View source for Korteweg–de Vries equation

{{short description|Mathematical model of waves on a shallow water surface}}
[[File:Cnoidal wave m=0.9.svg|thumb|right|upright=1.35|[[Cnoidal wave]] solution to the Korteweg–de Vries equation, in terms of the [[square (algebra)|square]] of the [[Jacobi elliptic function]] cn (and with value of the parameter {{nowrap|1=''m'' {{=}} 0.9}}).]]
[[File:KdV equation.gif|thumb|Numerical solution of the KdV equation {{math|<var>u</var><sub><var>t</var></sub> + <var>u</var><var>u</var><sub><var>x</var></sub> + &delta;<sup>2</sup><var>u</var><sub><var>x</var><var>x</var><var>x</var></sub> {{=}} 0}} ({{math|&delta; {{=}} 0.022}}) with an initial condition {{math|<var>u</var>(<var>x</var>, 0) {{=}} cos(&pi;<var>x</var>)}}. Its calculation was done by the Zabusky–Kruskal scheme.<ref>N.J. Zabusky and M. D. Kruskal, ''Phy. Rev. Lett.'', '''15''', 240 (1965)</ref> The initial cosine wave evolves into a train of solitary-type waves.]]

In [[mathematics]], the '''Korteweg–de Vries (KdV) equation''' 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  {{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>

==Definition==

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>

:<math>\partial_t \phi + \partial^3_x \phi - 6\, \phi\, \partial_x \phi =0\,</math>

with ∂<sub>''x''</sub> and ∂<sub>''t''</sub> denoting [[partial derivative]]s 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 <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.

==Soliton solutions==

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''&nbsp;&minus;&nbsp;''ct''&nbsp;&minus;&nbsp;''a'') = ''f''(''X''). Substituting it into the KdV equation gives the [[ordinary differential equation]]

:<math>-c\frac{df}{dX}+\frac{d^3f}{dX^3}-6f\frac{df}{dX} = 0,</math>

or, integrating with respect to ''X'',

:<math>-cf+\frac{d^2 f}{dX^2}-3f^2=A</math>

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]] of a particle of unit mass in a cubic potential

:<math>V(f) = -(f^3+\frac{1}{2}cf^2 + Af)</math>

If

:<math>A = 0,\,c > 0</math>

then the potential function ''V''(''f'') has [[local maximum]] at ''f''&nbsp;=&nbsp;0, there is a solution in which ''f''(''X'') starts at this point at 'virtual time' &minus;∞, 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&nbsp;∞.  In other words,  ''f''(''X'') approaches  0 as ''X''&nbsp;→&nbsp;±∞.  This is the characteristic shape of the ''[[soliton|solitary wave]]'' solution.

More precisely, the solution is

:<math>\phi(x,t)=-\frac12\, c\, \mathrm{sech}^2\left[{\sqrt{c}\over 2}(x-c\,t-a)\right]</math>

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=https://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]].

==Integrals of motion==

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

:<math>\int_{-\infty}^{+\infty} P_{2n-1}(\phi,\, \partial_x \phi,\, \partial_x^2 \phi,\, \ldots)\, \text{d}x\,</math>

where the polynomials ''P''<sub>''n''</sub> are defined recursively by

:<math>
\begin{align}
  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 \ge 2.
  \end{align}
</math>

The first few integrals of motion are:
* the mass <math>\int \phi\, \text{d}x,</math>
* the momentum <math>\int \phi^2\, \text{d}x,</math>
* the energy <math>\int 2 \phi^3 - \left( \partial_x \phi \right)^2\, \text{d}x.</math>
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}}.

==Lax pairs==

The KdV equation

:<math>\partial_t\phi = 6\, \phi\, \partial_x \phi - \partial_x^3 \phi</math>

can be reformulated as the [[Lax equation]]

:<math>L_t = [L,A] \equiv LA - AL \,</math>

with ''L'' a [[Sturm–Liouville operator]]:

:<math>
\begin{align}
  L &= -\partial_x^2 + \phi, 
  \\
  A &= 4 \partial_x^3 - 3 \left[ \phi\, \partial_x + \partial_x \phi \right]
\end{align}
</math>
and this accounts for the infinite number of first integrals of the KdV equation {{harv|Lax|1968}}.

