## 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!

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### 8 Responses to Exact solutions of nonlinear equations

1. Peter Morgan says:

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?

• mfrasca says:

Hi Peter,

You are wrong. These solutions have been checked till 3+1.

Marco

2. BlackGriffen says:

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.

• mfrasca says:

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

3. […] 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 […]

4. […] 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 […]

5. Daniel says:

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,

Ciao.

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