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

Abstract

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.

Optical solitons of the (2+1)-dimensional nonlinear Schrödinger equation with spatio-temporal dispersion in quadratic-cubic media

“…Whereas, not all forms of the exact solutions are acquired. Thereupon we use the method of complete discrimination system for polynomial proposed by Liu [39,[41][42][43][44] to solve equation (3) and obtain some new exact solutions, which are trigonometric function solutions, solitary wave solutions and elliptic functions double periodic solutions. In the meantime, the topological stability of all solutions is analyzed.…”

Optical wave solutions of highly dispersive nonlinear Schrödinger equation without the existence of inter-model dispersion

“…Kudryashov obtained several solitary wave solutions to the equation through traveling wave reductions [48]. Compared with previous studies by scholars, this article uses the trial equation method and the complete discrimination system for the polynomial method [49][50][51][52][53][54] to solve the model for the first time after using traveling wave transformation. These methods are easier to understand, simpler to calculate, and more effective than existing methods.…”

Optical solitons of Sasa-Satsuma model in birefringent fibers

“…[30,31], we can get new solutions, they are singular rational patterns and elliptic function patterns. This method is proposed by Liu [32][33][34][35][36][37][38][39] and are widely used to solve exact solutions of mathematical physics equations [40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55]. From the new patterns we obtained, we can better understand the dynamic behaviors of the model.…”