# Korteweg–de Vries equation

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 Boussinesq (1877, footnote on page 360) and rediscovered by Diederik Korteweg and Gustav de Vries (1895).[2]

Cnoidal wave solution to the Korteweg–de Vries equation, in terms of the square of the Jacobi elliptic function cn (and with value of the parameter m = 0.9).
Numerical solution of the KdV equation ut + uux + δ2uxxx = 0 (δ = 0.022) with an initial condition u(x, 0) = cos(πx). Its calculation was done by the Zabusky–Kruskal scheme.[1] The initial cosine wave evolves into a train of solitary-type waves.

## Definition

The KdV equation is a nonlinear, dispersive partial differential equation for a function ${\displaystyle \phi }$  of two real variables, space x and time t :[3]

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

with ∂x and ∂t denoting partial derivatives 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 ${\displaystyle \phi }$  by constants can be used to make the coefficients of any of the three terms equal to any given non-zero constants.

## Soliton solutions

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 ${\displaystyle \phi }$ (x,t) = f(x − ct − a) = f(X). Substituting it into the KdV equation gives the ordinary differential equation

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

or, integrating with respect to X,

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

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

${\displaystyle V(f)=-\left(f^{3}+{\frac {1}{2}}cf^{2}+Af\right)}$

If

${\displaystyle A=0,\,c>0}$

then the potential function V(f) has local maximum at f = 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.

More precisely, the solution is

${\displaystyle \phi (x,t)=-{\frac {1}{2}}\,c\,\mathrm {sech} ^{2}\left[{{\sqrt {c}} \over 2}(x-c\,t-a)\right]}$

where sech stands for the hyperbolic secant and a is an arbitrary constant.[4] This describes a right-moving soliton.

## Integrals of motion

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

Only the odd-numbered terms P(2n+1) result in non-trivial (meaning non-zero) integrals of motion (Dingemans 1997, p. 733).

## Lax pairs

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 L a Sturm–Liouville operator:

{\displaystyle {\begin{aligned}L&=-\partial _{x}^{2}+\phi ,\\A&=4\partial _{x}^{3}-3\left[\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).

## Least action principle

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, ${\displaystyle {\mathcal {L}}\,}$

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

with ${\displaystyle \phi }$  defined by

${\displaystyle \phi :={\frac {\partial \psi }{\partial x}}.\,}$
Derivation of Euler–Lagrange equations

Since the Lagrangian (eq (1)) contains second derivatives, the Euler–Lagrange equation of motion for this field is

${\displaystyle \partial _{\mu \mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu \mu }\psi )}}\right)-\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}\right)+{\frac {\partial {\mathcal {L}}}{\partial \psi }}=0.\quad \quad \quad \quad \quad \quad \quad (2)\,}$

where ${\displaystyle \partial }$  is a derivative with respect to the ${\displaystyle \mu }$  component.

A sum over ${\displaystyle \mu }$  is implied so eq (2) really reads,

${\displaystyle \partial _{tt}\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{tt}\psi )}}\right)+\partial _{xx}\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{xx}\psi )}}\right)-\partial _{t}\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{t}\psi )}}\right)-\partial _{x}\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{x}\psi )}}\right)+{\frac {\partial {\mathcal {L}}}{\partial \psi }}=0.\quad \quad (3)\,}$

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

${\displaystyle \partial _{tt}\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{tt}\psi )}}\right)=0\,}$
${\displaystyle \partial _{xx}\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{xx}\psi )}}\right)=\partial _{xx}\left(-\partial _{xx}\psi \right)\,}$
${\displaystyle \partial _{t}\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{t}\psi )}}\right)=\partial _{t}\left({\frac {1}{2}}\partial _{x}\psi \right)\,}$
${\displaystyle \partial _{x}\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{x}\psi )}}\right)=\partial _{x}\left({\frac {1}{2}}\partial _{t}\psi +3(\partial _{x}\psi )^{2}\right)\,}$
${\displaystyle {\frac {\partial {\mathcal {L}}}{\partial \psi }}=0\,}$

Remember the definition ${\displaystyle \phi =\partial _{x}\psi \,}$ , so use that to simplify the above terms,

