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Korteweg%E2%80%93de Vries Equation
Mathematical model of waves on a shallow water surface
Numerical solution of the KdV equation u_{t} + uu_{x} + δ^{2}u_{xxx} = 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.
The constant 6 in front of the last term is conventional but of no great significance: multiplying t, x, and $\phi$ 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 speedc. Such a solution is given by $\phi$(x,t) = f(x − ct − a) = f(X). Substituting it into the KdV equation gives the ordinary differential equation
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
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.
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:
shallow-water waves with weakly non-linear restoring forces,
Therefore, for the certain class of solutions of generalized GPE ($\lambda =4$ for the true one-dimensional condensate and
$\lambda =2$ while using the three dimensional equation in one dimension), two equations are one. Furthermore, taking the $\lambda =3$ case with the minus sign and the $\phi$ real, one obtains an attractive self-interaction that should yield a bright soliton.^{[]}
Variations
Many different variations of the KdV equations have been studied. Some are listed in the following table.
^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. S2CID14271642.
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.
Dingemans, M.W. (1997), Water wave propagation over uneven bottoms, Advanced Series on Ocean Engineering, 13, World Scientific, Singapore, ISBN981-02-0427-2, 2 Parts, 967 pages
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: Springer-Verlag, doi:10.1007/978-3-662-08054-2, ISBN978-3-540-02234-3, MR1997070
Lax, P. (1968), "Integrals of nonlinear equations of evolution and solitary waves", Communications on Pure and Applied Mathematics, 21 (5): 467-490, doi:10.1002/cpa.3160210503
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