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| 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 rich and interesting, and, in the broad sense, is a topic of active mathematical research. The KdV equation was first introduced by {{harvs|txt|last=Boussinesq|authorlink=Joseph Valentin Boussinesq|year=1877|loc=footnote on page 360}} and rediscovered by {{harvs|txt|first1=Diederik|last1=Korteweg|author1-link=Diederik Korteweg|first2=Gustav|last2=de Vries|author2-link=Gustav de Vries|year=1895}}.<ref>{{Citation | publisher = Oxford University Press | isbn = 9780198568438 | last = Darrigol | first = O. | title = Worlds of Flow: A History of Hydrodynamics from the Bernoullis to Prandtl | year = 2005 | page=84 }}</ref>
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| ==Definition==
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| The KdV equation is a nonlinear, dispersive [[partial differential equation]] for a [[function (mathematics)|function]] <math>\phi</math> of two [[real number|real]] variables, space ''x'' and time ''t'' :<ref>See e.g. {{citation | title=Solitons in mathematics and physics | first=Alan C. | last=Newell | publisher=SIAM | year=1985 | isbn=0-89871-196-7 }}, p. 6. Or Lax (1968), without the factor 6.</ref>
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| :<math>\partial_t \phi + \partial^3_x \phi + 6\, \phi\, \partial_x \phi =0,\,</math>
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| with ∂<sub>''x''</sub> and ∂<sub>''t''</sub> denoting [[partial derivative]]s with respect to ''x'' and ''t''.
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| The constant 6 in front of the last term is conventional but of no great significance: multiplying ''t'', ''x'', and <math>\phi</math> 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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| ==Soliton solutions==
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| 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 <math>\phi</math>(''x'',''t'') = ''f''(''x'' − ''ct'' − ''a'') = ''f''(''X''). Substituting it into the KdV equation gives the [[ordinary differential equation]]
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| :<math>-c\frac{df}{dX}+\frac{d^3f}{dX^3}+6f\frac{df}{dX} = 0,</math>
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| or, integrating with respect to ''X'',
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| :<math>-cf+\frac{d^2 f}{dX^2}+3f^2=A</math>
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| where ''A'' is a [[constant of integration]]. Interpreting the independent variable ''X'' above as a virtual time variable, this means ''f'' satisfies Newton's [[equation of motion]] in a [[cubic potential]]. If parameters are adjusted so that the potential function ''V''(''X'') has [[local maximum]] at ''X'' = 0, there is a solution in which ''f''(''X'') starts at this point at 'virtual time' −∞, eventually slides down to the [[local minimum]], then back up the other side, reaching an equal height, then reverses direction, ending up at the [[local maximum]] again at time ∞. In other words, ''f''(''X'') approaches 0 as ''X'' → ±∞. This is the characteristic shape of the ''[[solitary wave]]'' solution.
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| More precisely, the solution is
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| :<math>\phi(x,t)=\frac12\, c\, \mathrm{sech}^2\left[{\sqrt{c}\over 2}(x-c\,t-a)\right]</math>
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| where ''sech'' stands for the [[hyperbolic secant]] and ''a'' is an arbitrary constant.<ref name="Vakakis2002">{{cite book|author=Alexander F. Vakakis|title=Normal Modes and Localization in Nonlinear Systems|url=http://books.google.com/books?id=GAdkhFPq5HgC&pg=PA105|accessdate=27 October 2012|date=31 January 2002|publisher=Springer|isbn=978-0-7923-7010-9|pages=105–108}}</ref> This describes a right-moving [[soliton]].
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| ==Integrals of motion==
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| The KdV equation has infinitely many [[integral of motion|integrals of motion]] {{harv|Miura|Gardner|Kruskal|1968}}, which do not change with time. They can be given explicitly as
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| :<math>\int_{-\infty}^{+\infty} P_{2n-1}(\phi,\, \partial_x \phi,\, \partial_x^2 \phi,\, \ldots)\, \text{d}x\,</math>
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| where the polynomials ''P''<sub>''n''</sub> are defined recursively by
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| :<math>
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| \begin{align}
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| P_1&=\phi,
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| \\
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| P_n &= -\frac{dP_{n-1}}{dx} + \sum_{i=1}^{n-2}\, P_i\, P_{n-1-i}
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| \quad \text{ for } n \ge 2.
