New solutions of the Gardner equation: Analytical and numerical analysis of its dynamical understanding

In this paper, we deal with dynamical understanding of the Gardner equation known as the KdV–mKdV equation which studies various areas of physics including plasma physics, fluid dynamics, quantum field theory, solid state physics and others. By using the method of planar dynamical systems approach, in different parameter regions, we obtain the bifurcation of phase portraits of a traveling wave system. Deep analysis of phase portraits conduct to analytical and numerical solutions to the Gardner equation which stands for soliton solutions, kink solution and periodic solutions.