Nonlinear analysis studies equations and variational problems for which the superposition principle fails. If and solve an equation, their sum or a scalar multiple need not solve it. For the ordinary differential equation with , the solution exists for and becomes unbounded at . Multiplying this solution by two changes the two sides by different factors, so it does not produce another solution of the same equation.
The questions extend beyond finding a formula: does a solution exist, is it unique, how long does it remain regular, and does it stay near a stable shape? Nonlinear partial differential equations such as the Generalized Korteweg–De Vries equation and the mass-critical focusing nonlinear Schrödinger equation combine local evolution with conservation laws, compactness and functional inequalities.
A variational method can construct a ground state by minimizing an energy functional. A virial identity can instead show that a regular solution cannot persist indefinitely. Scaling and sharp constants identify the thresholds at which the same conserved quantity changes from controlling an evolution to permitting concentration.
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