Elliptic curves and traveling wave solutions to the KdV equation
What do elliptic curves
have to do with waves on shallow water surfaces?
It is a surprising and beautiful fact of mathematics that they are deeply related. Points on elliptic curves correspond with traveling wave solutions
of the KdV equation
, a nonlinear PDE that serves as a model of waves on shallow water surfaces.
Pick an elliptic curve using the interactive graph below. The inset graph shows the corresponding elliptic curve, while the animation shows the KdV solution associated with the curve (with ω=-i𝜋/2).
The point is initialized at a=4/3, b=-8/27, which coincides with the 1-soliton solution.