Finding solutions to nonlinear wave equations with time-periodic boundary conditions (closed timelike curves)

Question

Is it known how to solve nonlinear dispersive wave equations, such as the Klein-Gordon equation with a $\phi^4$ interaction with time-periodic boundary conditions? The motivation behind time-periodic boundary conditions is to provide a simplified model of spacetime with closed timelike curves (CTCs). I haven’t been able to find any substantial prior research on the problem. I am also interesting in how CTCs affect wave turbulence, such as in calculating the Kolmogorov-Zakharov spectra and diffusion equations or whether CTCs will cause PDEs that normally show weak wave turbulence to show strong wave turbulence.

So if you have the quartic interaction with Lagrangian $\mathcal{L}(\varphi)=\frac{1}{2} [\partial^\mu \varphi \partial_\mu \varphi -m^2 \varphi^2] -\frac{\lambda}{4!} \varphi^4$ with equations of motions $\partial^2\varphi+\mu_0^2\varphi+\lambda\varphi^3=0$ defined on $\mathbb{R}^n \times \mathbb{T}^1$, where $\mathbb{T}$is the torus, is there some standard way of finding solutions? Can an analog of the initial value problem be defined, perhaps with data defined on only part of a Cauchy hypersurface? Will the PDE defined on a spacetime with CTCs show finite time blow-up even when the PDE would not show blow-up without CTCs?

Answer 1

TL;DR. I believe no nontrivial time periodic dynamics exists for the equations you suggest. I do not have a full proof, but I do have some indicators in that sense.

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I will consider here the cubic wave equation $$ \tag{1} \frac{\partial^2 \phi}{\partial t^2}-\Delta \phi = \pm \phi^3.$$ This is a more mathematical notation for the "quartic interaction" you mentioned above with $\mu_0=0$ and $\lambda=\pm 1$.

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There are no nontrivial time-periodic solutions to (1) if the Cauchy initial data at $t=0$, i.e. $(\phi_0, \dot\phi_0)$ are assumed to belong to the critical Sobolev space $\dot{H}^{1/2}\times \dot{H}^{-1/2}$. Indeed, as proven by [1], any solution to (1) satisfying

$$\tag{2} \limsup_{t\to\infty}\lVert (\phi(t), \partial_t \phi(t))\rVert_{\dot{H}^{1/2}\times \dot{H}^{-1/2}} < \infty$$

must scatter to a linear solution as $t\to\infty$. This means that there must exist a linear solution $\phi_0$ such that

$$\lVert (\phi(t) - \phi_0(t), \partial_t \phi(t)-\partial_t\phi_0(t))\rVert_{\dot{H}^{1/2}\times \dot{H}^{-1/2}}\to 0.$$

This is of course in contrast with the time-periodic assumption. Indeed, any time-periodic solution to (1) obviously satisfies (2), hence it must scatter to some linear solution $\phi_0$, but then again by time periodicity $\phi=\phi_0$, hence $\phi=0$ since that is the only solution to (1) that also solves the linear wave equation.

Remark 1. To be fair, [1] also assumes that $\phi$ is radially symmetric. It is however widely conjectured that this assumption is redundant, and several partial results in that direction are available. Maybe there even is some new paper I am not aware of that proves this conjecture; I would not be surprised.

Remark 2. The critical Sobolev space is important because it is the only space $\dot{H}^s\times \dot{H}^{s-1}$ that is invariant under the natural scaling of (1). Here, $\lVert f\rVert_{\dot{H}^s}^2:= \int_{\mathbb R^3} \lvert \hat{f}(\xi)\rvert^2 \lvert \xi\rvert^{2s}\, d\xi$. So if there are no nontrivial solution in this space, it is enough for me to turn away from this problem. There might be nontrivial solutions in another class, though.

Remark 3. The Klein--Gordon case $\mu_0>0$ should also be considered. I expect similar dynamics in that case but I would not know how to answer precisely off the top of my head.

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[1] B. Dodson and A. Lawrie. “Scattering for the radial 3D cubic wave equation”. Anal. PDE 8.2 (2015), pp. 467–497.

Answer 2

I wanted to add some thoughts and commonly used terminology, I hope it is helpful.

Nonlinear dispersive equations like the nonlinear wave equation (say, on $\mathbb R^d$) sometimes admit ‘travelling wave solutions’ (in some cases called ‘solitons’) of the form $$ \varphi(x,t)=f(x-vt), $$ which can be seen as constant in time after a change of frame of reference. Actual time-periodic solutions analogous to solitons are called ‘breathers’, some equations admit them. I don’t know much about this, but for instance I have found this paper which discusses breathers of the Klein-Gordon equation. If you are looking for time-periodic solutions, maybe breathers are a starting point.

The above are all very special solutions, tipically for a given equation there is a finite number of parameters that describe all these solutions. It seems you want to know about more ‘generic’ solutions, with prescribed initial data in some sense. I have no idea on how that would work. One would expect that for a given initial datum at time zero, the solution is not periodic in time. Assigning an initial datum only on a partial region of space at time zero is one possibility, but I would have no intuition on such a problem and I don’t know of literature addressing something like that.