Korteweg-de Vries equation

The Korteweg-de Vries equation (KdVshort) isPDE forfunctiontwo real variables, xt.

Its solutions clump up into solitons.

To see how this works, consider solutionswhichfixed wave form (given by f(x)) maintains its shape astravels toright at speed c. Suchsolutiongiven by φ(x,t) = f(x-ct). This givesdifferential equation

or, integratingrespectx,

isconstantintegration. Interpretingindependent variable x above astime variable, this means f satisfies Newton's equationmotion incubic potential. If parametersadjusted so that f(x) has local maximum at x=0, there issolutionwhich f(x) starts at this point at 'time' -∞, eventually slides down tolocal minimum, then back upother side, reaching an equal height, then reverses direction, ending up atlocal maximum again at time ∞. In other words, f(x) approaches 0 as x→±∞. This ischaracteristic shape ofsolitary wave solution.