Korteweg–de Vries equation
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 nonlinear 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]}
Definition
The KdV equation is a nonlinear, dispersive partial differential equation for a function of two real variables, space x and time t :^{[3]}
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 by constants can be used to make the coefficients of any of the three terms equal to any given nonzero 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 (x,t) = f(x − ct − a) = f(X). Substituting it into the KdV equation gives the ordinary differential equation
or, integrating with respect to X,
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(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
where sech stands for the hyperbolic secant and a is an arbitrary constant.^{[4]} This describes a rightmoving 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
where the polynomials P_{n} are defined recursively by
The first few integrals of motion are:
 the mass
 the momentum
 the energy
Only the oddnumbered terms P_{(2n+1)} result in nontrivial (meaning nonzero) integrals of motion (Dingemans 1997, p. 733).
Lax pairs
The KdV equation
can be reformulated as the Lax equation
with L a Sturm–Liouville operator:
and this accounts for the infinite number of first integrals of the KdV equation (Lax 1968).
Least action principle
The Korteweg–de Vries equation
is the Euler–Lagrange equation of motion derived from the Lagrangian density,
with defined by
Since the Lagrangian (eq (1)) contains second derivatives, the Euler–Lagrange equation of motion for this field is
where is a derivative with respect to the component.
A sum over is implied so eq (2) really reads,
Evaluate the five terms of eq (3) by plugging in eq (1),
Remember the definition , so use that to simplify the above terms,
Finally, plug these three nonzero terms back into eq (3) to see
which is exactly the KdV equation
Longtime 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 FPU 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 problem in the continuum limit, it approximately describes the evolution of long, onedimensional waves in many physical settings, including:
 shallowwater waves with weakly nonlinear restoring forces,
 long internal waves in a densitystratified 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 nonlinear Schrödinger equation.
KdV equation and the Gross–Pitaevskii equation
Considering the simplified solutions of the form
we obtain the KdV equation as
or
or integrating for special case i.e. putting the integration constant to zero:
one obtains the generalized stationary Gross–Pitaevskii equation (GPE)
Therefore, for the certain class of solutions of generalized GPE ( for the true onedimensional condensate and while using the three dimensional equation in one dimension) two equations are one. While with with the minus sign and the real the selfinteraction is attractive one should expect the excess in value (bright) soliton solutions.
Variations
Many different variations of the KdV equations have been studied. Some are listed in the following table.
Name  Equation 

Korteweg–de Vries (KdV)  
KdV (cylindrical)  
KdV (deformed)  
KdV (generalized)  
KdV (generalized)  
KdV (Lax 7th) Darvishi, Kheybari & Khani (2007)  
KdV (modified)  
KdV (modified modified)  
KdV (spherical)  
KdV (super)  ,

KdV (transitional)  
KdV (variable coefficients)  
Korteweg–de Vries–Burgers equation^{[10]} 
See also
 Benjamin–Bona–Mahony equation
 Boussinesq approximation (water waves)
 Cnoidal wave
 Dispersion (water waves)
 Dispersionless equation
 Fifthorder Korteweg–de Vries equation
 Kadomtsev–Petviashvili equation
 Modified KdV–Burgers equation
 Novikov–Veselov equation
 Seventhorder Korteweg–de Vries equation
 Ursell number
 Vector soliton
Notes
 ↑ N.J. Zabusky and M. D. Kruskal, Phy. Rev. Lett., 15, 240 (1965)
 ↑ Darrigol, O. (2005), Worlds of Flow: A History of Hydrodynamics from the Bernoullis to Prandtl, Oxford University Press, p. 84, ISBN 9780198568438
 ↑ See e.g. Newell, Alan C. (1985), Solitons in mathematics and physics, SIAM, ISBN 0898711967, p. 6. Or Lax (1968), without the factor 6.
 ↑ Alexander F. Vakakis (31 January 2002). Normal Modes and Localization in Nonlinear Systems. Springer. pp. 105–108. ISBN 9780792370109. Retrieved 27 October 2012.
 ↑ See e.g. Grunert & Teschl (2009)
 ↑ 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.
