Consider the partial differential equation:
$$ t\frac{\partial^2}{\partial t^2} \sum_{n=1}^k \Phi_n(x,t)=-x\frac{\partial}{\partial x}\sum_{n=1}^k \Phi_n(x,t)+\sum_{n=2}^{k-1}a(n)\Phi_n(x,t) $$
where $a(n)$ is some prescribed sequence of natural numbers. For now let $a(n)$ be the oblong numbers.
I'm wondering how much power the sequence $a(n)$ has on constraining the solution space.
For example, if you specify an $a(n),$ does this imply the solution is unique, if it exists?
Fix $a(n)$ to be the oblong numbers. Then using an exponential ansatz, I found a solution to the equation:
$$ \Phi_n(x,t)=e^{\frac{nt}{\log x}} $$
Are there other solutions, or is this unique?
Edit:
Letting $k \to \infty$ we obtain:
$$ t\frac{\partial^2}{\partial t^2}\Psi(x,t)=-x\frac{\partial}{\partial x}\Psi(x,t)+h(x,t)$$
where $$\Psi(x,t)=-\frac{e^{\frac{t}{\log x}}}{e^{\frac{t}{\log x}}-1}=\sum_{n=1}^\infty\Phi_n(x,t)$$
and $$h(x,t)= -\frac{2e^{\frac{t}{\log x}}}{(e^{\frac{t}{\log x}}-1)^3}=\sum_{n=2}^{k-1}a(n)\Phi_n(x,t) $$
Fix $k=4$ for now. We therefore have the equation $$ (t \partial_{tt}+x \partial_x)(\Phi_1+\Phi_2+\Phi_3+\Phi_4)=a(2)\Phi_2+a(3)\Phi_3 $$ You can now proceed to solve the equations $$ (t \partial_{tt}+x \partial_x)\Phi_2=a(2)\Phi_2 \\ (t \partial_{tt}+x \partial_x)\Phi_3=a(3)\Phi_3 $$ These are solvable and admit a non-trivial solution of the form $$ \Phi_n=f_n+ x^{a(n)/2}, $$ where $f_n=f_n(t)$ is obtained by minimizing the action functional associated to the Lagrangian $$ L=\frac{1}{2}\bigg( (\partial_tu)^2+\frac{au^2}{2t} \bigg) $$ for a suitable domain with, say $t \geq t_0 >0$.
Now note that $\Phi_1$ and $\Phi_4$ dont appear on the right hand side of the equation. Hence we can add any function $\Phi_1$ with $$ \Phi \in \ker T $$ Where $Tu=(t\partial_{tt}+x \partial_x)u$ is the differential operator for your equation. Affine functions only depending on $t$ are among these - and hence a (non-trivial) solution is not unique and exists.
However, if you are interested in passing to the limit $n \to \infty$, I suggest you open a new questions, since this probably becomes a more functional analytics problem. In this case, you only consider a subset of some suitable space of functions such that $\sum_{n=2}^{\infty}a(n) \phi_n < \infty$ in a suitable sense.