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| \end{align}
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| </math>
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| The first few integrals of motion are:
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| * the mass <math>\int \phi\, \text{d}x,</math>
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| * the momentum <math>\int \phi^2\, \text{d}x,</math>
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| * the energy <math>\int \frac{1}{3} \phi^3 - \left( \partial_x \phi \right)^2\, \text{d}x.</math>
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| Only the odd-numbered terms ''P''<sub>(2''n''+1)</sub> result in non-trivial (meaning non-zero) integrals of motion {{harv|Dingemans|1997|p=733}}.
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| ==Lax pairs==
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| The KdV equation
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| :<math>\partial_t\phi = 6\, \phi\, \partial_x \phi - \partial_x^3 \phi</math>
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| can be reformulated as the Lax equation
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| :<math>L_t = [L,A] \equiv LA - AL \,</math>
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| with ''L'' a [[Sturm–Liouville operator]]:
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| :<math>
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| \begin{align}
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| L &= -\partial_x^2 + \phi,
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| \\
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| A &= 4 \partial_x^3 - 3 \left[ 2\phi\, \partial_x + (\partial_x \phi) \right]
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| \end{align}
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| </math>
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| and this accounts for the infinite number of first integrals of the KdV equation. {{harv|Lax|1968}}
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| ==Least action principle==
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| The Korteweg–de Vries equation
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| :<math>\partial_t \phi - 6\phi\, \partial_x \phi + \partial_x^3 \phi = 0, \,</math>
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| is the [[Euler–Lagrange equation]] of motion derived from the [[Lagrangian density]], <math>\mathcal{L}\,</math> | |
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| :<math>\mathcal{L} = \frac{1}{2} \partial_x \psi\, \partial_t \psi
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| + \left( \partial_x \psi \right)^3
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| - \frac{1}{2} \left( \partial_x^2 \psi \right)^2 \quad \quad \quad \quad (1) \,</math>
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| with <math>\phi</math> defined by
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| :<math>\phi = \frac{\partial \psi}{\partial x} = \partial_x \psi. \,</math>
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| {{hidden begin|title=Derivation of Euler–Lagrange equations}}
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| Since the Lagrangian (eq (1)) contains second derivatives, the [[Euler–Lagrange equation]] of motion for this field is
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| :<math>\partial_{\mu\mu} \left( \frac{\partial \mathcal{L}}{\partial ( \partial_{\mu\mu} \psi )} \right) - \partial_\mu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \psi )} \right) + \frac{\partial \mathcal{L}}{\partial \psi} = 0 . \quad \quad \quad \quad \quad \quad \quad (2) \,</math>
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| where <math>\partial</math> is a derivative with respect to the <math>\mu</math> component.
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| A sum over <math>\mu</math> is implied so eq (2) really reads,
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| :<math>\partial_{tt} \left( \frac{\partial \mathcal{L}}{\partial ( \partial_{tt} \psi )} \right) + \partial_{xx} \left( \frac{\partial \mathcal{L}}{\partial ( \partial_{xx} \psi )} \right) - \partial_t \left( \frac{\partial \mathcal{L}}{\partial ( \partial_t \psi )} \right) - \partial_x \left( \frac{\partial \mathcal{L}}{\partial ( \partial_x \psi )} \right) + \frac{\partial \mathcal{L}}{\partial \psi} = 0 . \quad \quad (3) \,</math>
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| Evaluate the five terms of eq (3) by plugging in eq (1),
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| :<math>\partial_{tt} \left( \frac{\partial \mathcal{L}}{\partial ( \partial_{tt} \psi )} \right) = 0 \,</math>
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| :<math>\partial_{xx} \left( \frac{\partial \mathcal{L}}{\partial ( \partial_{xx} \psi )} \right) = \partial_{xx} \left( -\partial_{xx} \psi \right) \,</math>
