Korteweg–de Vries equation
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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 famous 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. The solutions in turn include prototypical examples of solitons. 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 equation is named for Diederik Korteweg and Gustav de Vries who studied it in (Korteweg & de Vries 1895), though the equation first appears in (Boussinesq 1877, p. 360).
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[edit] Definition
The KdV equation is a nonlinear, dispersive partial differential equation for a function φ of two real variables, space x and time t :
with ∂x and ∂t denoting partial derivatives with respect to x and t.
[edit] Solitons
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 φ(x,t) = f(x-ct). This 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 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 '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 a is an arbitrary constant. This describes a right-moving soliton.
[edit] 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 Pn are defined recursively by
The first few integrals of motion are:
- the momentum
- the energy
Only the odd-numbered terms P(2n+1) result in non-trivial (meaning non-zero) integrals of motion (Dingemans 1997, p. 733).
[edit] 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)
[edit] Lagrangian
With the form of the Korteweg–de Vries equation written
there is a Lagrangian density, , from which the KdV equation can be derived,
with φ defined by
[edit] 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). Development of the analytic solution by means of the inverse scattering transform was done in 1967 by Gardner, Greene, Kruskal and Miura.
[edit] 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, 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, and more.
The KdV equation can also be solved using the inverse scattering transform such as those applied to the non-linear Schrödinger equation.
[edit] 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 (cylinderical) | |
KdV (deformed) | |
KdV (generalized) | |
KdV (generalized) | |
KdV (modified) | |
KdV (modified modified) | |
KdV (spherical) | |
KdV (super) | , |
KdV (transitional) | |
KdV (variable coefficients) | |
Korteweg-de Vries-Burgers equation |
[edit] See also
- Benjamin–Bona–Mahony equation
- Kadomtsev–Petviashvili equation
- Dispersion (water waves)
- Dispersionless equation
[edit] 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
- Drazin, P. G. (1983), Solitons, vol. 85, London Mathematical Society Lecture Note Series, Cambridge: Cambridge University Press, pp. viii+136, MR0716135, ISBN 0-521-27422-2
- de Jager, E.M., On the Origin of the Korteweg–de Vries Equation, <http://arxiv.org/abs/math/0602661> Arxiv math.HO/0602661
- Dingemans, M.W. (1997), Water wave propagation over uneven bottoms, vol. 13, Advanced Series on Ocean Engineering, World Scientific, Singapore, ISBN 981 02 0427 2, 2 Parts, 967 pages
- Korteweg, D. J. & de Vries, F. (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: 422--443
- Lax, P. (1968), “Integrals of nonlinear equations of evolution and solitary waves”, Comm. Pure Applied Math. 21: 467-490, DOI 10.1002/cpa.3160210503
- 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: 1204--1209, MR0252826, DOI 10.1063/1.1664701
- Takhtadzhyan, L.A. (2001), “Korteweg–de Vries equation”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
- Zabusky, N. J. & Kruskal, M. D. (1965), “Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States”, Phys. Rev. Lett. 15: 240 - 243, doi:10.1103/PhysRevLett.15.240, <http://link.aps.org/abstract/PRL/v15/p240>
[edit] External links
- Korteweg–de Vries equation at EqWorld: The World of Mathematical Equations.
- Cylindrical Korteweg–de Vries equation at EqWorld: The World of Mathematical Equations.
- Modified Korteweg–de Vries equation at EqWorld: The World of Mathematical Equations.
- Eric W. Weisstein, Korteweg–deVries Equation at MathWorld.
- Derivation of the Korteweg-de Vries equation for a narrow canal.
- Three Solitons Solution of KdV Equation - [1]
- Three Solitons (unstable) Solution of KdV Equation - [2]
- 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.