How Does the Inverse Scattering Method Apply to Soliton Equations?

[test1234]

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! =)

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

Substituting these into the equation
L_t+LA-AL=0
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

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

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

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

 

[Marioeden]

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

 

[test1234]

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 L_t+LA-AL and if this equates to 0, then the lax pair is correct? If not then it's back to the drawing board again?

 

[Ackbach]

I don't think L_{t}+[L,A] is necessarily identically zero. The idea is that Lax's Equation, with the L and A 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 L, 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.

 

[test1234]

Thanks Ackbeet!

Your dissertation is indeed very helpful! Now I have a better understanding of what's happening. Thanks a lot! =)

 

[Nuel]

I don't think L_{t}+[L,A] is necessarily identically zero. The idea is that Lax's Equation, with the L and A 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 L, 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.

 

[Ackbach]

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.

 

[Nuel]

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?

 

[Ackbach]

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.

 

[Nuel]

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.

 

[Ackbach]

You're very welcome! Have a good one!