# Must solitary wave a product of weakly nonlinear and weakly dispersive?

• hanson
In summary, the Korteweg-de Vries (KdV) equation is a balance between nonlinearity and dispersion which allows for permanent propagation of wave profiles. While it is commonly referred to as being a balance between weakly nonlinear and weakly dispersive effects, it is possible for it to also occur with stronger nonlinearities and dispersions. This distinction is made in order to simplify the equation and make it more comprehensible. However, there are cases where the KdV equation may not follow this pattern. In terms of perturbation series, it is common to assume a starting point of O(epsilon) rather than including a constant velocity term, which may seem counter-intuitive.

#### hanson

Hi all!
I know that KdV is a balance of the nonlinearity and the dispersive effect and hence the wave profile propagates permenantly without disperse.
However, why one always mention that it is the balance of WEAKLY nonlinear and WEAKLY dispersive?
Can't it be a balance of STRONGLY nonlinear and STRONLY dispersive?
Will the KdV equation still exist if the shallow water parameter and the nonlinear parameter does not tend to zero but just with the same order of magnitude? Nnamely O(d^2) = O(epsilon), yet, not tends to zero.

Can someone kindly help?

Perhaps.
And then again, perhaps not.

Remember that the only reason why we split up problems as "weakly" non-liear, or "weakly" linearly dispersive is in order to justify chopping away some terms so that the remaining image becomes clear and comprehensible.

If all effects should be taken into account, then we are back in a morass of terms and possible outcomes.

oh...i see..
That's means weakly nonlinear and weakly linearly dispersive region is not the only region that a KdV balance may occur?

From what I know, you may indeed have KdV-like equations with "stronger" non-linearities than of the KdV-type.

arildno said:
From what I know, you may indeed have KdV-like equations with "stronger" non-linearities than of the KdV-type.

Thank you arildno.
One more question regarding KdV equation.
I am currently reading an article:
www-personal.engin.umich.edu/~jpboyd/op121_boydchennlskdv.pdf
and a paper. (Please see the attached figure)
Both of them assume the perturbation series of the velocity start from O(epsilon), how come?
It is somewhat counter-intuitive to me since we always assume something like
u ~ u0 + eu1+ e^2u2 +... isn't it?
why does this start from eu1 without the constant velocity u0?

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## 1. What is a solitary wave?

A solitary wave, also known as a soliton, is a type of wave that maintains its shape and speed while traveling through a medium without dispersing or changing form.

## 2. What does it mean for a solitary wave to be weakly nonlinear?

A solitary wave is considered weakly nonlinear when its amplitude is small enough that the nonlinear effects become negligible. This means that the behavior of the wave can be described using linear equations.

## 3. How is a solitary wave affected by weak dispersion?

A solitary wave is considered weakly dispersive when the dispersion effects are small enough to not significantly alter the shape or speed of the wave. In this case, the wave maintains its solitary nature and does not break into smaller waves.

## 4. Can a solitary wave exist in all types of mediums?

No, a solitary wave requires a nonlinear medium in order to maintain its shape and speed. This means that it cannot exist in linear materials, such as air or water.

## 5. What are some real-world applications of solitary waves?

Solitary waves have been observed in various natural phenomena, such as ocean waves, tsunamis, and even in the behavior of neurons in the brain. They also have practical applications in fields such as fiber optics, where they are used to transmit information without distortion.