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Some famous nonlinear equations

Mathematicians and Physicists like to give fancy names to their favorite equations, such as Monge-Ampère, Hamilton-Jacobi, Navier-Stokes, Cahn-Hilliard, Ginzburg-Landau, etc. Do you know the definition and context of the following  famous nonlinear partial differential equations of evolution?

The first two give rise to solitons, the third one to traveling waves, while the last one is connected to interface growth and Tracy-Widom (survey). You may find many other nonlinear equations with fancy names here.

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