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| :<math>\partial_t \left( \frac{\partial \mathcal{L}}{\partial ( \partial_t \psi )} \right) = \partial_t \left( \frac{1}{2} \partial_x \psi \right) \,</math>
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| :<math>\partial_x \left( \frac{\partial \mathcal{L}}{\partial ( \partial_x \psi )} \right) = \partial_x \left( \frac{1}{2} \partial_t \psi + 3 (\partial_x \psi)^2 \right) \,</math>
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| :<math>\frac{\partial \mathcal{L}}{\partial \psi} = 0 \,</math>
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| Remember the definition <math>\phi = \partial_x \psi \,</math>, so use that to simplify the above terms,
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| :<math>\partial_{xx} \left( - \partial_{xx} \psi \right) = - \partial_{xxx} \phi \,</math>
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| :<math>\partial_t \left( \frac{1}{2} \partial_x \psi \right) = \frac{1}{2} \partial_t \phi \,</math>
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| :<math>\partial_x \left( \frac{1}{2} \partial_t \psi + 3 (\partial_x \psi)^2 \right) = \frac{1}{2} \partial_t \phi + 3 \partial_x (\phi)^2 = \frac{1}{2} \partial_t \phi + 6 \phi \partial_x \phi \,</math>
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| Finally, plug these three non-zero terms back into eq (3) to see
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| :<math>\left(- \partial_{xxx} \phi \right) - \left(\frac{1}{2} \partial_t \phi \right) - \left( \frac{1}{2} \partial_t \phi + 6 \phi \partial_x \phi \right) = 0, \,</math>
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| which is exactly the KdV equation
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| :<math>\partial_t \phi + 6 \phi\, \partial_x \phi + \partial_x^3 \phi = 0 .\,</math>
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| {{hidden end}}
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| ==Long-time asymptotics==
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| 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 {{harvtxt|Zabusky|Kruskal|1965}} and can be rigorously proven using the nonlinear [[Method of steepest descent|steepest descent]] analysis for oscillatory [[Riemann–Hilbert problem]]s.<ref>See e.g. {{harvtxt|Grunert|Teschl|2009}}</ref>
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| ==History==
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| 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. | |
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| The KdV equation was not studied much after this until
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| {{harvtxt|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 problem|Fermi, Pasta, and Ulam]] by showing that the KdV equation was the continuum limit of the [[Fermi–Pasta–Ulam problem|FPU]] system. Development of the analytic solution by means of the [[inverse scattering transform]] was done in 1967 by Gardner, Greene, Kruskal and Miura.<ref>{{Citation | doi=10.1103/PhysRevLett.19.1095 | first1=C.S. | last1=Gardner | first2=J.M. |last2=Greene | first3=M.D. | last3=Kruskal | first4=R.M | last4=Miura | title=Method for solving the Korteweg–de Vries equation | journal=Physical Review Letters | volume=19 | issue=19 | year=1967 | pages=1095–1097 | postscript=. | bibcode=1967PhRvL..19.1095G}}</ref><ref>{{Citation|last1=Dauxois|first1=Thierry |last2=Peyrard|first2=Michel|title=Physics of Solitons|year=2006|publisher=Cambridge University Press|isbn=0-521-85421-0}}</ref>
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| ==Applications and connections==
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| The KdV equation has several connections to physical problems. In addition to being the governing equation of the string in the [[Fermi–Pasta–Ulam problem]] in the continuum limit, it approximately describes the evolution of long, one-dimensional waves in many physical settings, including:
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| * shallow-water waves with weakly [[non-linear]] restoring forces,
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| * long [[internal waves]] in a density-stratified [[ocean]],
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| * [[ion acoustic wave]]s in a [[plasma (physics)|plasma]],
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| * [[Acoustics|acoustic]] waves on a [[crystal lattice]].
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| The KdV equation can also be solved using the [[inverse scattering transform]] such as those applied to the [[non-linear Schrödinger equation]].
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| ==Variations==
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| Many different variations of the KdV equations have been studied. Some are listed in the following table.
