The equation $r^2 \frac{\partial ^3\Psi(r,s)}{\partial r^3}=s^2 \frac{\partial \Psi(r,s)}{\partial s}.$ Possible connections in physics and math?

Question

Recently I wrote down the following linear third order partial differential equation:

$$r^2 \frac{\partial ^3\Psi(r,s)}{\partial r^3}=s^2 \frac{\partial \Psi(r,s)}{\partial s} \tag{1}$$

The particular solution I obtained for $(1)$ is:

$$\Psi(r,s)=2 \sqrt{\frac{r}{s}}K_1(2\sqrt{r s})$$

where $K_1$ is the modified Bessel function of the second kind and $r,s>0.$ I'll be honest, I guessed the solution, so I don't know the normal method.

I searched online for a list of most common partial differential equations but didn't see anything that resembled the form of $(1).$

I also talked to some people knowledegable in physics but most of the differential equations applied are second order. I learned from them that $(1)$ is dimensionally inconsistent and that one can make it dimensionally consistent using physical constants with the right units.

At this moment I'm not interested in the steps to obtain my solution. I would like to know if the PDE or the analytic solution I found are connected to other areas of math or physics.

I tried thinking of an ansatz that would help but couldn't come up with anything. I actually don't see anything of this form appearing in the literature so I can't just learn from someone else's solution and explanation.

I thought, Bessel functions are applied in physics...the Bessel differential equation. Some solutions find applications in vibrating membranes. However, there's some crucial differences in my case because I'm dealing with a modified Bessel function, of fixed index and it's $2$ dimensional i.e. is a function of $r,s.$

Does $(1)$ appear in the literature anywhere? Is $(1)$ or the solution $\Psi(r,s)$ connected to other problems in math or physics?

Thanks for the possible resources and/or feedback.

Answer 1

(Too long for a comment)

I'm not aware of such a differential equation in the scope of physics or mathematics, so that I don't have any references to provide. Nevertheless, which methods did you try apart from guessing ? Because a lot can be done by yourself. Here is a small list.

Obviously, all those methods can be mixed up and I probably forgot some of them. Nevertheless, there is no need for a detailed litterature when one shows ressourcefulness ;)

1. Ansatz

You have already determined one solution by guessing, which I haven't checked to be honest, but simpler solutions are still to be found, such as $\Psi(r,s) = a + br + cr^2$ or $\Psi(r,s) = r\ln r - r + \frac{1}{s}$. Stationary solutions constitute some interesting particular cases too.

2. Series expansion

You might try using a bivariate Taylor expansion, i.e. $\Psi(r,s) = \displaystyle \sum_{n,m} a_{nm} \frac{r^n}{n!} \frac{s^m}{m!}$, in order to convert the differential equation into a recurrence relation for the coefficients $a_{nm}$. You can also use Frobenius method to consider non-integer exponents or even less standard expansions by working with a plane wave expansion (i.e. Fourier series) or within the basis of Bessel functions (since they seem to have a link with this equation).

3. Eigenvalue decomposition

You can study this linear equation from the angle of an eigenvalue problem. You may find the eigenvalues of its differential operators in order to expand the solution in their eigenbasis.

4. Change/separation of variables

The equation can be simplified thanks to a change of variables with respect to $r$, $s$ or even $\Psi$ itself. Here, an evident change is done by setting $\sigma := 1/s$, so that $s^2\frac{\partial}{\partial s} \mapsto -\frac{\partial}{\partial\sigma}$, which allows to get rid of a non-constant coefficient. You can even introduce auxiliary variables in order to restructure the equation, as in the method of separation of the variables, i.e. $\Psi(r,s) = R(r)S(s) = f(r) + g(s)$ in the most simple cases.

5. Fourier/Laplace transform

I don't have to present such linear transforms anymore. In the present case, the Fourier transform permits for example to reduce this third-order differential equation to second-order, since $r^2\frac{\partial^3}{\partial r^3} \mapsto i\rho^3\frac{\partial^2}{\partial\rho^2}$. After that, all the methods dedicated to second-order PDEs in the litterature can be deployed. Of course, other transforms might be helpful.

6. Lagrangian and symmetry

This equation of motion may be derived from a Lagrangian. If so, you can then study its symmetries and deduce some conserved quantities in order to circumvent a direct resolution of the equation.

7. Lie theory

Symmetry is also a key point of Lie theory, which often turns out to be a powerful tool. You can explicit the flow of the equation by exponentiating its differential operator $-$ in physics, we would talk about the "propagator". You can also consider jet bundles in order to apply a generalized version of the method of characteristics for higher-order PDEs.

8. Stochastic calculus

Since the equation can be recasted as a second-order PDE, it may be reformulated as a Fokker-Planck equation or a stochastic process, so that the whole machinery of stochastic calculus may become handy. Even the Feynman-Kac formula might be useful.

9. Distributions

You can broaden the space of work by considering distributional solutions for instance. It can be deduced straight from inspection that a solution is given by $\Psi(r,s) = (Ar + Br^2)\theta(r) + C\theta(s) + D\delta(s)$, where $\theta$ is the Heaviside step function and $\delta = \theta'$ the Dirac delta function.

10. Numerical resolution

Depending on what you need next or by default, an analytical solution might be unnecessary, so that a numerical resolution of the equation is sufficient (to study asymptotical behaviours for instance).