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.
External links
- Korteweg-de Vries equation (http://eqworld.ipmnet.ru/en/solutions/npde/npde5101.pdf) at EqWorld: The World of Mathematical Equations.
- Cylindrical Korteweg-de Vries equation (http://eqworld.ipmnet.ru/en/solutions/npde/npde5102.pdf) at EqWorld: The World of Mathematical Equations.
- Modified Korteweg-de Vries equation (http://eqworld.ipmnet.ru/en/solutions/npde/npde5103.pdf) at EqWorld: The World of Mathematical Equations.