KdV Equation - Modelling Soliton

In summary: The Korteweg-de Vries equation is used to solve for soliton formation in a shallow water channel. The equation relates the topography (depth, width, shape), pressure, and velocity of the body moving through the water. Solutions to the equation can be found by substituting values into the equation and solving for various variables. In order to simplify the process, the equation can be reduced to a set of characteristic functions. These functions describe the shape, amplitude, and velocity of a soliton at a given depth and time.
  • #1
quid
19
0
Hi all,

I am attempting to model soliton formation numerically. The solitons will be formed by moving a body of some sort through a shallow channel of water with the free surface subject to atmospheric pressure.

My goal would be to numerically predict wave amplitudes, wavelengths, velocities etc.

I have read a bit about the Korteweg-de Vries equation however I have no idea where to start in terms of solving it to find the desired unknowns.

Could someone please help shed some light on where to start?

Thanks
 
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  • #2
google 'the Hirota Method'

go well
 
  • #3
Studiot said:
google 'the Hirota Method'

go well

Thanks.

What I am really struggling with is the implementation of the KdV solutions to real life applications. What I mean by that is I have no idea what the variables are/mean and how it applies to my situation.

Surely the equation relates somehow to the topography, the pressure source, velocity etc. but HOW??
 
  • #4
What form do you have the KDV in?

If you think back to the ordinary wave equation, it is written as spatial displacement in terms of amplitude, time and a constant that has the dimensions of velocity squared.

In the derivation of the ordinary (linear) wave equation additional properties of the medium are needed. These might be thermodynamic equations of state, elastic equations, continuity etc.
In order to recover other physical properties such as pressure you have to return to these equations.

The same is true with non linear equations and their solutions. KDV is not the only NL equation leading to soliton solutions, but at least it is couched in terms of some physical properties ( mean depth, displacement, and time etc).
 
  • #5
I have KdV in the form:

uxxx + 6uux + ut = 0;

I'm finding that there is many solutions to this equation. Alot of them I barely understand the derivation. Do I need to derive solutions myself or can I use solutions of others.

In my case I require a wave speed c of 5m/s and all other variables can be changed. What solution should I use? The hirota method?

Sorry but I'm really struggling with this topic.

Thanks so much Studiot
 
  • #6
Should your equation not be

Ut - 6UUxx + Uxxx = 0

ie -6, not +6

Incidentally even without TEX you can use the very convenient subscript and superscript functions directly from the icon on the full reply box.
 
  • #7
Studiot said:
Should your equation not be

Ut - 6Uxx + Uxxx = 0

ie -6, not +6

Sorry yes you are right,

Ut - 6UUxx + Uxxx = 0
 
  • #8
OK take this equation and use the method of characteristics to assume a solution

[tex]u(x,t) = f(\eta )[/tex]

where [tex]\eta [/tex] is parameter and c is a constant.

substitute

[tex] - cf' - 6ff + f''' = 0[/tex]

integrate once A is a constant

[tex] - cf - 3{f^2} + f'' = A[/tex]

Integrate again, B is another constant

[tex]\frac{1}{2}{\left( {f'} \right)^2} = {f^3} + \frac{1}{2}c{f^2} + Af + B[/tex]

for a single wave we need the solution to die away to zero in both directions so imposing boundary conditions

[tex]f,f',f' \to 0\;as\;\eta \to \pm \infty [/tex]

the equation becomes

[tex]{\left( {f'} \right)^2} = {f^2}(2f + c)[/tex]

rearrange and integrate

[tex]\int {\frac{{df}}{{f\sqrt {(2f + c)} }}} = \int {d\eta } [/tex]

Use substitution

[tex]f = \frac{1}{2}c\sec {h^2}\theta [/tex]

to end up with the standard solution for a water wave

[tex]f(x - ct) = - \frac{1}{2}c\sec {h^2}\{ \frac{{\sqrt c }}{2}\left( {x - ct - {x_0}} \right)[/tex]

does this help?
 
  • #9
Sorry mate, still pretty lost. Can you elaborate?
 
  • #10
I'd better draw a diagram.

Are you studying fluid mechanics or maths or computing?
 
  • #11
Fluid mechanics. My maths obviously isn't up to scratch.

I can use MatLab too if that helps.
 
  • #12
So do you need the derivation or can we work in a more 'fluid mechanicsy' format?

Why are you going for partial diff shorthand?
 
  • #13
Hirota's method works for this, but is a linearisation mthod.

You mentioned trying to work a numerical method. Are you trying to develop a numerical calculation 'molecule' or just to follow Hirota?
 
  • #14
No the derivation is not too important.

The fluid mechanics is what I need to get my head around.

I plan to run and experiment by running a pressure source through an open channel of water to generate solitons. What I need to predict (to certain accuracy) is the soliton height, velocity, profile, wavelength etc for certain froude depth numbers.

I have been advised that solving KdV was the way to go about it as any linear analysis cannot predict solitons accurately?
 
