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 behaviors 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).^[2] 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 Boussinesq (1877, footnote on page 360) and rediscovered by Diederik Korteweg and Gustav de Vries (1895),^[3][4] 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 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 ${\displaystyle \phi }$ of two dimensionless real variables, ${\displaystyle x}$ and ${\displaystyle t}$ which are proportional to space and time respectively:^[5]

${\displaystyle \partial _{t}\phi +\partial _{x}^{3}\phi -6\,\phi \,\partial _{x}\phi =0\,}$

with ${\displaystyle \partial _{x}}$ and ${\displaystyle \partial _{t}}$ denoting partial derivatives with respect to ${\displaystyle x}$ and ${\displaystyle t}$. For modelling shallow water waves, ${\displaystyle \phi }$ is the height displacement of the water surface from its equilibrium height.

The constant ${\displaystyle 6}$ in front of the last term is conventional but of no great significance: multiplying ${\displaystyle t}$, ${\displaystyle x}$, 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.

The ${\displaystyle \partial _{x}^{3}\phi }$ introduces dispersion while ${\displaystyle \phi \partial _{x}\phi }$ is an advection term.

The KdV equation has infinitely many integrals of motion (Miura, Gardner & Kruskal 1968), which do not change with time. They can be given explicitly as

${\displaystyle \int _{-\infty }^{+\infty }P_{2n-1}(\phi ,\,\partial _{x}\phi ,\,\partial _{x}^{2}\phi ,\,\ldots )\,{\text{d}}x\,}$

where the polynomials ${\displaystyle P_{n}}$ are defined recursively by

${\displaystyle {\begin{aligned}P_{1}&=\phi ,\\P_{n}&=-{\frac {dP_{n-1}}{dx}}+\sum _{i=1}^{n-2}\,P_{i}\,P_{n-1-i}\quad {\text{ for }}n\geq 2.\end{aligned}}}$

The first few integrals of motion are:

• the mass ${\displaystyle \int \phi \,\mathrm {d} x,}$
• the momentum ${\displaystyle \int \phi ^{2}\,\mathrm {d} x,}$
• the energy ${\displaystyle \int \left[2\phi ^{3}-\left(\partial _{x}\phi \right)^{2}\right]\,\mathrm {d} x}$.

Only the odd-numbered terms ${\displaystyle P_{2n+1}}$ result in non-trivial (meaning non-zero) integrals of motion (Dingemans 1997, p. 733).

The KdV equation

${\displaystyle \partial _{t}\phi =6\,\phi \,\partial _{x}\phi -\partial _{x}^{3}\phi }$

can be reformulated as the Lax equation

${\displaystyle L_{t}=[L,A]\equiv LA-AL\,}$

with ${\displaystyle L}$ a Sturm–Liouville operator:

${\displaystyle {\begin{aligned}L&=-\partial _{x}^{2}+\phi ,\\A&=4\partial _{x}^{3}-3\left[2\phi \,\partial _{x}+\partial _{x}\phi \right]\end{aligned}}}$

and this accounts for the infinite number of first integrals of the KdV equation (Lax 1968).

In fact, ${\displaystyle L}$ is the time-independent Schrödinger operator (disregarding constants) with potential ${\displaystyle \phi (x,t)}$. It can be shown that due to this Lax formulation that in fact the eigenvalues do not depend on ${\displaystyle t}$.

Zero-curvature representation

Setting the components of the Lax connection to be

$${\displaystyle L_{x}={\begin{pmatrix}0&1\\\phi -\lambda &0\end{pmatrix}},L_{t}={\begin{pmatrix}-\phi _{x}&2\phi +4\lambda \\2\phi ^{2}-\phi _{xx}+2\phi \lambda -4\lambda ^{2}&\phi _{x}\end{pmatrix}},}$$

the KdV equation is equivalent to the zero-curvature equation for the Lax connection,

$${\displaystyle \partial _{t}L_{x}-\partial _{x}L_{t}+[L_{x},L_{t}]=0.}$$

The Korteweg–De Vries equation

${\displaystyle \partial _{t}\phi +6\phi \,\partial _{x}\phi +\partial _{x}^{3}\phi =0,}$

is the Euler–Lagrange equation of motion derived from the Lagrangian density, ${\displaystyle {\mathcal {L}}\,}$

${\displaystyle {\mathcal {L}}:={\frac {1}{2}}\partial _{x}\psi \,\partial _{t}\psi +\left(\partial _{x}\psi \right)^{3}-{\frac {1}{2}}\left(\partial _{x}^{2}\psi \right)^{2}}$

(1)

with ${\displaystyle \phi }$ defined by

${\displaystyle \phi$ :={\frac {\partial \psi }{\partial x}}.}

It can be shown that any sufficiently fast decaying smooth solution will eventually split into a finite superposition of solitons travelling to the right plus a decaying dispersive part travelling to the left. This was first observed by Zabusky & Kruskal (1965) and can be rigorously proven using the nonlinear steepest descent analysis for oscillatory Riemann–Hilbert problems.^[8]

The history of the KdV equation started with experiments by John Scott Russell in 1834, followed by theoretical investigations by Lord Rayleigh and Joseph Boussinesq around 1870 and, finally, Korteweg and De Vries in 1895.

