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 are the prototypical examples of solitons; these may be found 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.

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It is a nonlinear, dispersive partial differential equation for a function φ of two real variables, x and t:

\partial_t\phi+\partial^3_x\phi+6\phi\partial_x\phi=0

Consider solutions in which a fixed wave form (given by f(x)) maintains its shape as it travels to the right at speed c. Such a solution is given by φ(x,t) = f(x-ct). This gives the ordinary differential equation

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

or, integrating with respect to x,

3f^2+\frac{d^2 f}{dx^2}-cf=A

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

\phi(x,t)=\frac{c}{2}\frac{1}{\cosh ^2\left[{\sqrt{c}\over 2}(x-ct-a)\right ]}

where a is an arbitrary constant. This describes a right-moving soliton.

Three Solitons Solution of KdV Equation - [1]

Three Solitons (unstable) Solution of KdV Equation - [2]

With the form of the Korteweg–de Vries equation written

\frac{\partial \phi}{\partial t} + \alpha \phi \frac{\partial \phi}{\partial x} + \nu \frac{\partial^3 \phi}{\partial x^3} = 0, \,

there is a Lagrangian, L, from which the KdV equation can be derived,

\mathcal{L} = \frac{1}{2} \frac{\partial \psi}{\partial x} \frac{\partial \psi}{\partial t} + \frac{\alpha}{6} \left( \frac{\partial \psi}{\partial x} \right)^3 - \frac{\nu}{2} \left( \frac{\partial^2 \psi}{\partial x^2} \right)^2 \quad \quad \quad \quad (1) \,
with φ defined by
\phi = \frac{\partial \psi}{\partial x} = \partial_x \psi. \,

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

\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 \partial is a derivative with respect to the μ component.

A sum over μ is implied so eq (2) really reads,

\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),

\partial_{tt} \left( \frac{\partial \mathcal{L}}{\partial ( \partial_{tt} \psi )} \right) = 0 \,
\partial_{xx} \left( \frac{\partial \mathcal{L}}{\partial ( \partial_{xx} \psi )} \right) = \partial_{xx} \left( -\nu \partial_{xx} \psi \right) \,
\partial_t \left( \frac{\partial \mathcal{L}}{\partial ( \partial_t \psi )} \right) = \partial_t \left( \frac{1}{2} \partial_x \psi \right) \,
\partial_x \left( \frac{\partial \mathcal{L}}{\partial ( \partial_x \psi )} \right) = \partial_x \left( \frac{1}{2} \partial_t \psi + \frac{\alpha}{2} (\partial_x \psi)^2 \right) \,
\frac{\partial \mathcal{L}}{\partial \psi} = 0 \,

Remember the definition \phi = \partial_x \psi \,, so use that to simplify the above terms,

\partial_{xx} \left( -\nu \partial_{xx} \psi \right) = - \nu \partial_{xxx} \phi \,
\partial_t \left( \frac{1}{2} \partial_x \psi \right) = \frac{1}{2} \partial_t \phi \,
\partial_x \left( \frac{1}{2} \partial_t \psi + \frac{\alpha}{2} (\partial_x \psi)^2 \right) = \frac{1}{2} \partial_t \phi + \frac{\alpha}{2} \partial_x (\phi)^2 = \frac{1}{2} \partial_t \phi + \alpha \phi \partial_x \phi \,

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

\left(- \nu \partial_{xxx} \phi \right) - \left(\frac{1}{2} \partial_t \phi \right) - \left( \frac{1}{2} \partial_t \phi + \alpha \phi \partial_x \phi \right) = 0, \,

which is exactly the KdV equation

\frac{\partial \phi}{\partial t} + \alpha \phi \frac{\partial \phi}{\partial x} + \nu \frac{\partial^3 \phi}{\partial x^3} = 0 .\,

The history of the KdV equation spans a period of about sixty years, starting 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. In the nineteenth century, when the study of water waves was of vital interest for applications in naval architecture and for the knowledge of tides and floods, this equation raised a wide-spread interest.

Soliton solutions were described by Norman Zabusky and Martin Kruskal in 1965. Development of the analytic solution by means of the inverse scattering transform was done in 1967 by Gardner, Greene, Kruskal and Miura.

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.

  • D. J. Korteweg and F. de Vries, "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, 1895.
  • P. G. Drazin. Solitons. Cambridge University Press, 1983.
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