Suppose $u_1, u_2: \mathbb{R}^2 \rightarrow \mathbb{R}$ are two solutions to the wave equation
\begin{equation}
\partial_t^2u_i = \partial_x^2u_i
\end{equation}
obeying the restriction
$\partial_x \left(u_1^2 + u_2^2\right) = 0$.
Is it so that $u_1$ and $u_2$ must be of the form
\begin{equation}
...
@gage if you check his profile he has a non trivial account on MO so I assume he understands the rules? No?
I assume you are trying to prove existence? Or the work done by the wave is constant? I have a background in nonlinear MHD and I haven't come across this formulation of conservation? Also do you have other initial conditions ?
No, in fact I come on this in the problem of nonlinear alfven waves in an ideal MHD setting where the velocity field is restricted to $\vec{v}=(v_x(z,t),v_y(z,t),0)$. If you check the $z$ component of the momentum equation and you let $u_1=B_x$, $u_2=B_y$ and take $z$ in stead of $x$ you come to this restriction, while also these components of the magn. field must satisfy the wave equation
Existence is then not my problem (because that question is already solved positively). Uniqueness is my problem.
Ok :), can you tell me anything about existence of axisymmetric solutions to the ideal MHD equations with velocity field $\vec{v}=r\Omega(r,z,t) \hat{\phi}$ (cylindrical coordinates) with $\Omega$ nontrivially time-dependent? That question is also in my mind lately
Yeah, ideal MHD, rotating plasmas ($\Omega$ is a rotational speed)
Because I know that in this setting things simplify a lot: poloidal field must be time independent, \Omega satisfies linear equations when regarding the induction equations and the $phi$-momentum eq.
You should check a paper by Ken mcclements and chippy dealing with the this very problem where the toroidal rotation is much. Larger than and poloidal so can be neglected in expansion. So are you working on tokamak dynamics? As I was looking at rotational plasma with the grad shafranov equation?
One more question whilst working thought model I am assuming you only have variation in the z axis? For all variables? I.e $B_x(z,t)$ and similarly for pressure?
Ah ok. So I only dealt with equilbria for rotating plasmas in the two fluid limit. So which Department are you atracked to? I only know two personally warwick (place of study) and leuven (having given a talk and had my an examiner for my viva from there goosens)
Discussion between Chinny84 and Thiba…
Discussion between Chinny84 and Thiba…