Korteweg-de Vries equation

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:

Related Topics:
Partial differential equation - Function - Real

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:partial_tphi+partial^3_xphi+6phipartial_xphi=0

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

Related Topics:
Soliton - Diederik Korteweg - Gustav de Vries

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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

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:-crac{df}{dx}+rac{d^3f}{dx^3}+6frac{df}{dx} = 0,

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or, integrating with respect to x,

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:3f^2+rac{d^2 f}{dx^2}-cf=A

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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.

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More precisely, the solution is

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:phi(x,t)=rac{c}{2}rac{1}{cosh ^2left}

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

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