KdV equation: Difference between revisions
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This is the Korteweg de Vries equation for a quantity <math>A(x,t)</math> in physical form | This is the Korteweg de Vries equation for a quantity <math>A(x,t)</math> in physical form | ||
<br> | <br> | ||
<math>A_t = -c A_x + \alpha A A_x + \beta A_{xxx}</math> | <math>A_t = -c A_x + \alpha A A_x + \beta A_{xxx}\,</math> | ||
<br> | <br> | ||
where subscripts denote partial derivatives. | where subscripts denote partial derivatives. | ||
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The parameter c denotes the linear advective wave speed, the parameter <math>\alpha</math> denotes the nonlinear effects while the parameter <math>\beta</math> denotes the dispersive effect. In practice these parameters depend on the physical situation considered. | The parameter c denotes the linear advective wave speed, the parameter <math>\alpha</math> denotes the nonlinear effects while the parameter <math>\beta</math> denotes the dispersive effect. In practice these parameters depend on the physical situation considered. | ||
To understand the meaning of the nonlinear and dispersive terms consider them one at a time. | To understand the meaning of the nonlinear and dispersive terms consider them one at a time. | ||
* The nonlinear term: <math>A_t = -(c-\alpha A) A_x.\, </math> | |||
<math>A_t = -(c-\alpha A) A_x. </math> | ** You can see that if <math>A>0</math> and <math>\alpha</math> is negative then larger waves have faster propagation speeds. | ||
* The Fourier transform of the dispersive term: <math>\bar{A}_t = -ik(c-\beta k^2) \bar{A}. </math> (where <math>k</math> is the wavenumber) | |||
You can see that if <math>A>0</math> and <math>\alpha</math> is negative then larger waves have faster propagation speeds. | ** Thus when <math>\beta</math> is positive shorter waves (with larger k) propagate slower. | ||
<math>\bar{A}_t = -ik(c-\beta k^2) \bar{A}. </math> | The [[Wave Envelope]] page uses the KdV equation to illustrate the difference between group velocity and phase speed via wave envelopes. | ||
where <math>k</math> is the wavenumber | |||
Latest revision as of 13:12, 24 July 2020
This is the Korteweg de Vries equation for a quantity in physical form
where subscripts denote partial derivatives.
The parameter c denotes the linear advective wave speed, the parameter denotes the nonlinear effects while the parameter denotes the dispersive effect. In practice these parameters depend on the physical situation considered.
To understand the meaning of the nonlinear and dispersive terms consider them one at a time.
- The nonlinear term:
- You can see that if and is negative then larger waves have faster propagation speeds.
- The Fourier transform of the dispersive term: (where is the wavenumber)
- Thus when is positive shorter waves (with larger k) propagate slower.
The Wave Envelope page uses the KdV equation to illustrate the difference between group velocity and phase speed via wave envelopes.