Korteweg-de-Vries equation

The Korteweg-de-Vries equation (KdV) is a third order nonlinear partial differential equation . It was proposed in 1895 by Diederik Korteweg and Gustav de Vries for the analysis of shallow water waves in narrow canals, but was previously investigated by Boussinesq in 1877. It describes solitons that were first observed in water channels by John Scott Russell in 1834 . In 1965, Norman Zabusky and Martin Kruskal were able to explain the quasi-periodic behavior in the Fermi-Pasta-Ulam experiment by showing that the KdV equation represents the continuous borderline case.

Mathematical formulation

The KdV equation is formulated as a partial differential equation in one dimension and time . It is a third order equation. Originally it was in the form of Korteweg and de Vries ${\displaystyle x}$ ${\displaystyle t}$

${\displaystyle {\frac {\partial \eta }{\partial t}}={\frac {3}{2}}\cdot {\sqrt {\frac {g}{l}}}\cdot {\frac {\partial \left({\frac {1}{2}}\eta ^{2}+{\frac {2}{3}}\alpha \cdot \eta +{\frac {1}{3}}\sigma \cdot {\frac {\partial ^{2}\eta }{\partial x^{2}}}\right)}{\partial x}}}$

formulated with explicitly for waves in channels, where ${\displaystyle \sigma ={\tfrac {l^{3}}{3}}-{\tfrac {T\cdot l}{\rho \cdot g}}}$

• ${\displaystyle l}$ the depth
• ${\displaystyle g}$the acceleration due to gravity
• ${\displaystyle T}$the surface tension
• ${\displaystyle \rho }$indicates the density of the liquid.

In today's specialist literature, however, the equation is mostly found in the abstract form

${\displaystyle {\frac {\partial u}{\partial t}}+6u\cdot {\frac {\partial u}{\partial x}}+{\frac {\partial ^{3}u}{\partial x^{3}}}=0,}$

which can be derived from the original equation through several transformation steps.

One of the important properties is the existence of soliton solutions. The simplest of these is

${\displaystyle u(x,t)={\frac {1}{2}}\cdot c\cdot \mathrm {sech} ^{2}\left({\frac {\sqrt {c}}{2}}\cdot (x-c\cdot t-a)\right),}$

in which

• ${\displaystyle a,\,c>0}$are arbitrary constants describing a single right moving soliton with velocity , and ${\displaystyle c}$
• ${\displaystyle \mathrm {sech} }$stands for the hyperbolic secant .

Mathematical Methods

The KdV equation is an example of a fully integrable system . The solutions can be given exactly in closed form. This is related to the fact that they can be understood as an infinite-dimensional Hamiltonian system with an infinite number of conserved quantities (constants of motion), which can also be specified explicitly.

The KdV equation can be solved with the inverse scattering transformation developed by Clifford Gardner , John Greene , Martin Kruskal and Robert Miura : To do this, a one-dimensional Schrödinger operator is assigned to a solution ${\displaystyle u(x,t)}$

${\displaystyle L(t)=-{\frac {d^{2}}{dx^{2}}}+u(x,t)}$

to. This forms together with the operator

${\displaystyle P(t)=-{\frac {d^{3}}{dx^{3}}}+3\left(u(x,t){\frac {d}{dx}}+{\frac {d}{dx}}u(x,t)\right)}$

the Lax pair of the KdV equation. In other words, solve the KdV equation if and only if: ${\displaystyle u(x,t)}$

${\displaystyle {\frac {d}{dt}}L(t)=\left[P(t),L(t)\right]\equiv P(t)L(t)-L(t)P(t)}$

The scattering data ( reflection coefficient and eigenvalues plus normalization constants) can also be assigned to the Schrödinger operator . The eigenvalues ​​are time-independent due to the Lax equation. The reflection coefficient and normalization constants fulfill linear differential equations , which can be solved explicitly. Then the solution is then reconstructed using inverse scattering theory . ${\displaystyle L(t)}$ ${\displaystyle u(x,t)}$

This has some interesting consequences. On the one hand one obtains that solutions of the KdV equation exist for all times, on the other hand one obtains that the solitons correspond exactly to the eigenvalues. One can even show that arbitrary, sufficiently strongly decreasing initial conditions are given asymptotically for large times  t by a finite number of solitons running to the right and a dispersive part running to the left .

In addition to the inverse scattering transformations, there are other solution methods, in particular the direct method of Ryōgo Hirota and the method of the Bäcklund transformations (with which a whole hierarchy of solutions can be generated).

See also

• Kadomtsev-Petviashvili equation

literature

• J. Boussinesq, Essai sur la theorie des eaux courantes, Memoires presentes par divers savants , l'Acad. of the Sci. Inst. Nat. France, XXIII (1877), pp. 1-680
• CS Gardner, JM Green, MD Kruskal, RM Miura, A method for solving the Korteweg-de Vries equation , Phys. Rev. Letters 19 (1967), pp. 1095-1097
• Diederik Korteweg, Gustav de Vries: On the Change of Form of Long Waves advancing in a Rectangular Canal and on a New Type of Long Stationary Waves. In: Philosophical Magazine. 5th series, No. 36, 1895, pp. 422-443
• K. Grunert, G. Teschl , “Long-Time Asymptotics for the Korteweg-de Vries Equation via Nonlinear Steepest Descent”, Math. Phys. Anal. Geom. 12 (2009), 287-324, arxiv : 0807.5041 , doi: 10.1007 / s11040-009-9062-2
• P. Lax , Integrals of nonlinear equations of evolution and solitary waves , Comm. Pure Applied Math. 21 (1968), pp. 467-490
• VA Marchenko , Sturm-Liouville Operators and Applications , Birkhäuser, Basel, 1986
• NJ Zabusky and MD Kruskal, Interaction of solitons in a collisionless plasma and the recurrence of initial states , Phys. Rev. Lett. 15 (1965), pp. 240-243

Web links

• Korteweg de Vries Equation , Mathworld