Nonlinear partial differential equations and superposition principle

Abstract

Unlike linear differential equations, with nonlinear differential equations the principle of superposition (which states that linear combination of solutions of equation is again the solution of this equation) does not hold. With some classes of equations such as equations of Navier-Stokes, Burgers, HamiltonJacobi type, pseudo-linear combination of solutions (in which the operations of addition and multiplication are replaced by pseudooperations) is again a solution. We also consider the solutions and weak solutions of Cauchy's problem for these equations. © 2007 IEEE.