Gaussian waves can exist for high degree logarithmic nonlinear wave equations if the balance between the dispersion and logarithmic nonlinearity is kept. Our mathematical tool is the logarithmic trial equation method.

Automated summary

In this paper, we investigate the mechanism of the existence of Gaussian waves in high degree logarithmic nonlinear wave motions by constructing model equations and proving the balance between dispersion and logarithmic nonlinearity. We use the logarithmic trial equation method as our mathematical tool in the study.