Inverse Scattering Method (help please)

In summary, the conversation was about using the inverse scattering method to solve nonlinear PDEs. The participants discussed the process of deriving coefficients using the LAX method and also mentioned the AKNS scheme. They also talked about potential research topics related to the IST, such as solving the Navier-Stokes equations or studying exactly solvable models. The conversation also included a recommendation to consult with a specific advisor for further guidance.
  • #1
test1234
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Hi there, I'm learning about solitons and chanced upon this pdf talking about the inverse scattering method. However, I'm stuck trying to derive the coefficients using the LAX method (pg 5 of the attached pdf or from http://arxiv.org/pdf/0905.4746.pdf). Hope that someone can help shed some light on it...Thanks in advance! =)

[itex]L=-\partial_x^{\phantom{0}2}+u(x,t)[/itex]
[itex]A=\alpha_3\partial_x^{\phantom{0}3}+\alpha_2\partial_x^{\phantom{0}2}+ \alpha_1\partial_x+\alpha_0,[/itex]
where [itex]\alpha_j (j=0,1,2,3)[/itex] may depend on x and t.

Substituting these into the equation
[itex]L_t+LA-AL=0[/itex]
[itex]u_t+[-\partial_x^{\phantom{0}2} +u(x,t)][\alpha_3\partial_x^{\phantom{0}3}+\alpha_2\partial_x^{\phantom{0}2}+ \alpha_1\partial_x+ \alpha_0]-[\alpha_3\partial_x^{\phantom{0}3}+\alpha_2\partial_x^{\phantom{0}2}+ \alpha_1\partial_x+\alpha_0][-\partial_x^{\phantom{0}2} +u(x,t)]=0[/itex]

Focusing on the LHS,
[itex]u_t[/itex]
[itex]-\partial_x^{\phantom{0}2}(\alpha_3)-\partial_x^{\phantom{0}5}-\partial_x^{\phantom{0}2}(\alpha_2)-\partial_x^{\phantom{0}4}-\partial_x^{\phantom{0}2}(\alpha_1)-\partial_x^{\phantom{0}3}-\partial_x^{\phantom{0}2}(\alpha_0)[/itex]
[itex]+u\alpha_3\partial_x^{\phantom{0}4}+u\alpha_2\partial_x^{\phantom{0}2}+u\alpha_1\partial_x+u\alpha_0[/itex]
[itex]+\alpha_3\partial_x^{\phantom{0}5}+\alpha_2\partial_x^{\phantom{0}4}+ \alpha_1\partial_x^{\phantom{0}3}+\alpha_0\partial_x^{\phantom{0}2}[/itex]
[itex]-\alpha_3u_{xxx}-\alpha_2u_{xx}-\alpha_1u_{x}[/itex]

Rearranging in terms of [itex]\partial_x^{\phantom{0}j} (j=0,1,2,3)[/itex] terms,
[itex]u_t[/itex]
[itex]+(\alpha_3 -1)\partial_x^{\phantom{0}5}+(\alpha_2 -1)\partial_x^{\phantom{0}4}+(\alpha_1+u\alpha_3-1)\partial_x^{\phantom{0}3}+(\alpha_0+u \alpha_2)\partial_x^{\phantom{0}2}+(u \alpha_1)\partial_x[/itex]
[itex]-\alpha_3u_{xxx}-\alpha_2u_{xx}-\alpha_1u_{x}[/itex]
[itex]-\partial_x^{\phantom{0}2} (\alpha_3+\alpha_2+\alpha_1+\alpha_0)[/itex]

I'm not sure what to do with the last string of terms which involve partially differentiating the [itex]\alpha[/itex] terms by x and as such how to obtain the expression for the coefficients in eqn (4.6). Any help is much appreciated. Thanks! =)
 

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  • Inverse Scattering Transform and the Theory of Solitons.pdf
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  • #2
It's been a while since I did this, but as far as I remember you don't necessarily have to derive the operator A, you just need to find one which works.

I think I always used A = 4[(d/dx)^3] - 3 (u[d/dx] + [d/dx]u), or something along those lines
 
  • #3
Thanks Marioeden!

Hmm...forgive me for the next qns, how do you find one which works? Is there a general form depending on the type of differential eqn you have?
So after guessing the certain form that A should be, we can just substitute A into [itex]L_t+LA-AL[/itex] and if this equates to 0, then the lax pair is correct? If not then it's back to the drawing board again?
 
