Painlevé Analysis and Hybrid N-soliton Solutions of the Gardner Equation with Time Dependent Coefficients

In this paper, we present the novel Gardner equation with variable coefficients. The notions of Painlevé analysis are utilized to discuss the integrability of the considered variable coefficients Gardner equation. The Cole-Hope transformation and Hirota bilinear approach is used to derive Hirota bilinear form of the Gardner equation with time dependent coefficients. Via Hirota bilinear form, the N-soliton solution is obtained. Graphical results are demonstrated to show the first, second, and third-order for different values of variable coefficient. The results presented here are new and have not been observed in the literature.