Exact solutions of nonlinear equations

Recently, I have posted on the site of Terry Tao and Jim Colliander, Dispersive Wiki. I am a regular contributor to this beautiful effort to collect all available knowledge about differential equations and dispersive phenomena. Of course, I can give contributions as a solver of differential equations in the vein of a pure physicist. But mathematicians are able to give rigorous theorems on the behavior of the solutions without really solving them. I invite you to take some time to look at this site and, if you are an expert, to register and contribute to it.

My recent contribution is about exact solutions of nonlinear equations. This is a really interesting field and most of the relevant results come from soliton theory. Terry posted on his blog about Liouville equation (see here). This equation is exactly solvable and is widely known to people working in string theory. But also one of the most known equations in physics literature can be solved exactly. My preprint shows this. Indeed I have to update it as, working on KAM theorem, I have obtained the exact solution to the following equation (check here on Dispersive Wiki):

$\Box\phi +\mu_0^2\phi+\lambda\phi^3 = 0$

that can be written as

$\phi(x) = \pm\sqrt{\frac{2\mu^4}{\mu_0^2 + \sqrt{\mu_0^4 + 2\lambda\mu^4}}}{\rm sn}\left(p\cdot x+\theta,\sqrt{\frac{-\mu_0^2 + \sqrt{\mu_0^4 + 2\lambda\mu^4}}{-\mu_0^2 - \sqrt{\mu_0^4 + 2\lambda\mu^4}}}\right)$

being now the dispersion relation

$p^2=\mu_0^2+\frac{\lambda\mu^4}{\mu_0^2+\sqrt{\mu_0^4+2\lambda\mu^4}}.$

As always $\mu$ is an arbitrary parameter with the dimension of a mass. You can see here an example of mass renormalization due to interaction. Indeed, from the dispersion relation we can recognize the following renormalized mass

$m^2(\mu_0,\mu,\lambda)=\mu_0^2+\frac{\lambda\mu^4}{\mu_0^2+\sqrt{\mu_0^4+2\lambda\mu^4}}$

that depends on the coupling. This class of solutions clearly show how the nonlinearities produce contributions to mass. Either by modifying it or by generating it. So, it is not difficult to imagine that Nature may have adopted them to display mass wherever there is not.

As a by-product, I am now able to give a consistent quantum field theory in the infrared for the scalar field (always thank to my work on KAM theorem), obtaining the needed corrections to the propagator and the spectrum. I hope to find some time in the next days to add all this new material to my preprint. Meanwhile, enjoy Dispersive Wiki!

This entry was posted on Wednesday, July 15th, 2009 at 10:59 am and is filed under Applied Mathematics, mathematics, Physics, Wikis. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

8 Responses to Exact solutions of nonlinear equations

Peter Morgan says:

15/07/2009 at 5:01 pm

The solutions in this post, and the dn solution on the Wiki, are effectively 1+1 solutions, in that there are no superpositions of waves in 3+1 dimensions. Presumably that means that your “I am now able to give a consistent quantum field theory in the infrared for the scalar field” is effectively only for 1+1 dimensions, insofar as your approach depends on the superpositions given here?

Reply
- mfrasca says:
  
  15/07/2009 at 8:12 pm
  
  Hi Peter,
  
  You are wrong. These solutions have been checked till 3+1.
  
  Marco
  
  Reply
BlackGriffen says:

15/07/2009 at 11:28 pm

Correct me if I’m wrong, but isn’t this the solution to the “classical equations of motion” for phi^4 theory? This would imply, if I understand correctly, that you’ve managed to sum the tree level diagrams, yes? How does this include renormalization then? I mean, the way I recall it, renormalization kicked in when you started evaluating loop diagrams.

Also, when you mention an infrared consistent quantum field theory does that mean that you know how to evaluate amplitudes as a perturbation about this classical solution instead of the free field one?

Also, I’m not clear whether you’re claiming that this is all of the exact solutions or if this is just a class of exact solutions.

Reply
- mfrasca says:
  
  15/07/2009 at 11:40 pm
  
  Hi BlackGriffen,
  
  Your comment made me clear that a point I did not emphasize: These are just classical solutions. At this stage I am not claiming nothing else and I cannot say nothing about quantum field theory. You put them into the equation of motion and you will get out zero, provided dispersion relation holds.
  
  About quantum field theory with these solutions, as I said in the post, I am working on it. I can do infrared quantum field theory with a gradient expansion. I hope to have some time to work out a quantum field theory with these solutions.
  
  Finally, this is just a class of exact solutions. They appear interesting as produce a dispersion relation with massive contributions coming from nonlinearities.
  
  Marco
  
  Reply
Exact solutions on arxiv « The Gauge Connection says:

24/07/2009 at 12:03 pm

[…] solutions on arxiv As promised in my preceding post (see here), I have posted yesterday a preprint on this matter on arxiv (see here). I have presented all the […]

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Exact solutions on arxiv | Phasing says:

24/07/2009 at 5:21 pm

[…] promised in my preceding post (see here), I have posted yesterday a preprint on this matter on arxiv (see here). I have presented all the […]

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Daniel says:

28/07/2009 at 10:50 pm

Hi Marco,

If i’m not mistaken, i believe the solutions to your differential equation above, namely,

$(\box + g + \phi^{2})\, \phi = 0 \; ;$

where $g = \mu^{2}/\lambda$ (we can always divide by $\lambda$ , for if $\lambda = 0$ we can just drop the last term and work with the well-known wave eq), are the well-known Parabolic Cylinder Functions.

To see this more clearly and explicitly done, you can check the following articles,

• Complexified Path Integrals and the Phases of Quantum Field Theory;
• Symmetry Breaking: A New Paradigm for Non-Perturbative QFT and Topological Transitions.

Ciao.

Reply
- mfrasca says:
  
  28/07/2009 at 11:01 pm
  
  Hi Daniel,
  
  They are Jacobi elliptic functions.
  
  You can check my latest paper on arxiv.
  
  Ciao,
  
  Marco
  
  Reply