Korteweg–de Vries equation

In mathematics, the Korteweg–de Vries equation (KdV equation for short) is a mathematical model of waves on shallow water surfaces. It is particularly notable 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. KdV can be solved by means of the inverse scattering transform. The mathematical theory behind the KdV equation is a topic of active research. The KdV equation was first introduced by Boussinesq (1877, footnote on page 360) and rediscovered by Diederik Korteweg and Gustav de Vries (1895).

The KdV equation is a nonlinear, dispersive partial differential equation for a function ${\displaystyle \phi }$ of two real variables, space x and time t :

with ∂_x and ∂_t denoting partial derivatives with respect to x and t.

The constant 6 in front of the last term is conventional but of no great significance: multiplying t, x, and ${\displaystyle \phi }$ by constants can be used to make the coefficients of any of the three terms equal to any given non-zero constants.

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