In this chapter we discuss nonlinear problems. We are interested in smooth solutions. No general theory for nonlinear differential equations is available. Instead, we ask the following questions. Assume that we know a solution U for a particular set of data. Is the problem still solvable if we make small perturbations of the data? Does the solution depend continuously on the perturbation; that is, do small perturbations in the data generate small changes in the solution?
We can linearize the nonlinear equations around the known solution U and we will see that the properties of this linear system often determine the answer to the questions above.
In practice, one often solves nonlinear problems numerically without having any knowledge as to whether the differential equations have a solution. If the numerical solution is smooth in the sense that it varies slowly with respect to the mesh, we can interpolate the numerical solution. The interpolant solves a nearby problem and the solution of the original problem can be considered a perturbation of the numerically constructed solution. Therefore, the questions above are of interest.
For the sake of simplicity, in this chapter we treat initial value problems for partial differential equations with 1-periodic boundary conditions. Conclusions analogous to those derived in this chapter can be drawn for more general initial boundary value problems.