My research has brought me to the following linear parabolic second order PDE:

$$ \frac{\partial^2}{\partial t^2}\Psi(t,x)=c(t,x)\frac{\partial}{\partial x}\Psi(t,x) $$

for $c(t,x)=-\frac{x}{t}$ and $x\in (0,1)$ and time $t\in (0, \infty).$

Using the ansatz $\Psi(t,x)=X(x)Y(t)$ this can be simplified to ODE's and solved.

However, an important solution class is realized in the following way:

Let $(M,g)$ be the Minkowski plane with null coordinates. Take the isometry $f:M \to \zeta $ with $f(x,y)=(e^x,e^y).$ Take the natural Cauchy foliation of $\zeta$ given by $\log x \log y = t.$ Essentially when you furnish $\zeta$ with $g$ and the induced measure by means of the volume form, a natural solution to the above equation becomes clear:

$$ \Psi(t,x)=\exp \frac{t}{\log x} $$

The solution looks like this:

Just like the free Schrodinger equation can be seen as a heat equation with an imaginary constant, there is an analogous PDE here too:

$$ \frac{\partial^2}{\partial t^2}\Psi(t,x)=\frac{x}{it} \frac{\partial}{\partial x}\Psi(t,x) $$

We get a similar looking solution:

$$ \Psi(t,x)=\exp \frac{it}{\log x} $$

Are there any papers in the literature that deal with these specific PDE's where I can read more about them?

If there are no papers in the literature, as I suspect, what is the interpretation of these equations and my solution, and how might they be physically relevant?

My research has brought me to the following linear parabolic second order PDE:

$$ \frac{\partial^2}{\partial x^2}\Psi(t,x)=c(t,x)\frac{\partial}{\partial t}\Psi(t,x) $$

for $c(t,x)=-\frac{t}{x}$ and $x\in (0,\infty)$ and time $t\in (0, 1).$

Using the ansatz $\Psi(t,x)=X(x)Y(t)$ this can be simplified to ODE's and solved.

However, an important solution class is realized in the following way:

Let $(M,g)$ be the Minkowski plane with null coordinates. Take the isometry $f:M \to \zeta $ with $f(x,y)=(e^x,e^y).$ Take the natural Cauchy foliation of $\zeta$ given by $\log x \log y = s.$ Essentially when you furnish $\zeta$ with $g$ and the induced measure by means of the volume form, a natural solution to the above equation becomes clear:

$$ \Psi(t,x)=\exp \frac{x}{\log t} $$

Just like the free Schrodinger equation can be seen as a heat equation with an imaginary constant, there is an analogous PDE here too:

$$ \frac{\partial^2}{\partial x^2}\Psi(t,x)=\frac{t}{ix} \frac{\partial}{\partial t}\Psi(t,x) $$

We get a similar looking solution:

$$ \Psi(t,x)=\exp \frac{ix}{\log t} $$

Are there any papers in the literature that deal with these specific PDE's where I can read more about them?

If there are no papers in the literature, as I suspect, what is the interpretation of these equations and my solution, and how might they be physically relevant?

My research has brought me to the following equation:

$$ \frac{\partial^2}{\partial t^2}\Psi(t,x)=c(t,x)\frac{\partial}{\partial x}\Psi(t,x) $$

for $c(t,x)=-\frac{x}{t}$ and $x\in (0,1)$ and time $t\in (0, \infty).$

An important solution class is realized in the following way:

Let $(M,g)$ be the Minkowski plane with null coordinates. Take the isometry $f:M \to \zeta $ with $f(x,y)=(e^x,e^y).$ Take the natural Cauchy foliation of $\zeta$ given by $\log x \log y = t.$ Essentially when you furnish $\zeta$ with $g$ and the induced measure by means of the volume form, a natural solution to the above equation becomes clear:

$$ \Psi(t,x)=\exp \frac{t}{\log x} $$

The solution looks like this:

Are there any papers in the literature that deal with this PDE where I can read more about it?

I'm actually unaware that this PDE has even been solved in the literature. It seems to me that nobody knows how to solve it non-perturbatively (please correct me if I'm wrong).

My research has brought me to the following linear parabolic second order PDE:

$$ \frac{\partial^2}{\partial t^2}\Psi(t,x)=c(t,x)\frac{\partial}{\partial x}\Psi(t,x) $$

for $c(t,x)=-\frac{x}{t}$ and $x\in (0,1)$ and time $t\in (0, \infty).$

Using the ansatz $\Psi(t,x)=X(x)Y(t)$ this can be simplified to ODE's and solved.

However, an important solution class is realized in the following way:

Let $(M,g)$ be the Minkowski plane with null coordinates. Take the isometry $f:M \to \zeta $ with $f(x,y)=(e^x,e^y).$ Take the natural Cauchy foliation of $\zeta$ given by $\log x \log y = t.$ Essentially when you furnish $\zeta$ with $g$ and the induced measure by means of the volume form, a natural solution to the above equation becomes clear:

$$ \Psi(t,x)=\exp \frac{t}{\log x} $$

The solution looks like this:

Just like the free Schrodinger equation can be seen as a heat equation with an imaginary constant, there is an analogous PDE here too:

$$ \frac{\partial^2}{\partial t^2}\Psi(t,x)=\frac{x}{it} \frac{\partial}{\partial x}\Psi(t,x) $$

We get a similar looking solution:

$$ \Psi(t,x)=\exp \frac{it}{\log x} $$

Are there any papers in the literature that deal with these specific PDE's where I can read more about them?

If there are no papers in the literature, as I suspect, what is the interpretation of these equations and my solution, and how might they be physically relevant?