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Concept
Korteweg–De Vries equation
Summary
In mathematics, the Korteweg–De Vries (KdV) equation is a partial differential equation (PDE) which serves as a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an integrable PDE and exhibits many of the expected behaviours for an integrable PDE, such as a large number of explicit solutions, in particular soliton solutions, and an infinite number of conserved quantities, despite the nonlinearity which typically renders PDEs intractable. The KdV can be solved by the inverse scattering method (ISM). In fact, Gardner, Greene, Kruskal and Miura developed the classical inverse scattering method to solve the KdV equation.
The KdV equation was first introduced by and rediscovered by , who found the simplest solution, the one-soliton solution. Understanding of the equation and behaviour of solutions was greatly advanced by the computer simulations of Zabusky and Kruskal in 1965 and then the development of the inverse scattering transform in 1967.
The KdV equation is a nonlinear, dispersive partial differential equation for a function of two dimensionless real variables, and which are proportional to space and time respectively:
with and denoting partial derivatives with respect to and . For modelling shallow water waves, is the height displacement of the water surface from its equilibrium height.
The constant in front of the last term is conventional but of no great significance: multiplying , , and by constants can be used to make the coefficients of any of the three terms equal to any given non-zero constants.
The introduces dispersion while is an advection term.
Consider solutions in which a fixed wave form (given by ) maintains its shape as it travels to the right at phase speed . Such a solution is given by . Substituting it into the KdV equation gives the ordinary differential equation
or, integrating with respect to ,
where is a constant of integration.
Official source
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