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I am trying to put down some code to get numerically the solution of the following PDE: $$ \partial^2_t\phi-\partial^2_x\phi+\lambda\phi^3=\delta(x)\delta(t). $$ Of course, there are several problems already starting with a proper numerical definition of the Dirac deltas and a proper setting of the initial conditions. Last but not least this equation is nonlinear.

My question is this: Is there any library of known numerical recipes for this kind of problems, also for the linear case? I have no prejudice about the code and also well-known packages as Matlab or Mathematica are good for my aims.

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$\phi$ must be a distribution, hence the real problem is how to give a meaning to $\phi^3$. What is your theoretical framework to handle this? –  Arnold Neumaier Oct 23 '12 at 13:51
    
@ArnoldNeumaier: You are correct. This is my theoretical published work. I am trying to approach it numerically if possible. –  Jon Oct 23 '12 at 14:05
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It is nearly impossible to do good numerics for a problem where you don't even know how to check whether something is a solution. Let alone use packages which do it for you. Hyperbolic solvers will probably just produce artifacts which will be difficult to analyze for their meaning. –  Arnold Neumaier Oct 23 '12 at 14:11
    
Thanks, Arnold. I think at least this is the proper answer. –  Jon Oct 23 '12 at 14:14

2 Answers 2

To begin, I would probably advice to consider the steady problem by looking for ground states $\phi = \Phi(x)e^{i\omega t}$. In this way you reduce your equation to the nonlinear Helmholtz-type problem: the elliptic operator $\omega\phi + \phi_{xx}$ + nonlinear part $\lambda\phi^3$.

Now, I don't see what you can do theoretically with the nonlinear term. Since the works of L. Schwartz (50s) we know that the product of distributions cannot be consistently defined...

Jon, what is your initial problem? What would you like to do?

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But also when one neglect the nonlinear term, does it exist a numerical algorithm to solve a pde for its Green function? –  Jon Oct 23 '12 at 14:11
    
In the steady case you can approximate the Dirac-delta function by its regularized version (something smooth, but localized). You can have a look on equation (11) in this paper: citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.4.2702 –  Denys Oct 23 '12 at 14:24
    
Thank you for your answer and the paper you cite that seems a proper one for this problem. I think that, with this regularized version of Dirac distribution, the numerical problem should not have to cope with the question of multiplying distributions. Rather, it seems well defined. –  Jon Oct 25 '12 at 6:45

Create some mesh in $(x,t)$-space, take out the elements containing the origin, and impose nonzero boundary conditions on the resulting boundary that you believe is appropriate for the solution near the singularity. Then solve the problem without the forcing term in a domain with the resulting hole. This will give you an idea how the solution will look like.

The quality will be as good or bad as your guess of the boundary. Maybe your theoretical published work mentioned in the discussion helps to tell you what this guess should be.

Alternatively, you could work with the Fourier transform of your equation in space-time, and try to solve it by collocation in frequency-momentum space.

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Thank you for the answer. I think that a way to manage the singularity is well exposed in the paper cited in a comment by @Denys. The paper is freely downloadable. –  Jon Oct 25 '12 at 6:43

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