# When is it useful to treat the time variable in a evolution problem as a spatial variable?

This question is a bit informally formulated and loosely stated, but I hope someone can make sense of it and guide me in some way. With my limited experience in PDEs, I have always seen the time variable being treated differently than the space variables.

A bit of background: As I'm studying a differential equation, 2nd order in space, 1st order in time, whose solution is a function $$u=u(\mathbf{x},t)$$ where $$\mathbf{x}=(x,y)\in \Omega\subset\mathbb{R}^2$$ where $$\Omega=[a,b]\times[c,d]$$ is a rectangular domain and $$t\in [0,\infty)$$. Using common reasoning in functional analysis, I was looking for solutions in Bochner spaces like $$H^1((0,\infty),H^2(\Omega))$$.

However, a certain change of variables $$z=y-t$$ and $$w=x$$ (I'm looking for traveling waves) reduced the original equation to the following $$-\Delta \phi(w,z)+\phi(w,z)-w=F(\phi(w,z)),$$ where $$u(x,y,t)=\phi(w,z)$$ and $$F$$ is an arbitrary (to be determined) function. In the $$(w,z)$$ variables, the new domain $$\tilde{\Omega}=[a,b]\times (\infty,b]$$ in question is an infinite semi strip containing $$\Omega$$ and $$\phi$$ inherits whichever boundary conditions $$u$$ satisfies. So now I realized in need to look for solutions in a new space $$H(\tilde{\Omega})$$ where $$H$$ is an appropriate function space.

Bottom line: To study the behavior of my solution $$u(x,y,t)$$ in time $$t$$ on a bounded domain $$\Omega$$, I now study the asymptotic behavior of $$\phi(w,z)$$ in the new "spatial variable" $$z=y-t$$.

The question: Is there an advantage of studying time $$t$$ as a spacial variable $$w=y-t$$?

If the question doesn't make any sense please let me know in the comment so that I make needed clarifications.

Added: Thank you for the very helpful comments. However, I should have emphasized that I'm interested in the nature of our new function space $$H(\tilde{\Omega})$$. Do I get an advantage in finding solutions in $$H(\tilde{\Omega})$$ vs $$H^1((0,\infty),H^2(\Omega))$$? In which space is time behavior easier to study?

• isn't this what we always do when we study traveling waves: time in a fixed coordinate frame can be exchanged with space in a moving frame; in physics we refer to this as Lorentz invariance. – Carlo Beenakker Jan 13 at 10:34
• A big issue is that time goes in one direction. While many evolutions are reversible, many are not. – Steve Huntsman Jan 13 at 13:39
• en.wikipedia.org/wiki/Problem_of_time – Steve Huntsman Jan 13 at 13:40
• If $x,y,z,t$ are just formal independant variables, then $w=y-t$ is no stranger than $v=x+y$. But if $x,y,z$ are supposed to have physical units (e.g. length), and $t$ is supposed to have units of time, then $w=y-t$ is not numerically defined. This is why Einstein's postulate that light propagates en vacuo at constant velocity $c$ is so important, for it formally permits space- and time- variables to be compared. Using $c$, we find $w=y-ct$ is numerically well defined with units of length. But, e.g., $w=e^y$ has no physical units, and expressions containing $w=e^y$ lose their physical content. – JHM Jan 14 at 0:58
• If a PDE represents a physical model, with physical variables, then I think it's important to express all transformations in the appropriate units (so-called "dimensional analysis"). However it is frequently very convenient (and extremely tempting) for the mathematicians to perform formally convenient change of variables, without regard to physical units. For example, if $r$ is a radius, then is $R:=1+r$ also a radius? I would argue not. And any intepretations of solutions which misidentified $R$ with a radius would be error prone. – JHM Jan 14 at 1:09