This paper deals with fractal Van der Pol damped nonlinear oscillators equation having nonlinearity. By combining the techniques of the Laplace transform and the variational iteration method, we establish approximate periodic solutions for the fractal damped nonlinear systems. In this simple way, nonlinear differential equations can be easily converted into linear differential equations. Illustrative examples including the Van der Pol damped nonlinear oscillator reveal that this method is very effective and convenient for solving fractal nonlinear differential equations. Finally, comparison of the obtained results with those of the other achieved method, also reveals that this coupling method not only suggests an easier method due to the Lagrange multiplier but also can be easily extended to other nonlinear systems.

In this article, various exact solutions of the fifth-order variable coefficient KdV equation with higher-order dispersion term are studied. Because of the complexity of the exact solution of the variable coefficient t, it has a certain influence on the tension waves at the fluid interface on the gravity surface. First, the bilinear KdV equation is derived by using the Hirota bilinear method, and four mixed solutions consisting of positive quartic function, quadratic function, exponential function, and hyperbolic function are constructed. Second, the linear superposition principle is used to obtain the resonance multisoliton solution, and two cases are taken as examples to illustrate the study of resonance multi soliton solution. In addition, 3D images and contour images are drawn by mathematical symbol calculation and appropriate parameters, and the process of tension fluctuation is vividly explained by physical phenomena. The results obtained greatly expand the exact solution of the KdV equation in the existing literature and enable us to understand nonlinear dynamical systems more deeply.

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