text: This equation is connected to the "modified" version of this equation via the Miura transform.
full_question: This equation is connected to the "modified" version of this equation via the Miura transform. While this equation possesses an infinite number of conservation laws, a related equation proposed by Benjamin, Bona, and Mahony only possesses three. This equation is integrable via the inverse scattering transform for the 1-dimensional Schrodinger equation. A 2+1 dimensional analogue of this equation is the Kadomtsev-Petviashvili equation. For a function f of one spatial dimension and time, this equation sets f-sub-t plus alpha f times f-sub-x plus f-sub-x-x-x to zero. A hyperbolic-secant-squared solution to this equation is one of the first discovered examples of soliton behavior. For 10 points, name this doubly eponymous nonlinear PDE used to model shallow water waves.
first_sentence: This equation is connected to the "modified" version of this equation via the Miura transform.
raw_answer: Korteweg-de Vries equation [or KdV equation]