I am modelling a heat distribution in optical element irradiated by laser. System is radially symmetric, and element is thin, i.e. heat value depends only on distance from center. Heat is received via laser radiation and is removed from upper and lower sides of the element. A significant characteristic of the problem is that heat absorbtion coefficient depends on the current temperature of an element in a complex periodic way. I'm researching the stationary distributions by running simulation until equilibrium appears. An interesting feature of a model is that there are several stationary states possible, depending on initial distribution (which is a consequence of heat absorbtion depending on current temperature). So, the system is described by parabolic partial differential equation with some coefficients depending on current temperature. I solve it with implicit numerical scheme.

So, the question is: if parameters of a system do not change in time, is it possible to get not a stationary distribution, but distribution changing in time (possibly, oscillating around some state). All initial distributions which I've tried after some time developed into one or several possible for given parameters stationary distributions.

Edit: This is the PDE of the problem:

$ \begin{aligned} C\frac{\partial u}{\partial t} = k\left(\frac{\partial^2 u}{\partial r^2} + \frac{1}{r}\frac{\partial u}{\partial r}\right) - \frac{\alpha}{l}u + \frac{A(u)}{l}I(r),\\r \in (0,R), \ t \in (0,T \ ]; \\ C\frac{\partial u}{\partial t} = 2u\frac{\partial^2 u}{\partial r^2} - \frac{\alpha}{l}u + \frac{A(u)}{l}I(r),\\ r=0; \\ \left. \frac{\partial u}{\partial r} \right|_{r=R}=0; \\ \left. \frac{\partial u}{\partial r} \right|_{r=0}=0; \end{aligned} $


$n=n_0+\alpha_n u$

$l=l_0(1+\alpha_l u)$

$k=\frac{2\pi n}{\alpha}$

$A(n,l)=4\chi k l \frac{1+n^2}{4n^2+(1-n^2)^2\sin^2\xi}$

$I(r)=\frac{P}{\pi a^2}\exp(-\frac{r^2}{a^2})$

If $\left. u\right|_{t=0}=0$ is the initial condition, then the system evolves into some "minimal" stationary distribution. Then it is possible to increase the laser power P, wait and decrease it back. Then, depending on coefficients, the system can evolve into new stationary state. The reason is the nonconstant and periodic radiation absorbtion function A which depends on current temperature of the element. This leads to equilibrium between heat absorbtion and removal for several distributions, not one.

So, while nonuniquiness of stationary distributions is the proven fact, I failed to achieve oscillating distribution by the simulation. But I cannot be sure that it doesn't exist - that's why I asked the question.

I should mention that I don't know anything about solving this PDE (with nonlinear A) in exact way, so numerical methods are the only way I can use.


1 Answer 1


It depends on the differential equation. Some parabolic PDEs should, given constant boundary conditions, converge to a steady state. Some will oscillate.

An example of an oscillating PDE is the Time-Dependent Schroedinger Equation (TDSE). If one selects an initial state which is a superposition of energy eigenstates, then the system will oscillate forever.


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