# Testing symmetry properties of nonlinear governing equations

1. Mar 31, 2013

### 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