Korteweg–De Vries equation



In mathematics, the Korteweg–De Vries (KdV) equation is a partial differential equation (PDE) which serves as a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an integrable PDE and exhibits many of the expected behaviors for an integrable PDE, such as a large number of explicit solutions, in particular soliton solutions, and an infinite number of conserved quantities, despite the nonlinearity which typically renders PDEs intractable. The KdV can be solved by the inverse scattering method (ISM). In fact, Gardner, Greene, Kruskal and Miura developed the classical inverse scattering method to solve the KdV equation.

The KdV equation was first introduced by  and rediscovered by Diederik Korteweg and Gustav de Vries in 1895, who found the simplest solution, the one-soliton solution. Understanding of the equation and behavior of solutions was greatly advanced by the computer simulations of Zabusky and Kruskal in 1965 and then the development of the inverse scattering transform in 1967.

Definition
The KdV equation is a partial differential equation that models (spatially) one-dimensional nonlinear dispersive nondissipative waves described by a function $$\phi(x,t)$$ adhering to:


 * $$\partial_t \phi + \partial^3_x \phi - 6\, \phi\, \partial_x \phi =0\, \quad x \in \mathbb{R}, \; t \geq 0,$$

where $$\partial^3_x \phi$$ accounts for dispersion and the nonlinear element $$\phi \partial_x \phi$$ is an advection term.

For modelling shallow water waves, $$\phi$$ is the height displacement of the water surface from its equilibrium height.

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.

One-soliton solution
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 $$\varphi(x,t)=f(x-ct-a)=f(X)$$. 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 of a particle of unit mass in a cubic potential
 * $$V(f) = -\left(f^3+\frac{1}{2}cf^2 + Af\right)$$.

If
 * $$A = 0,\,c > 0$$

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' $$-\infty$$, eventually slides down to the local minimum, then back up the other side, reaching an equal height, and then reverses direction, ending up at the local maximum again at time $$\infty$$. In other words, $$f(X)$$ approaches $$0$$ as $$X\to-\infty$$. This is the characteristic shape of the solitary wave solution.

More precisely, the solution is


 * $$\phi(x,t)= -\frac 1 2 \, c\, \operatorname{sech}^2\left[{\sqrt{c}\over 2}(x-c\,t-a)\right]$$

where $$\operatorname{sech}$$ stands for the hyperbolic secant and $$a$$ is an arbitrary constant. This describes a right-moving soliton with velocity $$c$$.

N-soliton solution
There is a known expression for a solution which is an $$N$$-soliton solution, which at late times resolves into $$N$$ separate single solitons. The solution depends on an decreasing positive set of parameters $$\chi_1, \cdots, \chi_N > 0$$ and a non-zero set of parameters $$\beta_1, \cdots, \beta_N$$. The solution is given in the form

where the components of the matrix $$A(x,t)$$ are given by $$A_{nm}(x,t) = \delta_{nm} + \frac{\beta_n e^{8\chi_n^3t}e^{-(\chi_n + \chi_m)x}}{\chi_n + \chi_m}.$$

This is derived using the inverse scattering method.

Integrals of motion
The KdV equation has infinitely many integrals of motion, 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} $$ The first few integrals of motion are: Only the odd-numbered terms $$P_{2n+1}$$ result in non-trivial (meaning non-zero) integrals of motion.
 * the mass $$\int \phi\, \mathrm{d}x,$$
 * the momentum $$\int \phi^2\, \mathrm{d}x,$$
 * the energy $$\int \left[ 2 \phi^3 - \left( \partial_x \phi \right)^2 \right] \, \mathrm{d}x$$.

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 - 6 \phi\, \partial_x - 3[\partial_x, \phi] \end{align} $$ where $$[\partial_x,\phi]$$ is the commutator such that $$[\partial_x,\phi]f=f\partial_x\phi$$. The Lax pair accounts for the infinite number of first integrals of the KdV equation.

In fact, $$L$$ is the time-independent Schrödinger operator (disregarding constants) with potential $$\phi(x,t)$$. It can be shown that due to this Lax formulation that in fact the eigenvalues do not depend on $$t$$.

Zero-curvature representation
Setting the components of the Lax connection to be $$L_x = \begin{pmatrix}0 & 1\\ \phi - \lambda & 0\end{pmatrix}, L_t = \begin{pmatrix}-\phi_x & 2\phi + 4\lambda \\ 2\phi^2 - \phi_{xx} + 2\phi\lambda - 4\lambda^2 & \phi_x\end{pmatrix},$$ the KdV equation is equivalent to the zero-curvature equation for the Lax connection, $$\partial_t L_x - \partial_x L_t + [L_x, L_t] = 0. $$

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, $$\mathcal{L}\,$$

with $$\phi$$ defined by


 * $$\phi := \frac{\partial \psi}{\partial x}. $$

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

where $$\partial$$ is a derivative with respect to the $$\mu$$ component.

A sum over $$\mu$$ is implied so eq (2) really reads,

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 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 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.

The KdV equation is now seen to be closely connected to Huygens' principle.

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

Integrating and taking the special case in which the integration constant is zero, we have:
 * $$ -\partial^2_x \phi - 3 \phi^2 =\pm \phi \,$$

which is the $$\lambda=1$$ special case of the generalized 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. 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.