Strong perturbation theory

In another post we have defined what a strongly perturbed physical system is. This implied that we know how to do strong perturbation theory, that is, one takes her preferred differential equation and get a solution series for the case of a large parameter. We show here that is indeed the case. So, let us consider as an example the following differential equation:

\dot y(t)+y(t)+\lambda y(t)^2=0.

We know the exact solution of this equation being

y(t) = \frac{1}{\lambda(e^t-1)+e^t}

to be compared to our approximated ones. The weak perturbation case, \lambda\rightarrow 0, asking for a solution in the form

y(t)=y_0(t)+\lambda y_1(t) + \lambda^2 y_2(t) + O(\lambda^3)

gives the set of equations

\dot y_0+y_0=0

\dot y_1 + y_1=-y_0^2

\dot y_2 + y_2=-2y_0y_1

and finally

y(t)=e^{-t}+\lambda e^{-2t}+\lambda^2 e^{-3t}+O(\lambda^3)

and from the numerical comparison for \lambda=0.005 we get the curves

that is really satisfactory. We now look for a solution series in the form

y=z_0+\frac{1}{\lambda}z_1+\frac{1}{\lambda^2}z_2+O\left(\frac{1}{\lambda^3}\right)

but a direct substitution into the original equation gives nonsense. There is one more step to do and this is a rescaling in time, that is we use instead of t the scaled variable \tau=\lambda t. After this substitution is accomplished we can put the strong perturbation series into the equation and obtain the meaningful set of differential equations

\dot z_0+z_0^2=0

\dot z_1+z_0+2z_0z_1=0

\dot z_2+z_1+z_1^2+2z_2z_0=0

where now “dot” means derivation with respect to \tau. We have finally the solution series, undoing the rescaling in time,

y(t)=\frac{1}{\lambda t+1}-\frac{1}{\lambda}\frac{\frac{\lambda^2 t^2}{2}+\lambda t}{(\lambda t+1)^2}+\frac{1}{\lambda^2}\frac{\frac{\lambda^3t^3}{12}+\frac{\lambda^2t^2}{4}+\frac{\lambda t}{4}+\frac{1}{4(\lambda t +1)}-\frac{1}{4}}{(\lambda t + 1)^2}+O\left(\frac{1}{\lambda^3}\right)

Numerical comparison with \lambda=100 gives

that is almost perfect in the coincidence between the exact and the approximate case. The method indeed works!

We note the following:

  • From the set of equations of the strong perturbation case we note that we have just interchanged the perturbation with respect to the weak perturbation case to obtain the series with inverted expansion parameter. This serves just as a bookeeper but it is not needed. This is duality in perturbation theory (look here and here). 
  • The strong perturbation series is a series in small times. Indeed the time scale is set by \lambda that decides how far can we go into the time scale for the comparison.

It is just curious that no mathematician in the history was able to get such a method out understanding that it was just a rescaling of the independent variable away. I was lucky as this did not happen!

So, as said at the start, you can take your preferred differential equation and sort out a solution series in a regime you have never seen before.

Have fun!

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8 Responses to Strong perturbation theory

  1. […] We hope to show in future posts how this machinery works for pdes. In case of odes we have already posted about (see here). […]

  2. […] The agreement is excellent confirming the fact that a strong coupling expansion is a gradient expansion. So, a large perturbation entering into a differential equation can be managed much in the same way one does for a small perturbation. In the case of ode look at this post. […]

  3. fzx says:

    Hi,

    What you have presented here is essentially boundary layer approximation, a well known method of singular perturbation for differential equations where the highest derivative is multiplied by a small parameter (in your case 1/lambda).

    • mfrasca says:

      Hi fzx,

      Let me explain in a few steps why this is not the case. The equation is

      \dot y+y+\lambda y^2=0.

      If I rescale time t\rightarrow\lambda t this becomes

      \lambda\dot y+y+\lambda y^2=0.

      Finally, dividing all by \lambda this yields

      \dot y +\frac{1}{\lambda}y+y^2=0.

      So, I have no small parameter multiplying the derivative, rather I have obtained that the perturbations have been exchanged and the new perturbation series has a meaning in inverse powers of \lambda (duality in perturbation theory).

      You can find an extended discussion in my Physical Review paper (a preprint is in arxiv here).

      Cheers,

      Marco

      • fzx says:

        Hi,

        Suppose a=1/lambda, your equation starts from a(y’ +y)+y^2=0. The highest time derivative is multiplied by a, which is a small parameter. The rescaling t->t/a is the inner variable used in boundary layer analysis; the same computation follows.

        • mfrasca says:

          Yes, but this has nothing to do with the content of the post and the given equation. Not even the method. I think what I stated should be clear enough to rule out what are you saying.

  4. fzx says:

    Hi,

    Look, I am not trying to criticize your work in anyway. I just want to point out that the method is essentially boundary layer approximation, not something entirely new that has been overlooked by all mathematicians in history, as your post suggests.

    • mfrasca says:

      Dear fzx,

      What you have just shown is that the problem with \lambda\rightarrow\infty can be solved with a known perturbation techniques in this particular case. You can do this only after the proper interpretation by me is given. It is this the missing logical leap and you cannot find any published paper in physics or mathematics where one treats the solution of a (partial or ordinary) differential equations from both sides. Just because this was not recognized. To have an idea of the cultural difficulty that this missed interpretation yields just check this answer of mine in mathoverflow.

      In order to have an idea of the matter, it is common wisdom to believe that problems with an infinitely strong perturbation were never treatable with perturbation techniques (just check standard literature in applied mathematics where my interpretation is never given). No duality whatsoever. For the Schroedinger equation this implied a lot of problems nobody was able to manage leaving a large room in the literature to occupy by my technique. The same is true for the Einstein equations and quantum field theory and I have never seen these solutions to appear in literature before. Indeed, I have got a lot of papers published in refereed journals for this reason.

      So, even if you can recover some known techniques obtaining the solution in particular cases, there is yet a logical gap to bridge and this is what I have done.

      Cheers,

      Marco

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