${\displaystyle \partial _{xx}\left(-\partial _{xx}\psi \right)=-\partial _{xxx}\phi \,}$
${\displaystyle \partial _{t}\left({\frac {1}{2}}\partial _{x}\psi \right)={\frac {1}{2}}\partial _{t}\phi \,}$
${\displaystyle \partial _{x}\left({\frac {1}{2}}\partial _{t}\psi +3(\partial _{x}\psi )^{2}\right)={\frac {1}{2}}\partial _{t}\phi +3\partial _{x}(\phi )^{2}={\frac {1}{2}}\partial _{t}\phi +6\phi \partial _{x}\phi \,}$

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

${\displaystyle \left(-\partial _{xxx}\phi \right)-\left({\frac {1}{2}}\partial _{t}\phi \right)-\left({\frac {1}{2}}\partial _{t}\phi +6\phi \partial _{x}\phi \right)=0,\,}$

which is exactly the KdV equation

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

## 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 Zabusky & Kruskal (1965) and can be rigorously proven using the nonlinear steepest descent analysis for oscillatory Riemann–Hilbert problems.[5]

## 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 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.[6][7]

The KdV equation is now seen to be closely connected to Huygens' principle.[8][9]

## 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:

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

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]

## Variations

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

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 (Lax 7th) Darvishi, Kheybari & Khani (2007) {\displaystyle {\begin{aligned}\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{aligned}}}
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[10] ${\displaystyle \displaystyle \partial _{t}u+\mu \partial _{x}^{3}u+2u\partial _{x}u-\nu \partial _{x}^{2}u=0}$
non-homogeneous KdV ${\displaystyle \partial _{t}u+\alpha u+\beta \partial _{x}u+\gamma \partial _{x}^{2}u=Ai(x),\quad u(x,0)=f(x)}$

### q-analogs

For the q-analog of the KdV equation, see Frenkel (1996) and Khesin, Lyubashenko & Roger (1997).

## Notes

1. ^ N.J. Zabusky and M. D. Kruskal, Phy. Rev. Lett., 15, 240 (1965)
2. ^ Darrigol, O. (2005), Worlds of Flow: A History of Hydrodynamics from the Bernoullis to Prandtl, Oxford University Press, p. 84, ISBN 9780198568438
3. ^ See e.g. Newell, Alan C. (1985), Solitons in mathematics and physics, SIAM, ISBN 0-89871-196-7, p. 6. Or Lax (1968), without the factor 6.
4. ^ Alexander F. Vakakis (31 January 2002). Normal Modes and Localization in Nonlinear Systems. Springer. pp. 105–108. ISBN 978-0-7923-7010-9. Retrieved 27 October 2012.
5. ^ See e.g. Grunert & Teschl (2009)
6. ^ Gardner, C.S.; Greene, J.M.; Kruskal, M.D.; Miura, R.M (1967), "Method for solving the Korteweg–de Vries equation", Physical Review Letters, 19 (19): 1095–1097, Bibcode:1967PhRvL..19.1095G, doi:10.1103/PhysRevLett.19.1095.
7. ^ Dauxois, Thierry; Peyrard, Michel (2006), Physics of Solitons, Cambridge University Press, ISBN 0-521-85421-0
8. ^ Fabio A. C. C. Chalub and Jorge P. Zubelli, "Huygens’ Principle for Hyperbolic Operators and Integrable Hierarchies"
9. ^ Berest, Yuri Y.; Loutsenko, Igor M. (1997). "Huygens' Principle in Minkowski Spaces and Soliton Solutions of the Korteweg–de Vries Equation". Communications in Mathematical Physics. 190: 113–132. arXiv:solv-int/9704012. doi:10.1007/s002200050235. S2CID 14271642.
10. ^ Shu, Jian-Jun (1987). "The proper analytical solution of the Korteweg-de Vries-Burgers equation". Journal of Physics A: Mathematical and General. 20 (2): 49–56. arXiv:1403.3636. Bibcode:1987JPhA...20L..49J. doi:10.1088/0305-4470/20/2/002.