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?

  • $\begingroup$ 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. $\endgroup$ – Carlo Beenakker Jan 13 at 10:34
  • $\begingroup$ A big issue is that time goes in one direction. While many evolutions are reversible, many are not. $\endgroup$ – Steve Huntsman Jan 13 at 13:39
  • $\begingroup$ en.wikipedia.org/wiki/Problem_of_time $\endgroup$ – Steve Huntsman Jan 13 at 13:40
  • $\begingroup$ 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. $\endgroup$ – JHM Jan 14 at 0:58
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    $\begingroup$ 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. $\endgroup$ – JHM Jan 14 at 1:09

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