The KdV equation was not studied much after this until Zabusky & Kruskal (1965) discovered numerically that its solutions seemed to decompose at large times into a collection of "solitons": well separated solitary waves. Moreover, the solitons seems to be almost unaffected in shape by passing through each other (though this could cause a change in their position). They also made the connection to earlier numerical experiments by Fermi, Pasta, Ulam, and Tsingou by showing that the KdV equation was the continuum limit of the FPUT system. Development of the analytic solution by means of the inverse scattering transform was done in 1967 by Gardner, Greene, Kruskal and Miura.^[9][10]

The KdV equation is now seen to be closely connected to Huygens' principle.^[11][12]

The KdV equation has several connections to physical problems. In addition to being the governing equation of the string in the Fermi–Pasta–Ulam–Tsingou problem in the continuum limit, it approximately describes the evolution of long, one-dimensional waves in many physical settings, including:

• shallow-water waves with weakly non-linear restoring forces,
• long internal waves in a density-stratified ocean,
• ion acoustic waves in a plasma,
• acoustic waves on a crystal lattice.

The KdV equation can also be solved using the inverse scattering transform such as those applied to the non-linear Schrödinger equation.

KdV equation and the Gross–Pitaevskii equation

This section does not cite any sources. (September 2015)

Considering the simplified solutions of the form

${\displaystyle \phi (x,t)=\phi (x\pm t)}$

we obtain the KdV equation as

${\displaystyle \pm \partial _{x}\phi +\partial _{x}^{3}\phi +6\,\phi \,\partial _{x}\phi =0\,}$

or

${\displaystyle \pm \partial _{x}\phi +\partial _{x}(\partial _{x}^{2}\phi +3\phi ^{2})=0\,}$

Integrating and taking the special case in which the integration constant is zero, we have:

${\displaystyle -\partial _{x}^{2}\phi -3\phi ^{2}=\pm \phi \,}$

which is the ${\displaystyle \lambda =1}$ special case of the generalized stationary Gross–Pitaevskii equation (GPE)

${\displaystyle -\partial _{x}^{2}\phi -3\phi ^{\lambda }\phi =\pm \phi \,}$

Therefore, for the certain class of solutions of generalized GPE (${\displaystyle \lambda =4}$ for the true one-dimensional condensate and ${\displaystyle \lambda =2}$ while using the three dimensional equation in one dimension), two equations are one. Furthermore, taking the ${\displaystyle \lambda =3}$ case with the minus sign and the ${\displaystyle \phi }$ real, one obtains an attractive self-interaction that should yield a bright soliton.^{[citation needed]}

Many different variations of the KdV equations have been studied. Some are listed in the following table.

More information , ...

Name	Equation
Korteweg–De Vries (KdV)	${\displaystyle \displaystyle \partial _{t}u+\partial _{x}^{3}u+6u\partial _{x}u=0}$
KdV (cylindrical)	${\displaystyle \displaystyle \partial _{t}u+\partial _{x}^{3}u-6u\partial _{x}u+{\tfrac {1}{2t}}u=0}$
KdV (deformed)	${\displaystyle \displaystyle \partial _{t}u+\partial _{x}\left({\frac {\partial _{x}^{2}u-2\eta u^{3}-3u(\partial _{x}u)^{2}}{2(\eta +u^{2})}}\right)=0}$
KdV (generalized)	${\displaystyle \displaystyle \partial _{t}u+\partial _{x}^{3}u=\partial _{x}^{5}u}$
KdV (generalized)	${\displaystyle \displaystyle \partial _{t}u+\partial _{x}^{3}u+\partial _{x}f(u)=0}$
KdV (modified)	${\displaystyle \displaystyle \partial _{t}u+\partial _{x}^{3}u\pm 6u^{2}\partial _{x}u=0}$
KdV (modified modified)	${\displaystyle \displaystyle \partial _{t}u+\partial _{x}^{3}u-{\tfrac {1}{8}}(\partial _{x}u)^{3}+(\partial _{x}u)(Ae^{au}+B+Ce^{-au})=0}$
KdV (spherical)	${\displaystyle \displaystyle \partial _{t}u+\partial _{x}^{3}u-6u\partial _{x}u+{\tfrac {1}{t}}u=0}$
KdV (super)	${\displaystyle \displaystyle {\begin{cases}\partial _{t}u=6u\partial _{x}u-\partial _{x}^{3}u+3w\partial _{x}^{2}w\\\partial _{t}w=3(\partial _{x}u)w+6u\partial _{x}w-4\partial _{x}^{3}w\end{cases}}}$
KdV (transitional)	${\displaystyle \displaystyle \partial _{t}u+\partial _{x}^{3}u-6f(t)u\partial _{x}u=0}$
KdV (variable coefficients)	${\displaystyle \displaystyle \partial _{t}u+\beta t^{n}\partial _{x}^{3}u+\alpha t^{n}u\partial _{x}u=0}$
Korteweg–De Vries–Burgers equation[13]	${\displaystyle \displaystyle \partial _{t}u+\mu \partial _{x}^{3}u+2u\partial _{x}u-\nu \partial _{x}^{2}u=0}$
non-homogeneous KdV	${\displaystyle \partial _{t}u+\alpha u+\beta \partial _{x}u+\gamma \partial _{x}^{2}u=Ai(x),\quad u(x,0)=f(x)}$

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