= Solution
Take the primitive state vector $\mathbf U=(\rho,p,u_x,u_y,u_z,B_x,B_y,B_z)^T$, with $\rho>0$. In the absence of gravity, the <ideal magnetohydrodynamic equations> are first order: expanding the <material derivatives>, <pressure> gradient and <Lorentz force> makes every term linear in a spatial derivative of $\mathbf U$, with coefficients depending on $\mathbf U$. Dividing <momentum> balance by $\rho$ therefore gives a <quasilinear partial differential equation> system $\partial_t\mathbf U+A_i(\mathbf U)\partial_i\mathbf U=0$ with eight-by-eight matrices. The <solenoidal magnetic-field constraint> is an additional constraint on initial data; the <ideal magnetohydrodynamic induction equation> preserves it because the divergence of a curl vanishes.
For a one-dimensional <simple wave in magnetohydrodynamics>, write $\mathbf U=\mathbf U(q(x,t))$ with a nonzero state-space tangent. Substitution gives $\mathbf U_q q_t+A_x\mathbf U_q q_x=0$. A nonconstant profile requires the tangent to be a right <eigenvector> of $A_x$:
$$
A_x\mathbf U_q=v(q)\mathbf U_q,\qquad q_t+v(q)q_x=0.
$$
The <chain rule> then gives \b[the wave-speed equation]:
$$
\boxed{v_t+v v_x=v_q(q_t+v q_x)=0.}
$$
This is the <Inviscid Burgers equation>, including the special case of constant $v$. Its <characteristic curves> are $x=a+v_0(a)t$, with $v=v_0(a)$. As long as the mapping remains invertible,
$$
v_x=\frac{v_0'(a)}{1+t v_0'(a)}.
$$
Thus a region with $v_0'<0$ steepens: faster characteristics catch slower ones, and the first gradient catastrophe occurs at $t_*=-1/\min_a v_0'(a)$ when the minimum is negative. A genuinely nonlinear compressive <simple wave> therefore forms a <shock wave>. Rarefactive profiles can spread instead; steepening is not inevitable for every initial profile.
To continue past this time, use <weak solutions> of the conservative mass, <momentum>, <fluid total-energy equation> and <ideal magnetohydrodynamic induction equation>. The <Rankine-Hugoniot conditions> fix the jumps and shock velocity, while physical entropy production selects admissible <magnetohydrodynamic shocks>. Diffusive <shock wave> layers are replaced by moving discontinuities, so their microscopic structure need not be explicitly resolved. In particular, the smooth adiabatic <pressure> equation must be replaced by total-energy conservation when imposing the <ideal magnetohydrodynamic shock conditions>.
For the <Alfvén waves>, impose the one-dimensional <solenoidal magnetic-field constraint>, so $B_x$ is constant and $\delta B_x=0$. Put $\mathbf B_\perp=(B_y,B_z)$ and $c=v-u_x$. The transverse components of the <linearized ideal magnetohydrodynamic equations> give
$$
-c\,\delta\mathbf u_\perp-\frac{B_x}{\mu_0\rho}\delta\mathbf B_\perp=0,\qquad
-c\,\delta\mathbf B_\perp-B_x\delta\mathbf u_\perp=0.
$$
The longitudinal <momentum> equation additionally gives $\delta p+\mathbf B_\perp\cdot\delta\mathbf B_\perp/\mu_0=0$. The Alfvén <eigenvectors> have $\delta\rho=\delta p=\delta u_x=0$, hence their transverse polarization is perpendicular to $\mathbf B_\perp$. For $s=\pm1$, their speed and explicit right <Alfvén characteristic eigenvectors> are
$$
\boxed{v_s=u_x+s\frac{B_x}{\sqrt{\mu_0\rho}},\qquad
r_s=\begin{pmatrix}0\\0\\0\\h_y\\h_z\\0\\-s\sqrt{\mu_0\rho}\,h_y\\-s\sqrt{\mu_0\rho}\,h_z\end{pmatrix},\quad B_yh_y+B_zh_z=0.}
$$
For $\mathbf B_\perp\ne0$ one may choose $(h_y,h_z)=(B_z,-B_y)$, up to a nonzero scalar factor. If $\mathbf B_\perp=0$, either transverse polarization is allowed and this characteristic speed is degenerate. These expressions describe the two propagating Alfvén branches for $B_x\ne0$; if $B_x=0$, they coalesce with advected, nonpropagating transverse disturbances.
Integrating along an Alfvén <simple wave> leaves $\rho,p,u_x,B_x$ unchanged and gives $d\mathbf u_\perp=-s\,d\mathbf B_\perp/\sqrt{\mu_0\rho}$ and $\mathbf B_\perp\cdot d\mathbf B_\perp=0$. Consequently \b[the finite-amplitude <nonlinear Alfvén wave> relations] are
$$
\boxed{f_1=C_y-\frac{s f_3}{\sqrt{\mu_0\rho}},\qquad f_2=C_z-\frac{s f_4}{\sqrt{\mu_0\rho}},\qquad f_3^2+f_4^2=B_\perp^2=\text{constant},\qquad v=v_s.}
$$
The constant-magnitude condition is essential: otherwise a varying <magnetic pressure> would drive longitudinal compression. An arbitrary smooth phase profile gives an explicit family, $f_3=B_\perp\cos\Theta(x-v_st)$ and $f_4=B_\perp\sin\Theta(x-v_st)$, with $f_1,f_2$ as above. Direct substitution in the transverse <momentum> and <ideal magnetohydrodynamic induction equation> gives $(u_x-v_s)\mathbf u_\perp'=B_x\mathbf B_\perp'/(\mu_0\rho)$ and $(u_x-v_s)\mathbf B_\perp'=B_x\mathbf u_\perp'$, confirming the solution without relying on the <eigenvector> argument. Longitudinal <momentum> holds because $p+B_\perp^2/(2\mu_0)$ is constant. The speed $v_s$ is independent of amplitude along this family: \b[these <linearly degenerate characteristic fields> translate without steepening].
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