  • #4
I don't think [itex]L_{t}+[L,A][/itex] is necessarily identically zero. The idea is that Lax's Equation, with the [itex]L[/itex] and [itex]A[/itex] you find, should reproduce whichever PDE you're trying to solve. There will be different operators depending on which PDE you're solving. So far as I know, there is no method for generating the right operators in this method. You have to dream them up, although [itex]L[/itex], as you've suggested, is typically chosen to be a Schrodinger operator. However, in the AKNS scheme, which ends up being equivalent to Lax's Method, there is a way to generate the appropriate matrices. See the wiki on this for more information. There are some good references there. You can also check out my Ph.D. dissertation.
 
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  • #5
Thanks Ackbeet!

Your dissertation is indeed very helpful! Now I have a better understanding of what's happening. Thanks a lot! =)
 
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  • #6
Ackbeet said:
I don't think [itex]L_{t}+[L,A][/itex] is necessarily identically zero. The idea is that Lax's Equation, with the [itex]L[/itex] and [itex]A[/itex] you find, should reproduce whichever PDE you're trying to solve. There will be different operators depending on which PDE you're solving. So far as I know, there is no method for generating the right operators in this method. You have to dream them up, although [itex]L[/itex], as you've suggested, is typically chosen to be a Schrodinger operator. However, in the AKNS scheme, which ends up being equivalent to Lax's Method, there is a way to generate the appropriate matrices. See the wiki on this for more information. There are some good references there. You can also check out my https://www.amazon.com/dp/3843363250/?tag=pfamazon01-20.
Please can upload your dissertation again the one there takes me to a textbook i need to pay to download.
 
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  • #7
You can google "Adrian Keister Dissertation" and get the version on Virginia Tech's servers. It's not as good as the book to which I linked, but it has most of the material in it.
 
  • #8
Thank you very much i have downloaded it. when i have money i will buy the book from amazon. I am trying to do my PhD on solitons and inverse scattering.Any suggestion for a topic sir?
 
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  • #9
Well, I think there could be more work done on what happens after the Forward Scattering step in the IST. You start with a nonlinear PDE, and then you do the forward scattering step, which takes the original nonlinear PDE into a simultaneous system of linear ODE's. Solving those is a bit more straight-forward than the original, to be sure! You could try solving the linear ODE's using more realistic potential functions (kind of like in Schrodinger's Equation - are you going to do the particle-in-a-box, or the hydrogen atom?). The result of solving the linear ODE's is the scattering data. Then you have to solve the Gelfand-Levitan-Marchenko integral equation to get the solution to the original nonlinear PDE (this is the inverse scattering step).

My dissertation was almost completely concerned with the Forward Scattering step, and deducing bounds on some of the scattering data based on characteristics of the system of linear ODE's you get after the Forward Scattering step.

Of all these steps, forward scattering is quite difficult, because you have to come up with the Lax pair, or the two corresponding operators in the AKNS scheme, such that Lax's equation (or its equivalent in the AKNS scheme) reproduces your original nonlinear PDE. You could think about the Navier-Stokes equations, and whether IST could solve them. That's an extremely difficult problem! You might try Navier-Stokes with a term or two left out, and see what you get. Other things to do: look at the exactly solvable models we have so far, see how the IST solves them, and then try the method on a different PDE.

My advisor, http://www.math.vt.edu/people/klaus/, would be a very good person to talk with, as he would be far more up on the latest research than I am.
 
  • #10
Sir Adrian thank you very much you do not know how much u have helped and given me hope on this IST thing today. I am extremely grateful. I will contact Martin Klaus right away.
 
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  • #11
You're very welcome! Have a good one!
 

1. What is the Inverse Scattering Method?

The Inverse Scattering Method is a mathematical technique used to reconstruct the physical properties of an object by analyzing the scattering patterns of waves that interact with the object.

2. How does the Inverse Scattering Method work?

The Inverse Scattering Method uses mathematical algorithms to analyze the scattering data and create a model that represents the physical properties of the object. This model can then be used to predict the scattering behavior for future interactions with the object.

3. What are the applications of the Inverse Scattering Method?

The Inverse Scattering Method has a wide range of applications in various fields, such as medical imaging, geophysical exploration, and non-destructive testing. It is particularly useful in situations where direct observation or measurement of an object is not possible.

4. What are the limitations of the Inverse Scattering Method?

One of the main limitations of the Inverse Scattering Method is that it requires accurate and complete data to produce an accurate model. In real-world scenarios, this may not always be possible, which can lead to errors in the reconstructed model.

5. How is the Inverse Scattering Method different from other imaging techniques?

The Inverse Scattering Method is different from other imaging techniques in that it does not rely on external sources of radiation or contrast agents. It uses only the scattering patterns of waves to create an image, making it a non-invasive and non-destructive imaging method.

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