# Differential equations: trying to connect a nonlinear equation to a linear one

The following is motivated by taking a product space $$\Omega$$ and splitting it into two parts via projections, whose subspaces, $$T$$ and $$X$$, are home to functions which satisfy a nonlinear PDE and a linear PDE respectively. I would like to understand if one can further take the Mellin followed by Fourier transforms of the nonlinear and linear PDE's respectively and arrive at the same resulting equation. I give some good evidence for this to be true. My question is at the end.

Begin by considering the nonlinear equation

$$c(t,x)\frac{\partial}{\partial t} \sqrt{g}=\Psi g \tag{1}$$

where $$\Psi g := \sum_{i=1}^n \sqrt{-\frac{\partial}{\partial x_i}g} \tag{2}$$

If we require that $$g(t_1,x_i)$$ satisfies $$\forall i$$

$$d(t_1,x_i)\frac{\partial^2}{\partial t_1^2}g(t_1,x_i)=- \frac{\partial}{\partial x_i} g(t_1,x_i) \tag{3}$$

and require that

$$g:=g(t_1,x_1,x_2,\cdot\cdot\cdot,x_n) :=\prod_{i=1}^ng(t_1,x_i) \tag{4}$$

Then a particular solution can be found. For example let $$c(t_1,x_1)=\frac{2\sqrt{t_1}}{\sqrt{x_1}}$$ and let $$d(t_1,x_i)={\frac{t_1}{x_i}}.$$ We recover the function which leads to our solution $$g$$

$$g(t_1,x_i)=e^{\frac{t_1}{\log x_i}} \tag{5}$$

for $$x_i \ne 0,1$$ and $$t_1>0$$.

We note that when $$n=1,$$ $$g$$ satisfies both the nonlinear $$(1)$$ and the linear $$(3).$$

We also note the following. If we suppose time is $$n$$-dimensional, then we have a generalization of $$(3)$$

$$\bigg( \sum_{i=1}^n t_i \bigg) \Delta h=- n x_1 \frac{\partial}{\partial x_1}h \tag{6}$$

where we may write $$h:=h(x_1,t_1,t_2,\cdot\cdot\cdot,t_n)=\prod_{i=1}^n e^{\frac{t_i}{\log x_1}}$$

There is a projection at work here. We either collapse $$n$$-dimensional time down to one dimension and let space be $$n$$ dimensional, or vice versa. To see this we can let time and space have the same dimensions. Both will have multiple dimensions now. We form the $$2n$$-dimensional product space

$$\Omega=\prod_{i=1}^n g(t_{i},x_i)=g(t_1,x_1)g(t_2,x_2)\cdot\cdot\cdot g(t_n,x_n) \tag{7}$$

Now $$\forall i$$ let $$t_i=t_1.$$ We recover $$g$$. Now $$\forall i$$ let $$x_i=x_1$$. We recover $$h$$. Note that $$g$$ satisfies $$(1)$$ and $$h$$ satisfies $$(6)$$.

In other words we have the time projection operator

$$P_{T}: \Omega \to T$$

such that $$(t_1,t_2,\cdot\cdot\cdot,t_n,x_1,x_2,\cdot\cdot\cdot,x_n)\mapsto (t_1,x_1,x_2,\cdot\cdot\cdot,x_n)$$

and we have the space projection operator

$$P_{X}: \Omega \to X$$

such that $$(t_1,t_2,\cdot\cdot\cdot,t_n,x_1,x_2,\cdot\cdot\cdot,x_n)\mapsto (t_1,t_2,\cdot\cdot\cdot,t_n,x_1)$$

I'm wondering why the time projection of the product space $$\Omega$$ yields a solution to a nonlinear PDE $$(1)$$ while the space projection of the product space $$\Omega$$ yields a solution to a linear PDE $$(6).$$

If $$n=1$$ then both $$(1)$$ and $$(6)$$ are simultaneously satisfied by $$g$$ and $$h.$$

One can transform $$(4)$$

$$\Phi(t_1,z_1,z_2,\cdot\cdot\cdot, z_n)=\int_{(0,1)^n}x_1^{z_1}x_2^{z_2}\cdot\cdot\cdot x_n^{z_n}~ g~ \frac{dx_1dx_2\cdot\cdot\cdot dx_n}{x_1x_2 \cdot\cdot\cdot x_n}$$

and obtain

$$\Phi(t_1,z_1,z_2,\cdot\cdot\cdot, z_n)= \prod_{i=1}^n \Phi(t_1,z_i)= \prod_{i=1}^n 2\sqrt{\frac{z_i}{t_1}}K_1\big(2\sqrt{z_i t_1}\big)$$

If $$z_i=z_1$$ $$\forall i$$ then we'll have a modified Bessel of the second kind to the $$n$$th power.

we can project down to

$$\Phi(t_1,z_i)=2\sqrt{\frac{z_i}{t_1}}K_1\big(2\sqrt{z_i t_1}\big)$$

which individually satisfy the linear third order PDE

$$z^2 \frac{\partial^3}{\partial z_i^3} \Phi(t_1,z_i)= t_1^2 \frac{\partial}{\partial t_1}\Phi(t_1,z_i) \tag{8}$$

We have a map given by the integral transform

$$f: g \to \Phi$$

Here we know $$f$$ is linear, bijective, continuous and smooth.

Question:

Does $$\Phi(t_1,z_1,z_2,\cdot\cdot\cdot, z_n)$$ satisfy some linear $$n$$-dimensional generalization of $$(8)?$$ (I would take a reference for this).

This will act like a bridge between $$(1)$$ and $$(6).$$

I formed the product space $$\Xi = \prod_{i=1}^n \Phi(t_i,z_i)$$

and tried to investigate projections like I did with $$\Omega.$$ But I'm not aware of any $$n$$-dimensional linear PDE's that products of modified Bessel functions of the second kind satisfy, which are generalizations of $$(8)$$.

I think perhaps using a computer program could help here in terms of deducing the differential equation. But I'm also confident it can be done theoretically.

• The notation $g(x_i)$ (particularly in equation (3)) seems to me to mean a function $g$ depending on some variable, with the value $x_i$ plugged in for the variable. But then, to my eyes, $g$ would not also depend on some variable $t$ and some other $x_j$ variables. Maybe it would help to use a clearer notation to see what variables $g$ depends on. Commented Sep 4, 2023 at 12:47
• In equation (6), you write for any $i$ chosen, and then write an equation with a sum over $i$ in it, so it looks like $i$ is not chosen. Commented Sep 4, 2023 at 12:50
• @BenMcKay thanks I fixed it Commented Sep 4, 2023 at 14:28