==Least action principle==

The Korteweg–de Vries equation

:<math>\partial_t \phi +  6\phi\, \partial_x \phi +  \partial_x^3 \phi = 0, \,</math>

is the [[Euler–Lagrange equation]] of motion derived from the [[Lagrangian density]], <math>\mathcal{L}\,</math>

:<math>\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  \quad \quad \quad \quad (1) \,</math>

with <math>\phi</math> defined by

:<math>\phi := \frac{\partial \psi}{\partial x}. \,</math>

{{hidden begin|title=Derivation of Euler–Lagrange equations}}
Since the Lagrangian (eq (1)) contains second derivatives, the [[Euler–Lagrange equation]] of motion for this field is

:<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>

where <math>\partial</math> is a derivative with respect to the <math>\mu</math> component.

A sum over <math>\mu</math> is implied so eq (2) really reads,

:<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>

Evaluate the five terms of eq (3) by plugging in eq (1),

:<math>\partial_{tt} \left( \frac{\partial \mathcal{L}}{\partial ( \partial_{tt} \psi )} \right) = 0 \,</math>
:<math>\partial_{xx} \left( \frac{\partial \mathcal{L}}{\partial ( \partial_{xx} \psi )} \right) = \partial_{xx} \left( -\partial_{xx} \psi \right) \,</math>
:<math>\partial_t \left( \frac{\partial \mathcal{L}}{\partial ( \partial_t \psi )} \right) = \partial_t \left( \frac{1}{2} \partial_x \psi \right) \,</math>
:<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>
:<math>\frac{\partial \mathcal{L}}{\partial \psi} = 0 \,</math>

Remember the definition <math>\phi = \partial_x \psi \,</math>, so use that to simplify the above terms,

:<math>\partial_{xx} \left( - \partial_{xx} \psi \right) = - \partial_{xxx} \phi \,</math>
:<math>\partial_t \left( \frac{1}{2} \partial_x \psi \right) = \frac{1}{2} \partial_t \phi \,</math>
:<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>

Finally, plug these three non-zero terms back into eq (3) to see

:<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>

which is exactly the KdV equation

:<math>\partial_t \phi + 6 \phi\, \partial_x \phi + \partial_x^3 \phi = 0 .\,</math>
{{hidden end}}

==Long-time asymptotics==

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>

==History==

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 {{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–Tsingou problem|Fermi, Pasta, Ulam, and Tsingou]] by showing that the KdV equation was the continuum limit of the [[Fermi–Pasta–Ulam–Tsingou problem|FPUT]] 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>

The KdV equation is now seen to be closely connected to [[Huygens' principle]].<ref>Fabio A. C. C. Chalub1 and Jorge P. Zubelli, "[http://cmaf.fc.ul.pt/arquivo/docs/preprints/pdf/2004/2004024.pdf Huygens’ Principle for Hyperbolic Operators and Integrable Hierarchies]"</ref><ref>Yuri Yu. Berest and Igor M. Loutsenko, "[https://arxiv.org/pdf/solv-int/9704012.pdf Huygens’ Principle in Minkowski Spaces and Soliton Solutions of the Korteweg–de Vries Equation]", {{arxiv|solv-int/9704012}} {{doi|10.1007/s002200050235}}</ref>

==Applications and connections==

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 wave]]s in a [[plasma (physics)|plasma]],
* [[Acoustics|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 ===
{{Unreferenced section|date=September 2015}}

Considering the simplified solutions of the form

:<math>\phi(x,t)=\phi(x \pm t) </math>

we obtain the KdV equation as

:<math> \pm \partial_x \phi + \partial^3_x \phi + 6\, \phi\, \partial_x \phi =0\,</math>

or

:<math> \pm \partial_x \phi +  \partial_x (\partial^2_x \phi + 3 \phi^2) =0\,</math>

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

:<math>  -\partial^2_x \phi - 3 \phi^2 =\pm \phi \,</math>

which is the <math>\lambda=1</math> special case of the generalized stationary [[Gross–Pitaevskii equation]] (GPE)

