Dispersion Relation KdV equation

In summary, the dispersion relation for a linear equation is found by dropping the non-linear term and inserting a plane wave solution. For a nonlinear equation like the KdV, the superposision principle does not hold and it is difficult to determine a precise dispersion relation. However, by inserting a plane wave solution into the linearized version, an approximation can be obtained.
  • #1
hanson
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Hi all.
I have some questions about the dispersion relation in the study of waves.
First of all, why do we always assume a plane wave solution when we want to obtain a dispersion relation?

Second, is "assuming a plane wave solution" a general way to obtian all dispersion relations? for both linear and nonlinear wave equations? So, what is the dispersion relation for the KdV equation? I can hardly see anyone deriving the dispersio relation for nonlinear equations like KdV and NLS etc, why?
 
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  • #2
Try this link for excerpts from "Oscillations and Waves"
http://books.google.com/books?id=ge...6VB&sig=pDI-xn5gOrVd_qmFKMPIsIl3xEA#PPA413,M1


I think we tend to start with the simplest form for the mathematical description of a physical phenonmenon and work up to more complex situations.

Working in one dimension is simple compared to two or more, and where possible, one would prefer a linear system to a nonlinear one.

Solutions to 'idealized' models are a first step before introducing more complex and difficult structures.
 
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  • #3
First, concerning LINEAR diff.eqs:

A typical feature of the solutions to these is that the SUM of two solutions is ALSO a solution.

Furthermore, it is generally true that ALL solutions of a given diff.eq can be regarded as generated by a sum of base solutions.

This is what is called the superposition principle, and it is immensely important in all of physics.

However, with NON-linear diff.eqs, the superposition principle does not in general hold, i.e it is NOT normally true that the sum of two solutions is also a solutions, nor can we assume there exists a set of basis solutions out of which we can generate all solutions.

But that means, esssentially, that for the non-linear case, "every particular problem has its own solution", rather than this solution being derivable from solution to other problem.
The problem with non-linear diff.eqs is that neither the solutions methods nor the solutions themselves for a particular problem has any generalizable value.
 
  • #4
Plane wave solutions are handy to work with since they of course are plane, hence if the wave is progressing in one direction, for example parallel to an x-axis, then for every point on x the wave motion can be described by a single number denoting the amplitude in the entire yz-plane. Now all periodic functions and following all waves can be described by a Fourier transform which in essence is a long combination of plane waves weighted by a coefficient function. Therefore the dispersion relation can be found by considering the plane component waves of this Fourier transform.

For a linear equation all component waves will be independent of each other due to the superposition principle and therefore it is relatively easy to find the dispersion realtion. For non-linear equation like the KdV the superposition principle does not apply and there is mixing between the different component plane waves. This makes it difficult to determine a exact dispersion relation.

An approximation to the dispersion relation can be deduced by inserting a plane wave solution in the linearized version( just dropping the non-linear term) and this yields:

w(k)=ck-ek^3

where c denotes the coefficient before the term involving the first derivative with respect to x and e is the coefficient of the dispersive term.
 

1. What is the Dispersion Relation KdV equation?

The Dispersion Relation KdV (Korteweg-de Vries) equation is a mathematical equation that describes the evolution of nonlinear, dispersive waves. It was first proposed by Diederik Korteweg and Gustav de Vries in 1895 as a model for shallow water waves in canals. It has since been applied to various physical phenomena, such as ocean waves, plasma physics, and optics.

2. What does the equation represent?

The equation represents the relationship between the temporal and spatial variations of a wave. It describes how the shape of a wave changes over time and space due to nonlinear and dispersive effects. This allows us to understand the behavior of waves in a wide range of physical systems.

3. How is the equation derived?

The Dispersion Relation KdV equation is derived from the Navier-Stokes equations, which describe the motion of a fluid. It is a simplification of these equations, and it assumes that the fluid is inviscid (no friction) and incompressible (constant density). This allows for a more manageable equation that still captures the essential dynamics of waves.

4. What is the significance of the dispersion relation in the equation?

The dispersion relation in the equation represents the relationship between the wave's frequency and its wavenumber. It determines how different frequencies and wavelengths are affected by nonlinear and dispersive effects. This is important in understanding how waves behave and interact with each other in different physical systems.

5. What are some real-world applications of the Dispersion Relation KdV equation?

The equation has been used in various fields, including oceanography, plasma physics, and optics. In oceanography, it has been used to study the dynamics of shallow water waves and their interactions with currents and tides. In plasma physics, it has been used to model nonlinear waves in plasma, which is important for understanding processes in the Sun and other stars. In optics, it has been applied to the propagation of light in nonlinear media, such as optical fibers. Overall, the equation has a wide range of applications in understanding and predicting the behavior of waves in different physical systems.

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