Korteweg–De Vries equation

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, exhibiting typical behaviors 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, Clifford Gardner, John M. Greene, Martin Kruskal and Robert Miura developed the classical inverse scattering method to solve the KdV equation. The KdV equation was first introduced by Joseph Valentin Boussinesq (1877, footnote on page 360) and rediscovered by Diederik Korteweg and Gustav de Vries in 1895, who found the simplest solution, the one-soliton solution. Understanding of the equation and behavior of solutions was greatly advanced by the computer simulations of Norman Zabusky and Kruskal in 1965 and then the development of the inverse scattering transform in 1967. In 1972, T. Kawahara proposed a fifth-order KdV type of equation, known as Kawahara equation, that describes dispersive waves, particularly in cases when the coefficient of the KdV equation becomes very small or zero. In 1970, a generalization of the one-dimensional Korteweg–de Vries (KdV) equation to two spatial dimensions, x and y, was introduced by Kadomtsev and Petviashvili (see Kadomtsev–Petviashvili equation). The stability and dynamics of the KdV type solitons with respect to nonlinear transverse perturbations of finite wavelength are described by the Shrira-Pesenson equation.

Definition The KdV equation is a partial differential equation that models (spatially) one-dimensional nonlinear dispersive nondissipative waves described by a function ϕ ( x , t ) {\displaystyle \phi (x,t)} adhering to:

∂ t ϕ + ∂ x 3 ϕ − 6 ϕ ∂ x ϕ = 0 x ∈ R , t ≥ 0 , {\displaystyle \partial _{t}\phi +\partial _{x}^{3}\phi -6\,\phi \,\partial _{x}\phi =0\,\quad x\in \mathbb {R} ,\;t\geq 0,}

where ∂ x 3 ϕ {\displaystyle \partial _{x}^{3}\phi } accounts for dispersion and the nonlinear element ϕ ∂ x ϕ {\displaystyle \phi \partial _{x}\phi } is an advection term. For modelling shallow water waves, ϕ {\displaystyle \phi } is the height displacement of the water surface from its equilibrium height. The constant 6 {\displaystyle 6} in front of the last term is conventional but of no great significance: multiplying t {\displaystyle t} , x {\displaystyle x} , and ϕ {\displaystyle \phi } by constants can be used to make the coefficients of any of the three terms equal to any given non-zero constants.

Soliton solutions

One-soliton solution Consider solutions in which a fixed waveform, given by f ( X ) {\displaystyle f(X)} , maintains its shape as it travels to the right at phase speed c {\displaystyle c} . Such a solution is given by φ ( x , t ) = f ( x − c t − a ) = f ( X ) {\displaystyle \varphi (x,t)=f(x-ct-a)=f(X)} . Substituting it into the KdV equation gives the ordinary differential equation

− c d f d X + d 3 f d X 3 − 6 f d f d X = 0 , {\displaystyle -c{\frac {df}{dX}}+{\frac {d^{3}f}{dX^{3}}}-6f{\frac {df}{dX}}=0,}

or, integrating with respect to X {\displaystyle X} ,

− c f + d 2 f d X 2 − 3 f 2 = A {\displaystyle -cf+{\frac {d^{2}f}{dX^{2}}}-3f^{2}=A}

where A {\displaystyle A} is a constant of integration. Interpreting the independent variable X {\displaystyle X} above as a virtual time variable, this means

f {\displaystyle f} satisfies Newton's equation of motion of a particle of unit mass in a cubic potential

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