Periodic Loop Solutions and Their Limit Forms for the Kudryashov‐Sinelshchikov Equation

Abstract

The Kudryashov-Sinelshchikov equation is studied by using the bifurcation method of dynamical systems and the method of phase portraits analysis. We show that the limit forms of periodic loop solutions contain loop soliton solutions, smooth periodic wave solutions, and periodic cusp wave solutions. Also, some new exact travelling wave solutions are presented through some special phase orbits.

A improved F-expansion method and its application to Kudryashov-Sinelshchikov equation

“…Some new exact solutions expressed by exponential function, hyperbolic functions for the Kudryashov–Sinelshchikov equation, are derived by using this method. The results of references have been enriched. From the derivation of the exact solutions, it is easy to see that the improved F‐expansion method is a useful tool to explore the exact solution for PDE.…”

Local conservation laws, symmetries, and exact solutions for a Kudryashov‐Sinelshchikov equation

“…First of all, many papers have been focused on the special case b =1, d = e =0. () Tu et al classified the Lie symmetries of Equation for those specific values of the parameters. In addition, they constructed the optimal system of subalgebras and obtained exact solutions by using the symmetry reductions.…”

“…These solutions include the form of Jacobi elliptic functions, hyperbolic functions, trigonometric, and rational functions. In He et al,() by using the bifurcation method of dynamical systems and the method of phase portraits analysis, the authors show the limit forms of periodic loop solutions contain loop soliton solutions, smooth periodic wave solutions, and periodic cusp wave solutions as well as the existence of peakons, solitary waves, and nonsmooth periodic waves. Randrüüt and Braun showed that the travelling wave solutions of the KS Equation can be derived from corresponding solutions of a generalized KdV equation.…”

Exact solutions of the Kudryashov-Sinelshchikov equation by modified exp-function method

“…Ryabov [6] obtained some exact solutions for β = −3 and β = −4 using a modification of the truncated expansion method [8]. Li and He discussed the equation by the bifurcation method of dynamical systems and the method of phase portraits analysis [9][10][11]. In [12], the equation is studied by the Lie symmetry method.…”