I am working on solving the differential equation on $\mathbb{R} \times S^1 \times S^1$ and would be very thankful for some assistance, particularly in understanding and applying Fourier expansion along the periodic dimensions, and working with orthogonal functions.

The equation:

$\partial_i \partial^i \phi + (x_i x^i) \phi + (x_3 \partial_2 \phi - x_2 \partial_3 \phi)=0$,

where $x_1 \in \mathbb{R}$, and $x_2 \sim x_2 + 2\pi$, $x_3 \sim x_3+2\pi$, $i=1,2,3$.

The boundary conditions are $\phi \rightarrow 0$ as $x_1 \rightarrow \infty$ and $\partial_1 \phi \rightarrow 0$ as $x_1 \rightarrow \infty$.

Fourier expansion along $x_2$ and $x_3$:

Expanding $\phi = \sum\limits_{m,n} \phi_{mn}(x_1) e^{i(mx_2 + n x_3)}$ where $\phi_{mn}$ is a function only of $x_1$ allows us to write the pde as one with only a second-order derivative with respect to $x_1$:

$ \left( \partial_1^2 + (x_i x^i) \right) \phi + \sum\limits_{m,n} \left( - (m^2+n^2) + i (mx_3 - n x_2) \right) \phi_{mn} e^{i(mx_2 + nx_3)} =0.$

Next steps:

What I would like to do is express this as a differential equation for one function $\phi_{mn}$ (i.e. for one combination of $m$ and $n$) rather than an infinite set of coupled differential equations for all the functions $\{\phi_{mn} \}$ since this appears unsolvable to me (although perhaps it is not).


Is there a way to apply orthogonality of the complex exponentials $f_{m}(y) = \frac{1}{\sqrt{2\pi}} e^{imy}$, $\int_{-\pi}^{\pi} f^*_m f_n dy = \delta_{mn}$ to write this diff eq in terms of a single function $\phi_{mn}$, for instance by multiplying the equation by an exponential $f_p(x_2) f_q(x_3)=\frac{1}{2\pi}e^{i(p x_2 + qx_3)}$ and integrating with respect to $x_2$ and $x_3$ in order to pick out just the terms $\phi_{pq}$? Rather than just pointing out an answer to my problem (although I would be very thankful since that is my ultimate goal), I'm really interested in a way of thinking about the basis of functions $\{ f_m \}$ that would allow me to grasp how the variable coefficients in the pde affect the orthogonality of each of the terms in the equation.

Of course, if you feel that I am haring off in the wrong direction entirely, I'd appreciate knowing that as well.

Thank you in advance for your time!


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