Nonlinear partial differential equations and superposition principle

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, Hamilton-Jacobi type, pseudo-linear combination of solutions (in which the operations of addition and multiplication are replaced by pseudo-operations) is again a solution. We also consider the solutions and weak solutions of Cauchy's problem for these equations.