:<math>  -\partial^2_x \phi - 3 \phi^{\lambda} \phi =\pm \phi \,</math>

Therefore, for the certain class of solutions of generalized GPE (<math>\lambda=4</math> for the true one-dimensional condensate and
<math>\lambda=2</math> while using the three dimensional equation in one dimension), two equations are one. Furthermore, taking the <math>\lambda=3</math> case with the minus sign and the <math>\phi</math> real, one obtains an attractive self-interaction that should yield a [[bright soliton]].{{Fact|date=August 2018}}

==Variations==

Many different variations of the KdV equations have been studied. Some are listed in the following table. 
{| class="wikitable"
!Name
!Equation
|-
|Korteweg–de Vries (KdV)
|<math>\displaystyle \partial_t u + \partial^3_x u + 6u \partial_x u=0</math>
|-
|KdV (cylindrical)
|<math>\displaystyle \partial_t u + \partial_x^3 u - 6u  \partial_x u +  \tfrac{1}{2t}u = 0</math>
|-
|KdV (deformed)
|<math>\displaystyle \partial_t u + \partial_x \left (\frac{\partial_x^2 u - 2 \eta u^3 - 3 u (\partial_x u)^2}{2(\eta+u^2)} \right ) = 0</math>
|-
|KdV (generalized)
|<math>\displaystyle \partial_t u + \partial_x^3 u = \partial_x^5 u </math>
|-
|[[Generalized Korteweg–de Vries equation|KdV (generalized)]]
|<math>\displaystyle \partial_t u + \partial_x^3 u + \partial_x f(u) = 0</math>
|-
|[[Lax 7th oder Korteweg–de Vries equation|KdV (Lax 7th)]] {{harvtxt|Darvishi|Kheybari|Khani|2007}}
|<math>\begin{align}
\partial_{t}u +\partial_{x} & \left\{
      35u^{4}+70\left(u^{2}\partial_{x}^{2}u+
      u\left(\partial_{x}u\right)^{2}\right)
      \right. \\ & \left. \quad
      +7\left ( 2u\partial_{x}^{4}u+
              3\left(\partial_{x}^{2}u\right)^{2}+4\partial_{x}\partial_{x}^{3}u\right )
      +\partial_{x}^{6}u
  \right\}=0 
\end{align}
</math>
|-
|[[Modified Korteweg–de Vries equation|KdV (modified)]]
|<math>\displaystyle \partial_t u + \partial_x^3 u \pm 6 u^2 \partial_x u  = 0</math>
|-
|KdV (modified modified)
|<math>\displaystyle  \partial_t u +  \partial_x^3 u - \tfrac{1}{8}(\partial_x u)^3 + (\partial_x u)(Ae^{au}+B+Ce^{-au}) = 0</math>
|-
|KdV (spherical)
|<math>\displaystyle \partial_t u + \partial_x^3 u - 6 u \partial_x u + \tfrac{1}{t}u = 0</math>
|-
|[[Super Korteweg–de Vries equation|KdV (super)]]
|<math>\displaystyle \begin{cases} \partial_t u = 6 u \partial_x u - \partial_x^3 u + 3 w \partial_x^2 w \\ \partial_t w = 3 (\partial_x u) w + 6 u \partial_x w - 4 \partial_x^3 w \end{cases}</math>
|-
|KdV (transitional)
|<math>\displaystyle \partial_t u + \partial_x^3 u - 6 f(t) u \partial_x u  = 0</math>
|-
|KdV (variable coefficients)
|<math>\displaystyle \partial_t u + \beta t^n \partial_x^3 u + \alpha t^nu \partial_x u=  0</math>
|-
| Korteweg–de Vries–Burgers equation<ref>{{cite journal | title = The proper analytical solution of the Korteweg-de Vries-Burgers equation | journal = Journal of Physics A: Mathematical and General | date = 1987 | first = Jian-Jun | last = Shu | volume = 20 | issue = 2 | pages = 49–56 | doi = 10.1088/0305-4470/20/2/002 |arxiv = 1403.3636 |bibcode = 1987JPhA...20L..49J }}</ref>
|<math>\displaystyle \partial_t u + \mu \partial_x^3 u + 2 u \partial_x u -\nu \partial_x^2 u = 0</math>
|-
|[http://www.spm.uem.br/bspm/pdf/next/Art13.pdf non-homogeneous KdV ( A.Aghili, H.Zeinali, )]
|<math>\partial_{t} u+\alpha u+\beta \partial_{x} u+\gamma \partial_{x}^2 u=Ai(x), \quad  u(x,0)=f(x)</math>
|}

===q-analogs===
For the [[q-analog]] of the KdV equation, see {{harvtxt|Frenkel|1996}} and {{harvtxt|Khesin|Lyubashenko|Roger|1997}}.

