Principle of superposition

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In a linear equation, the principle of superposition asserts that the sum (superposition) of two solutions is again a solution. More generally, any average or integral of solutions will again be a solution, after some suitable convergence hypotheses and a suitable interpretation of the average. This principle underlies the concept of a fundamental solution, as well as Duhamel's formula.

Nonlinear equations generally do not obey an exact principle of superposition (although completely integrable equations come close). On the other hand, one often has an approximate principle of superposition, in that the sum of two solutions which are sufficiently "non-interacting" will be an approximate solution, and thus (by some suitable stability theory) be close to an exact solution. Such approximate principles of superposition are particularly potent when combined with either an induction on energy or a concentration compactness argument.