Korteweg–de Vries equation

Cnoidal wave solution to the Korteweg–de Vries equation, in terms of the square of the Jacobi elliptic function cn (and with value of the parameter m = 0.9).

File:KdV equation.gif

Numerical solution of the KdV equation u_t + uu_x + δ²u_xxx = 0 (δ = 0.022) with an initial condition u(x, 0) = cos(πx). Its calculation was done by the Zabusky-Kruskal scheme. The initial cosine wave evolves into a train of solitary-type waves.

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 non-linear 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).

Definition

The KdV equation is a nonlinear, dispersive partial differential equation for a function $\phi$ of two real variables, space x and time t :

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

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 $\phi$ 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 speed c. 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

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

or, integrating with respect to X,

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

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

$\phi(x,t)=\frac12\, c\, \mathrm{sech}^2\left[{\sqrt{c}\over 2}(x-c\,t-a)\right]$

where sech stands for the hyperbolic secant and a is an arbitrary constant. This describes a right-moving 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

$\int_{-\infty}^{+\infty} P_{2n-1}(\phi,\, \partial_x \phi,\, \partial_x^2 \phi,\, \ldots)\, \text{d}x\,$

where the polynomials P_n are defined recursively by

$\begin{align} P_1&=\phi, \\ P_n &= -\frac{dP_{n-1}}{dx} + \sum_{i=1}^{n-2}\, P_i\, P_{n-1-i} \quad \text{ for } n \ge 2. \end{align}$

The first few integrals of motion are:

• the mass $\int \phi\, \text{d}x,$
• the momentum $\int \phi^2\, \text{d}x,$
• the energy $\int \frac{1}{3} \phi^3 - \left( \partial_x \phi \right)^2\, \text{d}x.$

Only the odd-numbered terms P_(2n+1) result in non-trivial (meaning non-zero) integrals of motion (Dingemans 1997, p. 733).

Lax pairs

The KdV equation

$\partial_t\phi = 6\, \phi\, \partial_x \phi - \partial_x^3 \phi$

can be reformulated as the Lax equation

$L_t = [L,A] \equiv LA - AL \,$

with L a Sturm–Liouville operator:

$\begin{align} L &= -\partial_x^2 + \phi, \\ A &= 4 \partial_x^3 - 3 \left[ 2\phi\, \partial_x + (\partial_x \phi) \right] \end{align}$

and this accounts for the infinite number of first integrals of the KdV equation (Lax 1968).

Least action principle

The Korteweg–de Vries equation

$\partial_t \phi + 6\phi\, \partial_x \phi + \partial_x^3 \phi = 0, \,$

is the Euler–Lagrange equation of motion derived from the Lagrangian density, $\mathcal{L}\,$

$\mathcal{L} = \frac{1}{2} \partial_x \psi\, \partial_t \psi + \left( \partial_x \psi \right)^3 - \frac{1}{2} \left( \partial_x^2 \psi \right)^2 \quad \quad \quad \quad (1) \,$

with $\phi$ defined by

$\phi = \frac{\partial \psi}{\partial x} = \partial_x \psi. \,$

Derivation of Euler–Lagrange equations

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 $\mu$ component.

A sum over $\mu$ 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),

$\partial_{tt} \left( \frac{\partial \mathcal{L}}{\partial ( \partial_{tt} \psi )} \right) = 0 \,$

$\partial_{xx} \left( \frac{\partial \mathcal{L}}{\partial ( \partial_{xx} \psi )} \right) = \partial_{xx} \left( -\partial_{xx} \psi \right) \,$

$\partial_t \left( \frac{\partial \mathcal{L}}{\partial ( \partial_t \psi )} \right) = \partial_t \left( \frac{1}{2} \partial_x \psi \right) \,$

$\partial_x \left( \frac{\partial \mathcal{L}}{\partial ( \partial_x \psi )} \right) = \partial_x \left( \frac{1}{2} \partial_t \psi + 3 (\partial_x \psi)^2 \right) \,$

$\frac{\partial \mathcal{L}}{\partial \psi} = 0 \,$

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

$\partial_{xx} \left( - \partial_{xx} \psi \right) = - \partial_{xxx} \phi \,$

$\partial_t \left( \frac{1}{2} \partial_x \psi \right) = \frac{1}{2} \partial_t \phi \,$

$\partial_x \left( \frac{1}{2} \partial_t \psi + 3 (\partial_x \psi)^2 \right) = \frac{1}{2} \partial_t \phi + 3 \partial_x (\phi)^2 = \frac{1}{2} \partial_t \phi + 6 \phi \partial_x \phi \,$

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

$\left(- \partial_{xxx} \phi \right) - \left(\frac{1}{2} \partial_t \phi \right) - \left( \frac{1}{2} \partial_t \phi + 6 \phi \partial_x \phi \right) = 0, \,$

which is exactly the KdV equation

$\partial_t \phi + 6 \phi\, \partial_x \phi + \partial_x^3 \phi = 0 .\,$

Long-time 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.