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| {| class="wikitable"
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| !Name
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| !Equation
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| |-
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| |Korteweg–de Vries (KdV)
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| |<math>\displaystyle \partial_t\phi + \partial^3_x \phi + 6\, \phi\, \partial_x\phi=0</math>
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| |-
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| |KdV (cylindrical)
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| |<math>\displaystyle \partial_t u + \partial_x^3 u - 6\, u\, \partial_x u + u/2t = 0</math>
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| |-
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| |KdV (deformed)
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| |<math>\displaystyle \partial_t u + \partial_x (\partial_x^2 u - 2\, \eta\, u^3 - 3\, u\, (\partial_x u)^2/2(\eta+u^2)) = 0</math>
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| |-
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| |KdV (generalized)
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| |<math>\displaystyle \partial_t u + \partial_x^3 u = \partial_x^5 u </math>
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| |-
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| |[[Generalized Korteweg–de Vries equation|KdV (generalized)]]
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| |<math>\displaystyle \partial_t u + \partial_x^3 u + \partial_x f(u) = 0</math>
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| |-
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| |[[Lax 7th oder Korteweg–de Vries equation|KdV (Lax 7th)]] {{harvtxt|Darvishi|Kheybari|Khani|2007}}
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| |<math>
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| \begin{align}
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| \partial_{t}u
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| +\partial_{x} & \left\{
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| 35u^{4}+70\left(u^{2}\partial_{x}^{2}u+
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| u\left(\partial_{x}u\right)^{2}\right)
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| \right. \\ & \left. \quad
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| +7\left[2u\partial_{x}^{4}u+
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| 3\left(\partial_{x}^{2}u\right)^{2}+4\partial_{x}\partial_{x}^{3}u\right]
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| +\partial_{x}^{6}u
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| \right\}=0
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| \end{align}
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| </math>
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| |-
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| |[[Modified Korteweg–de Vries equation|KdV (modified)]]
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| |<math>\displaystyle \partial_t u + \partial_x^3 u \pm 6\, u^2\, \partial_x u = 0</math>
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| |-
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| |KdV (modified modified)
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| |<math>\displaystyle \partial_t u + \partial_x^3 u - (\partial_x u)^3/8 + (\partial_x u)(Ae^{au}+B+Ce^{-au}) = 0</math>
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| |-
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| |KdV (spherical)
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| |<math>\displaystyle \partial_t u + \partial_x^3 u - 6\, u\, \partial_x u + u/t = 0</math>
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| |-
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| |[[Super Korteweg–de Vries equation|KdV (super)]]
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| |<math>\displaystyle \partial_t u = 6\, u\, \partial_x u - \partial_x^3 u + 3\, w\, \partial_x^2 w</math>,
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| <math>\displaystyle \partial_t w = 3\, (\partial_x u)\, w + 6\, u\, \partial_x w - 4\, \partial_x^3 w</math>
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| |-
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| |KdV (transitional)
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| |<math>\displaystyle \partial_t u + \partial_x^3 u - 6\, f(t)\, u\, \partial_x u = 0</math>
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| |-
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| |KdV (variable coefficients)
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| |<math>\displaystyle \partial_t u + \beta\, t^n\, \partial_x^3 u + \alpha\, t^nu\, \partial_x u= 0</math>
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| |-
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| | Korteweg–de Vries–Burgers equation
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| |<math>\displaystyle \partial_t u + \mu\, \partial_x^3 u + 2\, u\, \partial_x u -\nu\, \partial_x^2 u = 0</math>
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| |}
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| ==See also==
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| * [[Benjamin–Bona–Mahony equation]]
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| * [[Boussinesq approximation (water waves)]]
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| * [[Cnoidal wave]]
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| * [[Dispersion (water waves)]]
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| * [[Dispersionless equation]]
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| * [[Kadomtsev–Petviashvili equation]]
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| * [[Novikov–Veselov equation]]
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| * [[Ursell number]]
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| * [[Vector soliton]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{citation|last=Boussinesq|first= J.|title= Essai sur la theorie des eaux courantes|series= Memoires presentes par divers savants ` l’Acad. des Sci. Inst. Nat. France, XXIII|pages= 1–680|year=1877|url=http://gallica.bnf.fr/ark:/12148/bpt6k56673076}}
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| * {{cite arXiv | first=E.M. |last=de Jager | year=2006 | eprint=math/0602661 | version=v1 | title= On the origin of the Korteweg–de Vries equation | class=math.HO }}
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| * {{citation | title=Water wave propagation over uneven bottoms | first=M.W. | last=Dingemans | year=1997 | series=Advanced Series on Ocean Engineering | volume=13 | publisher=World Scientific, Singapore | isbn=981-02-0427-2 }}, 2 Parts, 967 pages
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| * {{citation|mr=0716135|last= Drazin|first= P. G.|title= Solitons|series= London Mathematical Society Lecture Note Series|volume= 85|publisher= Cambridge University Press|place= Cambridge|year= 1983|pages= viii+136 |isbn= 0-521-27422-2}}
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| * {{Citation| last = Grunert| first = Katrin| last2 = Teschl| first2 = Gerald| author2-link = Gerald Teschl | year = 2009| title = Long-Time Asymptotics for the Korteweg-de Vries Equation via Nonlinear Steepest Descent| periodical = Math. Phys. Anal. Geom.| volume = 12| issue = 3| pages = 287–324| arxiv = 0807.5041| doi = 10.1007/s11040-009-9062-2|bibcode = 2009MPAG...12..287G }}
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| * {{Citation | last1=Kappeler | first1=Thomas | last2=Pöschel | first2=Jürgen | title=KdV & KAM | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] | isbn=978-3-540-02234-3 | mr=1997070 | year=2003 | volume=45}}
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| * {{citation|first=D. J.|last= Korteweg|first2=G. |last2=de Vries|title=On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves|journal=Philosophical Magazine|volume=39|issue=240|pages= 422–443|year= 1895|doi=10.1080/14786449508620739}}
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| * {{citation|first=P.|last= Lax|authorlink=Peter Lax|title=Integrals of nonlinear equations of evolution and solitary waves|journal=Comm. Pure Applied Math.|volume=21|year=1968|pages= 467–490|doi=10.1002/cpa.3160210503|issue=5 }}
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| *{{Citation
| |
| | doi = 10.1017/S0022112081001559
| |
| | volume = 106
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| | year = 1981
| |
| | pages = 131–147
| |
| | last = Miles
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| | first = John W.
