In a nonlinear medium, waves may be self-trapped in structures called solitons. The solitons studied up to now propagate at a speed the order of the speed of the free wave. New solitons studied here are called 'static solitons? because they remain in a defined region of space. Maxwell's equations are particularly convenient to build these solitons. The creation of electron-positron pairs from electromagnetic waves in the vacuum shows a nonlinearity of the electromagnetic medium. Assuming a convenient form for these nonlinearities allows construction of a number of qualitatively different static solitons, which give an elegant solution to the wave-particle duality. The principles of quantum mechanics, which are in practice useless, may be replaced by de Broglie's ?double solution?, his and being, respectively, the waves in the nonlinear and nearly linear regions. The formalism of quantum mechanics is then seen as a phenomenological method of interpolation and extrapolation necessary to remedy the weakness of the methods presently available to study the nonlinear mechanics: the interactions of the static solitons are fundamentally nonlinear.