= Solution
For this geometry $\nabla\cdot\mathbf u=0$, $\nabla\cdot\mathbf B=0$ and $\mathbf u\cdot\nabla=0$. Consequently constant <mass density> and <pressure> satisfy the full nonlinear continuity and adiabatic pressure equations. Write $\mathbf B_\perp=(B_x,B_y)$ and $\mathbf u_\perp=(u_x,u_y)$. The <ideal magnetohydrodynamic induction equation> and the transverse <ideal magnetohydrodynamic momentum equation> become
$$
\partial_t\mathbf B_\perp=B_z\partial_z\mathbf u_\perp,\qquad
\rho\partial_t\mathbf u_\perp=\frac{B_z}{\mu_0}\partial_z\mathbf B_\perp.
$$
The longitudinal momentum equation is
$$
0=-\frac{1}{2\mu_0}\partial_z|\mathbf B_\perp|^2.
$$
Thus constant transverse magnetic magnitude eliminates the otherwise unavoidable <magnetic pressure> acceleration. Differentiating the <ideal magnetohydrodynamic induction equation> in time gives, for either transverse component,
$$
\boxed{\partial_t^2W=\frac{B_z^2}{\mu_0\rho}\partial_z^2W.}
$$
A constant-magnitude <nonlinear Alfvén wave> is obtained by taking $B_z\ne0$ and
$$
B_x=A\cos F(z-ct),\qquad B_y=A\sin F(z-ct),\qquad
c=\pm\frac{B_z}{\sqrt{\mu_0\rho}},\qquad
\mathbf u_\perp=\mathbf U_0-\frac{c}{B_z}\mathbf B_\perp,
$$
where $A$ and $\mathbf U_0$ are constants and $F$ is any sufficiently differentiable real function. Both first-order equations hold, and $B_x^2+B_y^2=A^2$. \b[Taking $F(\zeta)=k\zeta$ gives an exact finite-amplitude circularly polarized wave.] More general traveling rotations also work; an arbitrary superposition of oppositely traveling solutions of the <wave equation> need not preserve transverse magnetic magnitude and hence need not solve the full nonlinear system. If $B_z=0$, the coupled first-order equations instead require time-independent transverse fields and velocities, with spatially constant transverse magnetic magnitude; there is no propagating <Alfvén wave> in the $z$ direction.
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