Korteweg-de Vries equation

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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:

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

Its solutions clump up into solitons. It is named for Diederik Korteweg and Gustav de Vries.

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=A</math>

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

<math>\phi(x,t)=\frac{c}{2}\frac{1}{\cosh ^2\left[{\sqrt{c}\over 2}(x-ct-a)\right ]}</math>

where a is an arbitrary constant. This describes a right-moving soliton.

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Connections

The KdV equation has several connections to physical problems. It is the governing equation of the string in the Fermi-Pasta-Ulam problem in the continuum limit. The KdV equation also describes shallow-water waves with weakly non-linear restoring forces.

The KdV equation can also be solved using inverse scattering techniques like those applied to the Schrödinger equation.

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External links

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References

  • Korteweg, D. J. and de Vries, F. "On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves." Philosophical Magazine, 39, 422--443, 1895.
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