Classical scalar theory in D=1+1 and gradient expansion


As said before a pde with a large parameter has the spatial variations that are negligible. Let us see this for a very simple case. We consider the following equation

\frac{\partial^2\phi}{\partial t^2}-\frac{\partial^2\phi}{\partial x^2}-\lambda\phi^3=0

with the conditions \phi(0,t)=0, \phi(1,t)=0 and \phi(x,0)=x^2-x where the choice of a parabolic profile is arbitrary and can be changed. We also know that, if we can neglect the spatial part, the solution can be written down analytically as (see here and here):

\phi\approx (x^2-x){\rm sn}\left[(x^2-x)\sqrt{\frac{\lambda}{2}}t+x_0,i\right]

being x_0={\rm cn}^{-1}(0,i). Indeed, for \lambda = 5000 we get the following pictures

Numerical Curves - t is chosen as 0=red, 1/8=blue, 1/4=green, 0.3=yellow

Numerical Curves - t is chosen as 0=red, 1/8=blue, 1/4=green, 0.3=yellow

and

Analytical solution - t chosen as above

Analytical solution - t chosen as above

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.

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