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Korteweg–de Vries equation
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Everything about Kdv Equation totally explainedIn mathematics, the Korteweg–de Vries equation ( KdV equation for short) is a mathematical model of waves on shallow water surfaces. It is particularly famous as the prototypical example of an exactly solvable model, that is, a non-linear partial differential equation whose solutions can be exactly and precisely specified. The solutions in turn include prototypical examples of solitons. KdV can be solved by means of the inverse scattering transform. The mathematical theory behind the KdV equation is rich and interesting, and, in the broad sense, is a topic of active mathematical research. The equation is named for Diederik Korteweg and Gustav de Vries who studied it in, though the equation first appears in .
Definition
The KdV equation is a nonlinear, dispersive partial differential equation for a function φ of two real variables, space x and time t :
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with ∂ x and ∂ t denoting partial derivatives with respect to x and t.
Solitons
Consider solutions in which a fixed wave form (given by f( x)) maintains its shape as it travels to the right at phase speed c. Such a solution is given by φ( x, t) = f( x- ct). This gives the ordinary differential equation
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| KdV (spherical) |
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| KdV (super) |
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| KdV (transitional) |
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| KdV (variable coefficients) |
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| Korteweg-de Vries-Burgers equation |
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Further Information
Get more info on 'Kdv Equation'.
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