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} u_1(x,t)=C_1\sin(kz)\sin(kt+\phi_1) + C_2\cos(kz)\sin(kt+\phi_2) \end{equation} \begin{equation} u_2(x,t)=\pm \left( C_1\cos(kz)\sin(kt+\phi_1) - C_2\sin(kz)\sin(kt+\phi_2) \right) \end{equation} EDIT: as is evident from Robert Israel's answer, $u_1(z,t)=0, u_2(z,t)=t$ is a counterexample. But of course it is not of the kind I'm looking for. So impose the extra constraint that $u_1$ and $u_2$ must be globally bounded, what about then?

  • $\begingroup$ Welcome to MSE what have you tried or what are your thoughts? $\endgroup$ – user171177 Dec 22 '14 at 18:21
  • $\begingroup$ Ah my beloved MHD waves. :). $\endgroup$ – Chinny84 Dec 22 '14 at 18:51
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    $\begingroup$ @gage if you check his profile he has a non trivial account on MO so I assume he understands the rules? No? $\endgroup$ – Chinny84 Dec 22 '14 at 18:58
  • $\begingroup$ 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 ? $\endgroup$ – Chinny84 Dec 22 '14 at 19:02
  • $\begingroup$ 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 $\endgroup$ – Thibaut Demaerel Dec 22 '14 at 19:07

Counterexample: let $u_1(x,t) = 0$, $u_2(x,t) = t$.

  • $\begingroup$ In terms of MHD waves the solutions are periodic. With the waves being non degenerate (in general ideal .ie non resistive plasmas). $\endgroup$ – Chinny84 Dec 22 '14 at 20:28

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