 ↑ Dauxois, Thierry; Peyrard, Michel (2006), Physics of Solitons, Cambridge University Press, ISBN 0521854210
 ↑ Fabio A. C. C. Chalub1 and Jorge P. Zubelli, "Huygens’ Principle for Hyperbolic Operators and Integrable Hierarchies"
 ↑ Yuri Yu. Berest and Igor M. Loutsenko, "Huygens’ Principle in Minkowski Spaces and Soliton Solutions of the Korteweg–de Vries Equation", arXiv:solvint/9704012 DOI 10.1007/s002200050235
 ↑ Shu, JianJun (1987). "The proper analytical solution of the Kortewegde VriesBurgers equation". Journal of Physics A: Mathematical and General. 20 (2): 49–56. arXiv:1403.3636. Bibcode:1987JPhA...20L..49J. doi:10.1088/03054470/20/2/002.
References
 Boussinesq, J. (1877), Essai sur la theorie des eaux courantes, Memoires presentes par divers savants ` l’Acad. des Sci. Inst. Nat. France, XXIII, pp. 1–680
 de Jager, E.M. (2006). "On the origin of the Korteweg–de Vries equation". arXiv:math/0602661v1 [math.HO].
 Dingemans, M.W. (1997), Water wave propagation over uneven bottoms, Advanced Series on Ocean Engineering, 13, World Scientific, Singapore, ISBN 9810204272, 2 Parts, 967 pages
 Drazin, P. G. (1983), Solitons, London Mathematical Society Lecture Note Series, 85, Cambridge: Cambridge University Press, pp. viii+136, ISBN 0521274222, MR 0716135
 Grunert, Katrin; Teschl, Gerald (2009), "LongTime Asymptotics for the Kortewegde Vries Equation via Nonlinear Steepest Descent", Math. Phys. Anal. Geom., 12 (3), pp. 287–324, arXiv:0807.5041, Bibcode:2009MPAG...12..287G, doi:10.1007/s1104000990622
 Kappeler, Thomas; Pöschel, Jürgen (2003), KdV & KAM, 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], 45, Berlin, New York: SpringerVerlag, ISBN 9783540022343, MR 1997070
 Korteweg, D. J.; de Vries, G. (1895), "On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves", Philosophical Magazine, 39 (240): 422–443, doi:10.1080/14786449508620739
 Lax, P. (1968), "Integrals of nonlinear equations of evolution and solitary waves", Comm. Pure Applied Math., 21 (5): 467–490, doi:10.1002/cpa.3160210503
 Miles, John W. (1981), "The Korteweg–De Vries equation: A historical essay", Journal of Fluid Mechanics, 106: 131–147, Bibcode:1981JFM...106..131M, doi:10.1017/S0022112081001559.
 Miura, Robert M.; Gardner, Clifford S.; Kruskal, Martin D. (1968), "Korteweg–de Vries equation and generalizations. II. Existence of conservation laws and constants of motion", J. Mathematical Phys., 9 (8): 1204–1209, Bibcode:1968JMP.....9.1204M, doi:10.1063/1.1664701, MR 0252826
 Takhtadzhyan, L.A. (2001), "K/k055800", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 9781556080104
 Zabusky, N. J.; Kruskal, M. D. (1965), "Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States", Phys. Rev. Lett., 15 (6): 240–243, Bibcode:1965PhRvL..15..240Z, doi:10.1103/PhysRevLett.15.240
External links
Wikimedia Commons has media related to Korteweg–de Vries equation. 
 Korteweg–de Vries equation at EqWorld: The World of Mathematical Equations.
 Korteweg–de Vries equation at NEQwiki, the nonlinear equations encyclopedia.
 Cylindrical Korteweg–de Vries equation at EqWorld: The World of Mathematical Equations.
 Modified Korteweg–de Vries equation at EqWorld: The World of Mathematical Equations.
 Modified Korteweg–de Vries equation at NEQwiki, the nonlinear equations encyclopedia.
 Weisstein, Eric W. "Korteweg–deVries Equation". MathWorld.
 Derivation of the Korteweg–de Vries equation for a narrow canal.
 Three Solitons Solution of KdV Equation –
 Three Solitons (unstable) Solution of KdV Equation –
 Mathematical aspects of equations of Korteweg–de Vries type are discussed on the Dispersive PDE Wiki.
 Solitons from the Korteweg–de Vries Equation by S. M. Blinder, The Wolfram Demonstrations Project.
 Solitons & Nonlinear Wave Equations