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

Abstract

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.

Abundant Bounded and Unbounded Solitary, Periodic, Rogue‐Type Wave Solutions and Analysis of Parametric Effect on the Solutions to Nonlinear Klein–Gordon Model

“…Due to the sensitive effect of a rogue wave, scholars are being highly interested in deriving rogue and such type of colliding wave solutions in the recent exploration [33][34][35][36]. Moreover, dynamical researchers have investigated new rogon waves [37], optical M-shaped solitons with interaction with shock waves [38], orbital stability of solitary waves [39], and the nonexistence of global solutions of the time-fractional model [40] newly.…”