= Solution
For the circular <MHD wave polarization>, $B_x^2+B_y^2=a^2B_0^2$ is constant. Choose any constant $\rho_0>0,p_0>0$, put $v_a=B_0/\sqrt{\mu_0\rho_0}$ and take
$$
\boxed{\mathbf u=-a v_a(\cos\zeta,\sin\zeta,0),\qquad
\rho=\rho_0,\qquad p=p_0,\qquad \zeta=k(z-v_at)}.
$$
This gives a <circularly polarized nonlinear Alfvén wave> of arbitrary amplitude. All fields depend only on $z,t$, and $u_z=0$, so $\mathbf u\cdot\nabla\mathbf u=0$ and $\nabla\cdot\mathbf u=0$. Constant <mass density> and <pressure> then satisfy continuity and the adiabatic <pressure> equation, and $\nabla\cdot\mathbf B=0$ because $B_z$ is constant.
The remaining <ideal magnetohydrodynamic induction equation> and transverse <magnetohydrodynamic momentum equation> are
$$
\partial_t\mathbf B_\perp=B_0\partial_z\mathbf u_\perp,\qquad
\rho_0\partial_t\mathbf u_\perp=\frac{B_0}{\mu_0}\partial_z\mathbf B_\perp.
$$
Since $\partial_t=-v_a\partial_z$ on the traveling profiles and $\mathbf u_\perp=-(v_a/B_0)\mathbf B_\perp$, both identities hold exactly when $v_a^2=B_0^2/(\mu_0\rho_0)$. The longitudinal <magnetohydrodynamic momentum equation> holds because $p+|\mathbf B_\perp|^2/(2\mu_0)$ is constant. \b[The circularly polarized wave is therefore an exact nonlinear solution even though the fluid is compressible.] Compressibility permits <mass density> changes; it does not require every possible motion to compress the gas.
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