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.

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.

The KdV equation can also be solved using the inverse scattering transform such as those applied to the non-linear Schrödinger equation.

KdV equation and the Gross–Pitaevskii equation

Considering the simplified solutions of the form

$\phi(x,t)=\phi(x \pm t)$

we obtain the KdV equation as

$\pm \partial_x \phi + \partial^3_x \phi + 6\, \phi\, \partial_x \phi =0\,$

or

$\pm \partial_x \phi + \partial_x (\partial^2_x \phi + 3 \phi^2) =0\,$

or integrating for special case i.e. putting the integration constant to zero:

$-\partial^2_x \phi - 3 \phi^2 =\pm \phi \,$

one obtains the generalized $(\lambda=1)$ stationary Gross–Pitaevskii equation (GPE)

$-\partial^2_x \phi - 3 \phi^{\lambda} \phi =\pm \phi \,$

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. While with $3$ with the minus sign and the $\phi$ real the self-interaction 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)	$\displaystyle \partial_t\phi + \partial^3_x \phi + 6\, \phi\, \partial_x\phi=0$
KdV (cylindrical)	$\displaystyle \partial_t u + \partial_x^3 u - 6\, u\, \partial_x u + u/2t = 0$
KdV (deformed)	$\displaystyle \partial_t u + \partial_x (\partial_x^2 u - 2\, \eta\, u^3 - 3\, u\, (\partial_x u)^2/2(\eta+u^2)) = 0$
KdV (generalized)	$\displaystyle \partial_t u + \partial_x^3 u = \partial_x^5 u$
KdV (generalized)	$\displaystyle \partial_t u + \partial_x^3 u + \partial_x f(u) = 0$
KdV (Lax 7th) Darvishi, Kheybari & Khani (2007)	$\begin{align} \partial_{t}u +\partial_{x} & \left\{ 35u^{4}+70\left(u^{2}\partial_{x}^{2}u+ u\left(\partial_{x}u\right)^{2}\right) \right. \\ & \left. \quad +7\left[2u\partial_{x}^{4}u+ 3\left(\partial_{x}^{2}u\right)^{2}+4\partial_{x}\partial_{x}^{3}u\right] +\partial_{x}^{6}u \right\}=0 \end{align}$
KdV (modified)	$\displaystyle \partial_t u + \partial_x^3 u \pm 6\, u^2\, \partial_x u = 0$
KdV (modified modified)	$\displaystyle \partial_t u + \partial_x^3 u - (\partial_x u)^3/8 + (\partial_x u)(Ae^{au}+B+Ce^{-au}) = 0$
KdV (spherical)	$\displaystyle \partial_t u + \partial_x^3 u - 6\, u\, \partial_x u + u/t = 0$
KdV (super)	$\displaystyle \partial_t u = 6\, u\, \partial_x u - \partial_x^3 u + 3\, w\, \partial_x^2 w$, $\displaystyle \partial_t w = 3\, (\partial_x u)\, w + 6\, u\, \partial_x w - 4\, \partial_x^3 w$
KdV (transitional)	$\displaystyle \partial_t u + \partial_x^3 u - 6\, f(t)\, u\, \partial_x u = 0$
KdV (variable coefficients)	$\displaystyle \partial_t u + \beta\, t^n\, \partial_x^3 u + \alpha\, t^nu\, \partial_x u= 0$
Korteweg–de Vries–Burgers equation	$\displaystyle \partial_t u + \mu\, \partial_x^3 u + 2\, u\, \partial_x u -\nu\, \partial_x^2 u = 0$

See also

• Benjamin–Bona–Mahony equation
• Boussinesq approximation (water waves)
• Cnoidal wave
• Dispersion (water waves)
• Dispersionless equation
• Fifth-order Korteweg–de Vries equation
• Kadomtsev–Petviashvili equation
• Novikov–Veselov equation
• Seventh-order Korteweg–de Vries equation
• Ursell number
• Vector soliton

References

• , 2 Parts, 967 pages

External links

• 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 – [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.
• Solitons & Nonlinear Wave Equations