In this paper, we solve the Duffing equation for given initial conditions. We introduce the concept of the discriminant for the Duffing equation and we solve it in three cases depending on sign of the discriminant. We also show the way the Duffing equation is applied in soliton theory.

An elementary introduction to Maldacena's AdS/CFT correspondence is given, with some emphasis in the Fefferman–Graham construction. This is based on lectures given by one of us (E.A.) at the Universidad Autonoma de Madrid.

In this paper we show the way to apply some and analytical and numerical techniques in order to solve the forced Van der Pol oscillator. We illustrate the obtained results with examples. A comparison with Runge–Kutta numerical method is made in order to see the accuracy of the approximated analytical solution.

In this work, we give approximate expressions for Jacobian and elliptic Weierstrass functions and their inverses by means of the elementary trigonometric functions, sine and cosine. Results are reasonably accurate. We show the way the obtained results may be applied to solve nonlinear ODEs and other problems arising in nonlinear physics. The importance of the results in this work consists on giving easy and accurate way to evaluate the main elliptic functions cn, sn, and dn, as well as the Weierstrass elliptic function and their inverses. A general principle for solving some nonlinear problems through elementary functions is stated. No similar approach has been found in the existing literature.

Future scientific and technological evolution in many areas of applied mathematics and modern physics will necessarily depend on dealing with complex systems. Such systems are complex in both their composition and behavior, namely, dealing with complex dynamical systems using different types of Duffing equations, such as real Duffing equations and complex Duffing equations. In this paper, we derive an analytical solution to a complex Duffing equation. We extend the Krýlov–Bogoliúbov–Mitropólsky method for solving a coupled system of nonlinear oscillators and apply it to solve a generalized form of a complex Duffing equation.