  In this article, we follow an idea that is opposite to the idea of Hopf and
Cole: we use transformations in order to transform simpler linear or nonlinear
differential equations (with known solutions) to more complicated nonlinear
differential equations. In such a way, we can obtain numerous exact solutions
of nonlinear differential equations. We apply this methodology to the classical
parabolic differential equation (the wave equation), to the classical
hyperbolic differential equation (the heat equation), and to the classical
elliptic differential equation (Laplace equation). In addition, we use the
methodology to obtain exact solutions of nonlinear ordinary differential
equations by means of the solutions of linear differential equations and by
means of the solutions of the nonlinear differential equations of Bernoulli and
Riccati. Finally, we demonstrate the capacity of the methodology to lead to
exact solutions of nonlinear partial differential equations on the basis of
known solutions of other nonlinear partial differential equations. As an
example of this, we use the Korteweg--de Vries equation and its solutions.
Traveling wave solutions of nonlinear differential equations are of special
interest in this article. We demonstrate the existence of the following
phenomena described by some of the obtained solutions: (i) occurrence of the
solitary wave--solitary antiwave from the solution, which is zero at the
initial moment (analogy of an occurrence of particle and antiparticle from the
vacuum); (ii) splitting of a nonlinear solitary wave into two solitary waves
(analogy of splitting of a particle into two particles); (iii) soliton behavior
of some of the obtained waves; (iv) existence of solitons which move with the
same velocity despite the different shape and amplitude of the solitons.
