Korteweg–de Vries equation
In mathematics, the Korteweg–de Vries equation 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 and rediscovered by.
Definition
The KdV equation is a nonlinear, dispersive partial differential equation for a function of two real variables, space x and time t :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 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 maintains its shape as it travels to the right at phase speed c. Such a solution is given by = f = f. Substituting it into the KdV equation gives the ordinary differential equationor, integrating with respect to X,
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
If
then the potential function V has local maximum at f = 0, there is a solution in which f 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 approaches 0 as X → ±∞. This is the characteristic shape of the solitary wave solution.
More precisely, the solution is
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, which do not change with time. They can be given explicitly aswhere the polynomials Pn are defined recursively by
The first few integrals of motion are:
- the mass
- the momentum
- the energy
Lax pairs
The KdV equationcan be reformulated as the Lax equation
with L a Sturm–Liouville operator:
and this accounts for the infinite number of first integrals of the KdV equation.
Least action principle
The Korteweg–de Vries equationis the Euler–Lagrange equation of motion derived from the Lagrangian density,
with defined by
Since the Lagrangian contains second derivatives, the Euler–Lagrange equation of motion for this field is
where is a derivative with respect to the component.
A sum over is implied so eq really reads,
Evaluate the five terms of eq by plugging in eq,
Remember the definition, so use that to simplify the above terms,
Finally, plug these three non-zero terms back into eq to see
which is exactly the KdV equation
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. 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.
KdV equation and the Gross–Pitaevskii equation
Considering the simplified solutions of the formwe obtain the KdV equation as
or
Integrating and taking the special case in which the integration constant is zero, we have:
which is the special case of the generalized stationary Gross–Pitaevskii equation
Therefore, for the certain class of solutions of generalized GPE, two equations are one. Furthermore, taking the case with the minus sign and the 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.| Name | Equation |
| Korteweg–de Vries | |
| KdV | |
| KdV | |
| KdV | |
| KdV | |
| KdV | |
| KdV | |
| KdV | |
| KdV | |
| KdV | |
| KdV | |
| KdV | |
| Korteweg–de Vries–Burgers equation | |
| non-homogeneous KdV |