How to solve this Hermite Initial Value Problem with Fourier Transform

Question

So I have a Hermite-Gaussian profile given by the initial value problem $$ iu_z + u_{xx}=0,\quad (x,z)\in\mathbb{R}\times\mathbb{R}_+ $$ where the condition initially is given by this $$ u_0(x) = H_n\left(\sqrt{2b}x\right)e^{-bx^2} $$ where $H_n(x)$ is the Hermite polynomial of nth order

The problem is to find $u(x,z)$ and compute various moments of $x$ to determine the behavior of the profile with propagation distance.

I know that the Fourier Transform of this Hermite polynomial is given by thanks to this answer $$ \mathscr{F}(H_n(\sqrt{2b}x)e^{-bx^2}) =(-i)^n\sqrt{\frac{2\pi}{2b}}H_n\left(\frac{\xi}{\sqrt{2b}}\right)e^{-\xi^2/4b} $$ I took the Fourier Transform of the initial equation and got a new equation with two variables $\xi_1, \xi_2$ , namely $$ \mathscr{F}(iu_z + u_{xx}) = [i(i\xi_1) + i^2\xi^2_2]\widehat u = -(\xi_1+\xi_2^2)\widehat u = 0 $$

I can't for the life of me understand how to extend this initial condition to two variables (x,z) to find the inverse Fourier Transform. Any help is appreciated.

Answer 1

(Partial answer)

You only need to take the Fourier transform with respect to $x$, such that your differential equation becomes $i\hat{u}_z-\xi^2\hat{u}=0$, hence $\hat{u}(z,\xi) = A(\xi)e^{-iz\xi^2}$. The initial condition gives then $$ A(\xi) = \hat{u}(0,\xi) = \hat{u}_0(\xi) = (-i)^n\sqrt{\frac{\pi}{b}}H_n\left(\frac{\xi}{\sqrt{2b}}\right)e^{-\xi^2/4b}. $$ I let you taking the inverse Fourier transform $-$ N.B. : the fact that Hermite functions are eigenfunctions of the Fourier transform will help (see here).

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Addendum. I don't know in which context your differential equation arises (it looks like Schrödinger equation for a free particle), but it is to be noticed that it can take the form of the standard 1D heat equation, i.e. $u_t = u_{xx}$, after a Wick rotation $z \mapsto -it$.

Answer 2

@Abezhiko’s answer is great and solves the problem efficiently. Just to provide another perspective on the problem: You can view applying the Fourier transform as a form of separation of variables. If the transform does separate the variables, then you can write the solution in the form of a single integral where each component is a product of a Fourier component in $x$ and a Fourier component in $z$, with each product individually solving the differential equation. So

$$ u(x,z) = \int\mathrm d\xi\,\hat u(\xi)\mathrm e^{\mathrm i\xi x}\mathrm e^{\mathrm i\xi_z(\xi)z}\;, $$

where the function $\xi_z(\xi)$ is determined by the condition that $\mathrm e^{\mathrm i\xi x}\mathrm e^{\mathrm i\xi_z(\xi)z}$ solves the differential equation. Your derivation shows $\xi^2+\xi_z(\xi)=0$, so $\xi_z(\xi)=-\xi^2$. Thus the general solution has the form

$$ u(x,z) = \int\mathrm d\xi\,\hat u(\xi)\mathrm e^{\mathrm i\xi x}\mathrm e^{-\mathrm i\xi^2z}\;, $$

and since you know the Fourier transform of $u(x,0)$, you know $\hat u(\xi)$ and can obtain $u(x,z)$ by performing the inverse transform. This is of course the same inverse transform that you need to perform in @Abezhiko’s answer.