The Korteweg-de Vries equation (KdV equation for short) is the following partial differential equation for a function φ of two real variables, x and t:
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
or, integrating with respect to x,
where A 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.
More precisely, the solution is
![\phi(x,t)=\frac{c}{2}\frac{1}{\cosh ^2\left[{\sqrt{c}\over 2}(x-ct-a)\right ]}](a/../upload/math/a0670b9a78ae5168439efa7defaae1c0.png)
where a is an arbitrary constant. This describes a right-moving soliton.
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