KdV equation

The Korteweg-de Vries equation (KdV for short) is a PDE for a function of two real variables, x and t.

<math>\partial_t\phi+\partial^3_x\phi+6\phi\partial_x\phi=0<math>

Its solutions clump up into solitons.

To see how this works, consider solutions in which a fixed wave form (given by f(x)) maintains its shape as it travels to the right at speed c. Such a solution is given by φ(x,t) = f(x-ct). This gives the differential equation

<math>-c\frac{df}{dx}+\frac{d^3f}{dx^3}+6f\frac{df}{dx} = 0,<math>

or, integrating with respect to x,

<math>3f^2+\frac{d^2 f}{dx^2}-cf<math>

is a constant of integration. Interpreting the independent variable x above as a time variable, this means f satisfies Newton's equation of motion in a cubic potential. If parameters are adjusted so that f(x) has local maximum at x=0, there is a solution in which f(x) starts at this point at 'time' -∞, eventually slides down to the local minimum, then back up the other side, reaching an equal height, then reverses direction, ending up at the local maximum again at time ∞. In other words, f(x) approaches 0 as x→±∞. This is the characteristic shape of the solitary wave solution.