  • #15
Any derivation is fine. I just am getting confused as there is so many all involving different parameters. I'm hoping I can just use the solutions already discovered by other methods?
 
  • #16
Here is the wave in terms of some real world quantities.
a is the amplitude eta is the (wave) function which describes the action (shape), c0 is the velocity.
h0 is the undisturbed depth.
Note the usual wave (x-Vt) on the horiz axis ie a function of x and time.

The soliton is traveling left to right and we ignore the left hand (negative) half.

The KDV for water solitons is

[tex]\frac{{\partial \eta }}{{\partial t}} + {c_0}\frac{{\partial \eta }}{{\partial x}} + \frac{{3{c_0}}}{{2{h_0}}}\eta \frac{{\partial \eta }}{{\partial x}} + \frac{{1{c_0}h_0^2}}{6}\frac{{{\partial ^3}\eta }}{{\partial {x^3}}} = 0[/tex]

where

[STRIKE][tex]{c_0} = \sqrt {gh_0^2} [/tex][/STRIKE]

c0 = √gh0

edit see post#20


is the velocity of gravity waves.

A solution is

[tex]\eta = a\sec {h^2}\left\{ {\sqrt {\frac{{3a}}{{4h_0^3}}} } \right.\left. {\frac{{\left( {x - Vt} \right)}}{1}} \right\}[/tex]

Note I said 'A solution is'. As you note there are many, but we want one that decays away to infinity on either side as I said in my previous derivation.

Would you like the derivation in this format, rather than as previous?

Does this help?
 

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  • #17
I should perhaps mention that the solution presented depends upon an initial disturbance

[tex]{\eta _0}[/tex]

If you are going to be investigating the relation between the initial disturbance and the dispersive terms in the equation then you may need to rework the derivation.
 
  • #18
Studiot said:
[tex]{c_0} = \sqrt {gh_0^2} [/tex]
is the velocity of gravity waves.

Is is not:

[tex]{c_0} = \sqrt {gh_0} [/tex]??
 
  • #19
Sorry I'm absolutely struggling today. One with the equation format on the forum. and two just in general.

So I now sub in eta into the KdV equation for water waves in order to solve for amplitude for a certain wave velocity, time and displacement?
 
  • #20
Yes you are right

c0 = √gh0

Well spotted.

:blushing:
 
  • #21
No worries :)

So where do I go from here? ha I mean I don't really see what I can do with the solution when I don't know any of the variables?
 
  • #22
Surely you have the real world variables in the last version of the equation I posted plus the sketch?

If you want to consider the conditions under which a soliton will form then you need to go back to the KDV and consider varying each term. The third term is the nonlinear one which is added to include for dispersion.
 
Last edited:
  • #23
Sorry I was very new to the topic earlier and had no real idea. I have done some reading and I think I am progressing in terms of understanding.

My main remaining question is this:

Can I predict the amplitude of the solitons using the KdV equation? I mean the solution to the equation gives me the wave profile if I am not mistaken, that is assuming you know the variables required. In my case I have all the variables except the soliton amplitude (as I want to predict this) therefore the solution is of no help to me as I also do not have the wave profile.

I hope my question makes sence, and once again Thanks Studiot!
 
  • #24
hi I am going to start a small project on 'elementary solution of kdv equation'.
can someone tell me to do this project what should i learn first.
also tell me what are the necessary definitions that i should know.
 

1. What is the KdV equation?

The Korteweg-de Vries (KdV) equation is a mathematical model that describes the behavior of long, one-dimensional waves in certain nonlinear media, such as water or plasma. It was first proposed by Dutch scientist Johannes Martinus Burgers and later refined by Dutch mathematician Diederik Korteweg and physicist Gustav de Vries.

2. What is a soliton?

A soliton is a type of wave that maintains its shape and velocity while propagating through a medium. It is a nonlinear phenomenon that arises from the balance between dispersion and nonlinearity in the medium. Solitons were first observed by Scottish engineer John Scott Russell in the 19th century while studying water waves in a canal.

3. How does the KdV equation model solitons?

The KdV equation describes the evolution of a wave's amplitude and phase over time, taking into account the effects of dispersion and nonlinearity. When these two factors are balanced, the KdV equation predicts the formation of solitons - localized, solitary waves that maintain their shape and velocity as they travel.

4. What are some real-world applications of the KdV equation and solitons?

The KdV equation and solitons have been used to model a wide range of physical phenomena, including water waves, plasma waves, and even traffic flow. They are also important in the study of nonlinear systems and chaos theory. In addition, the KdV equation has been applied in fields such as optics, fluid dynamics, and quantum mechanics.

5. Are there any limitations to the KdV equation's ability to model solitons?

While the KdV equation is a powerful tool for describing solitons, it is not a complete model and has some limitations. For example, it does not take into account factors such as dissipation and higher-order nonlinear effects, which can affect the behavior of solitons in certain situations. Researchers continue to refine and improve upon the KdV equation in order to better understand and predict the behavior of solitons in different systems.

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