The Orbital Stability of Solitary Wave Solutions for the Generalized Gardner Equation and the Influence Caused by the Interactions between Nonlinear Terms

In this paper, the orbital stability of solitary wave solutions for the generalized Gardner equation is investigated. Firstly, according to the theory of orbital stability of Grillakis-Shatah-Strauss, a general conclusion is given to determine the orbital stability of solitary wave solutions. Furthermore, on the basis of the two bell-shaped solitary wave solutions of the equation, the explicit expressions of the orbital stability discriminants are deduced to give the orbitally stable and instable intervals for the two solitary waves as the wave velocity changing. Moreover, the influence caused by the interaction between two nonlinear terms is also discussed. From the conclusion, it can be seen that the influences caused by this interaction are apparently when 0<p<4, which shows the complexity of this system with two nonlinear terms. Finally, by deriving the orbital stability discriminant d′′(c) in the form of Gaussian hypergeometric function, the numerical simulations of several main conclusions are given in this paper.

On the classical and nonclassical symmetries of a generalized Gardner equation

In this paper, we consider a generalized Gardner equation from the point of view of classical and nonclassical symmetries in partial differential equations. We perform a complete analysis of the symmetry reductions by using the similarity variables and the similarity solutions which allow us to reduce our equation into an ordinary differential equation. Moreover, we prove that the nonclassical method applied to the equation leads to new symmetries, which cannot be obtained by using the Lie classical method. Finally, we calculate exact travelling wave solutions of the equation by using the simplest equation method.