Testing symmetry properties of nonlinear governing equations

[nickthequick]

Hi,

I'm a bit uncertain about the validity of my argument/approach to the following:

I'm trying to prove that the solution to a partial differential equation

$\frac{\partial u(x,t)}{\partial t} + N[u(x,t)] = 0$, where N is some nonlinear operator, CAN BE (not necessarily is) asymmetric about x=0. I do not know $u(x,t)$.

To that end, I've been examining the behavior of the system under the transformation $x\to -x$ (i.e. a reflection or inversion transformation) and can show that

$\frac{\partial u(x,t)}{\partial t} + N[u(x,t)] \neq\frac{\partial u(-x,t)}{\partial t} + N[u(-x,t)],$

so that we do not expect u(x,t) = u(-x,t).

I am wondering if this is the best way to test this reflection symmetry. Also, if I can find the terms in the operator N that violate this symmetry, is it coherent to call these 'symmetry breaking' terms?

Any references to relevant resources would be appreciated,

Thanks!

Nick

 

[jvicens]

Hi Nick,

Your approach to testing the reflection symmetry in your partial differential equation seems sound. By showing that the transformed equation is not equal to the original equation, you are demonstrating that the system is not symmetric. However, this does not necessarily prove that the solution is asymmetric about x=0. It is possible that the solution could still be symmetric, but the equation itself is not.

As for your question about calling the terms in N that violate symmetry "symmetry breaking," it may be more accurate to call them "asymmetry-inducing" terms. This is because the terms themselves are not necessarily breaking symmetry, but rather contributing to the overall asymmetric behavior of the equation.

In terms of references, you may find useful information in books or articles on nonlinear partial differential equations or symmetry breaking in physical systems. Additionally, consulting with colleagues or seeking guidance from a mentor in your field may provide valuable insights and resources.

Best of luck with your research!