==See also==
* [[Benjamin–Bona–Mahony equation]]
* [[Boussinesq approximation (water waves)]]
* [[Cnoidal wave]]
* [[Dispersion (water waves)]]
* [[Dispersionless equation]]
* [[Fifth-order Korteweg–de Vries equation]]
* [[Kadomtsev–Petviashvili equation]]
* [[Modified KdV–Burgers equation]]
* [[Novikov–Veselov equation]]
* [[Seventh-order Korteweg–de Vries equation]]
* [[Ursell number]]
* [[Vector soliton]]

==Notes==
{{reflist}}

==References==

*{{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}}
* {{cite arXiv | first=E.M. |last=de Jager | year=2006 | eprint=math/0602661v1 | title= On the origin of the Korteweg–de Vries equation }}
* {{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
* {{citation|mr=0716135|last= Drazin|first= P. G.| authorlink=Philip Drazin | 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|doi=10.1017/CBO9780511662843}}
* {{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 }}
* {{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 | doi=10.1007/978-3-662-08054-2}}
* {{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}}
* {{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 }}
*{{Citation
| doi = 10.1017/S0022112081001559
| volume = 106
| year = 1981
| pages = 131–147
| last = Miles
| 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. |author3link=Martin Kruskal|title=Korteweg–de Vries equation and generalizations. II. Existence of conservation laws and constants of motion|journal=  J. Math. Phys. |volume= 9 |year= 1968 |pages=1204–1209|doi=10.1063/1.1664701|bibcode = 1968JMP.....9.1204M|issue=8 }}
*{{springer|id=K/k055800|first=L.A.|last= Takhtadzhyan}}
*{{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}}

==External links==
{{Commons category}}
* [http://eqworld.ipmnet.ru/en/solutions/npde/npde5101.pdf Korteweg–de Vries equation] at EqWorld: The World of Mathematical Equations.
* [http://www.primat.mephi.ru/wiki/ow.asp?Korteweg-de_Vries_equation Korteweg–de Vries equation] at NEQwiki, the nonlinear equations encyclopedia.
* [http://eqworld.ipmnet.ru/en/solutions/npde/npde5102.pdf Cylindrical Korteweg–de Vries equation] at EqWorld: The World of Mathematical Equations.
* [http://eqworld.ipmnet.ru/en/solutions/npde/npde5103.pdf Modified Korteweg–de Vries equation] at EqWorld: The World of Mathematical Equations.
* [http://www.primat.mephi.ru/wiki/ow.asp?Modified_Korteweg-de_Vries_equation Modified Korteweg–de Vries equation] at NEQwiki, the nonlinear equations encyclopedia.
* {{mathworld|urlname=Korteweg–deVriesEquation |title=Korteweg–deVries Equation}}
* [https://web.archive.org/web/20081006234627/http://panda.unm.edu/courses/finley/P573/solitons/KdVDeriv.pdf Derivation] of the Korteweg–de Vries equation for a narrow canal.
*Three Solitons Solution of KdV Equation – [https://www.youtube.com/watch?v=H4rN3Wr4ctw]
*Three Solitons (unstable) Solution of KdV Equation – [https://www.youtube.com/watch?v=5z5SylS2QHE]
*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 [https://web.archive.org/web/20060812140140/http://tosio.math.toronto.edu/wiki/index.php/Main_Page Dispersive PDE Wiki].
* [http://demonstrations.wolfram.com/SolitonsFromTheKortewegDeVriesEquation/ Solitons from the Korteweg–de Vries Equation] by  S. M. Blinder, [[The Wolfram Demonstrations Project]].
* [http://lie.math.brocku.ca/~sanco/solitons/ Solitons & Nonlinear Wave Equations]

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