Abstract
In this paper the classification of single traveling wave solutions of (1+1) dimensional Gardner equation with variable coefficients is obtained by applying the complete discrimination system to the polynomial and trial equation methods. In particular, the corresponding solutions for the concrete parameters are constructed to show that each solution in the classification can be realized. Moreover, numerical simulations shown in the paper could help us better understand the nature of each solution.
Highlights
Nonlinear differential equations have been widely applied to describe physical phenomena in many scientific fields, such as physics, electronics and other engineering and applied sciences
Finding the exact solutions is of great significance
In some situations, the modified KdV equation should be applied when we discuss the effect of surface tension
Summary
Nonlinear differential equations have been widely applied to describe physical phenomena in many scientific fields, such as physics, electronics and other engineering and applied sciences. The trial equation method [9,10,11,12,13,14] and the complete discrimination system for polynomial method [15,16,17,18,19,20] are applied to the generalized (1 + 1) dimensional Gardner equation with variable coefficients. Few methods can get all the traveling wave solutions to the nonlinear equations. Substituting Eq (5) into Eq (4), an ordinary equation system is obtained, and the trial function F(u) could be a polynomial, rational function, or some other irrational function.
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