| |
| | authorlink = John W. Miles
| |
| | title = The Korteweg–De Vries equation: A historical essay
| |
| | journal = Journal of Fluid Mechanics
| |
| |bibcode = 1981JFM...106..131M
| |
| | postscript = . }}
| |
| *{{citation|mr=0252826|last= Miura|first= Robert M.|last2= Gardner|first2= Clifford S.|last3= Kruskal|first3= Martin D. |title=Korteweg–de Vries equation and generalizations. II. Existence of conservation laws and constants of motion|journal= J. Mathematical Phys. |volume= 9 |year= 1968 |pages=1204–1209|doi=10.1063/1.1664701|bibcode = 1968JMP.....9.1204M|issue=8 }}
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| *{{springer|id=K/k055800|first=L.A.|last= Takhtadzhyan}}
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| *{{citation|title=Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States
| |
| |first= N. J.|last= Zabusky
| |
| |first2= M. D.|last2= Kruskal
| |
| |journal=Phys. Rev. Lett. |volume=15|pages= 240–243 |year=1965
| |
| |doi= 10.1103/PhysRevLett.15.240|bibcode=1965PhRvL..15..240Z|issue=6}}
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| ==External links==
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| {{Commons category}}
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| * [http://eqworld.ipmnet.ru/en/solutions/npde/npde5101.pdf Korteweg–de Vries equation] at EqWorld: The World of Mathematical Equations.
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| * [http://www.primat.mephi.ru/wiki/ow.asp?Korteweg-de_Vries_equation Korteweg–de Vries equation] at NEQwiki, the nonlinear equations encyclopedia.
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| * [http://eqworld.ipmnet.ru/en/solutions/npde/npde5102.pdf Cylindrical Korteweg–de Vries equation] at EqWorld: The World of Mathematical Equations.
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| * [http://eqworld.ipmnet.ru/en/solutions/npde/npde5103.pdf Modified Korteweg–de Vries equation] at EqWorld: The World of Mathematical Equations.
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| * [http://www.primat.mephi.ru/wiki/ow.asp?Modified_Korteweg-de_Vries_equation Modified Korteweg–de Vries equation] at NEQwiki, the nonlinear equations encyclopedia.
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| * {{mathworld|urlname=Korteweg–deVriesEquation |title=Korteweg–deVries Equation}}
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| * [http://panda.unm.edu/Courses/Finley/p573/solitons/KdVDeriv.pdf Derivation] of the Korteweg–de Vries equation for a narrow canal.
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| *Three Solitons Solution of KdV Equation – [http://www.youtube.com/watch?v=H4rN3Wr4ctw]
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| *Three Solitons (unstable) Solution of KdV Equation – [http://www.youtube.com/watch?v=5z5SylS2QHE]
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| *Mathematical aspects of equations of [http://tosio.math.toronto.edu/wiki/index.php/Korteweg-de_Vries_equation Korteweg–de Vries type] are discussed on the [http://tosio.math.toronto.edu/wiki/index.php/Main_Page Dispersive PDE Wiki].
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| * [http://demonstrations.wolfram.com/SolitonsFromTheKortewegDeVriesEquation/ Solitons from the Korteweg–de Vries Equation] by S. M. Blinder, [[The Wolfram Demonstrations Project]].
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| * [http://lie.math.brocku.ca/~sanco/solitons/index.html Solitons & Nonlinear Wave Equations]
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| {{DEFAULTSORT:Korteweg-De Vries Equation}}
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| [[Category:Partial differential equations]]
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| [[Category:Exactly solvable models]]
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| [[Category:Solitons]]
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| [[Category:Equations of fluid dynamics]]
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