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| == Nonlinear PDE and Perturbation Methods ==
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| The application of the perturbation methods described above to PDE gives an interesting result, i.e. the dual series to the small parameter solution series is a gradient expansion [[FraA2006]].
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| This can be seen by considering a NLKG equation
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| <math>\Box\phi+\lambda V(\phi)=0.</math>
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| The choice of the perturbation term to compute a small or a large parameter series depends also on the way the derivatives of the field are managed.
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| In order to see this we apply the computation given in the previous section by rescaling time as <math>t\rightarrow \lambda t</math> and take a solution series with a large parameter
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| <math>\phi = \sum_{n=0}^{\infty}\frac{1}{\lambda^n}\phi_n.</math>
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| The non trivial set of equations is so obtained
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| <math>\partial_\tau^2\phi_0+V(\phi_0)=0</math>
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| <math>\partial_\tau^2\phi_1+V'(\phi_0)\phi_1=\Delta^2\phi_0</math>
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| <math>\vdots</math>
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| where <math>\tau=\lambda t</math>. Indeed, this is a gradient expansion.
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| An interesting problem that applies to a given PDE is
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| <math>\Box\phi+\lambda V(\phi)=j</math>
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| where <math>j</math> is a driving term. When a small parameter series has to be computed we obtain that at the leading order one has generally to solve
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| <math>\Box\phi_0=j</math>
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| that is very easy to solve by the Green function method
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| <math>\phi_0=\phi_{H}+\int d^Dx'G(x-x')j(x')</math>
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| where
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| <math>\Box\phi_{H}=0</math>
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| and
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| <math>\Box G=\delta^D</math>
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| being $\delta^D$ a Dirac distribution of the given dimensionality <math>D</math>.
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| A similar result holds also for the large parameter series [[FraA2007]],[[FraB2007]]. We note that the leading order of the gradient expansion is now
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| <math>\partial_\tau^2\phi_0+V(\phi_0)=j.</math>
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| A small time series holds that has the form
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| <math>\phi_0=\sum_{n=0}^{\infty}a_n\int d\tau'G(\tau-\tau')(\tau-\tau')^nj(\tau')</math>
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| being
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| <math>\partial_\tau^2 G+V(G)=\delta</math>
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| and the coefficients <math>a_n</math> are computed by deriving the equation we started from and with the initial conditions and will generally depend on the values of the source and its derivatives at the intial time. The success of the method relies on the ability to obtain analitically the Green function.
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| An interesting case is given by <math>V(\phi)=\phi^3</math> [[FraB2006]]. In this case one has
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| <math>G(t)=\theta(t)\left(\frac{2}{\lambda}\right)^\frac{1}{4}sn\left[\left(\frac{\lambda}{2}\right)^\frac{1}{4} \Lambda t,i\right]</math>I
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| where <math>\theta</math> is the step function, <math>sn</math> is a Jacobi elliptic function and <math>\Lambda</math> is a scale parameter being the theory scale invariant. This gives immediately the mass spectrum of the quantum theory as [[FraB2006]]
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| <math>\mu_n=(2n+1)\frac{\pi}{2K(i)}\left(\frac{\lambda}{2}\right)^\frac{1}{4}\Lambda</math>
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| where <math>n</math> is an integer and <math>K(i)=\int_0^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{1+\sin^2\theta}}</math>.
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