= Solution
Use the standard uniform-density <incompressible flow> version of <ideal magnetohydrodynamics>, in which $\nabla\cdot\mathbf u=0$ and <pressure> enforces that constraint. The finite-$\gamma$ compressible adiabatic <pressure> equation is replaced by this <pressure> constraint. The consequences of retaining that cover-sheet equation literally as an additional condition are given below.
Write $F=f(R,z+v(R)t)$, $G=g(R,z-v(R)t)$, $U=F+G$ and $b=F-G$, where $b$ is the toroidal <Alfvén velocity>. The azimuthal <velocity> is $U$, while the <Alfvén velocity> vector is $b\mathbf e_\phi+v\mathbf e_z$. Both solenoidal constraints hold automatically. Since there is no meridional flow, the azimuthal <magnetohydrodynamic momentum equation> and <ideal magnetohydrodynamic induction equation> reduce to
$$
U_t=v\,b_z,\qquad b_t=v\,U_z.
$$
Differentiating the traveling arguments verifies both equations for arbitrary smooth $f$, $g$ and $v$. In particular, no condition $v'=0$ is needed.
The remaining force balance is essential. Define the total <pressure> per unit mass, including the <gravitational potential>, by
$$
\Pi=\frac p\rho+\Phi+\frac12(b^2+v^2).
$$
The radial acceleration is $-U^2/R$, and magnetic curvature tension contributes $-b^2/R$. The radial and axial momentum equations are consequently
$$
\boxed{\Pi_R=\frac{U^2-b^2}{R}=\frac{4FG}{R},\qquad \Pi_z=0.}
$$
They are compatible if and only if
$$
\boxed{\partial_z(FG)=0.}
$$
This is the <pressure compatibility of nonlinear torsional Alfvén profiles>. It is also sufficient locally: when $FG$ is independent of $z$, choose
$$
\Pi(R,t)=C(t)+\int_{R_0}^R\frac{4F(R',z,t)G(R',z,t)}{R'}\,dR',\qquad
p=\rho\left[\Pi-\Phi-\frac12(b^2+v^2)\right].
$$
A stationary axisymmetric $\Phi$ satisfying the Poisson equation can be absorbed in this <pressure>. Gravity need not vanish. If the axis is included, smoothness requires $U,b=O(R)$ and a regular axial field; $f,g=O(R)$ and smooth even $v$ give the usual regular extension. The <pressure> constant may be chosen to make $p$ positive on a bounded region where the other terms are bounded.
The compatible profiles have the following interpretation. At a cylinder with $v\ne0$, the variables $a=z+vt$ and $c=z-vt$ are independent as $z,t$ vary. Compatibility reads $f_a(R,a)g(R,c)+f(R,a)g_c(R,c)=0$. If both profiles are nonzero, division gives $f_a/f=-g_c/g=\kappa(R)$, independent of both traveling arguments. Therefore
$$
\boxed{f(R,a)=A(R)e^{\kappa(R)a},\qquad
g(R,c)=C(R)e^{-\kappa(R)c}}
$$
locally on such a region. Alternatively, one profile vanishes identically and the other is unrestricted. Choosing a point where one profile is nonzero shows that a nontrivial opposite profile obeys the same linear <ordinary differential equation> everywhere on a connected axial interval, so isolated zeros do not create additional overlapping branches.
If $v>0$, a pure $f$ profile travels towards negative $z$, and a pure $g$ profile towards positive $z$, each at the local <Alfvén speed> $v(R)$. Their <velocity> and toroidal-field perturbations have equal magnitudes, with respectively the same or opposite signs. They are finite-amplitude <torsional Alfvén waves>. For a single profile, centrifugal force and magnetic curvature tension cancel, leaving spatially constant $\Pi$; <gas pressure> compensates the varying <magnetic pressure>. The waveform is transported without axial steepening. Different cylinder speeds shear its phase radially, <magnetohydrodynamic phase mixing>.
At linear order both wave families may be superposed, since their interaction product is second order in amplitude. Two arbitrary opposing profiles cannot simply be superposed at finite amplitude: their product would generate an axial variation in the radial force imbalance, incompatible with purely azimuthal <velocity>. The exponential pair is a formal exception, but is unbounded in an axial direction when $\kappa\ne0$. If both profiles must be bounded on the whole axial line, $\kappa=0$, so $f=A(R)$ and $g=C(R)$ describe a steady rotating, twisted-field equilibrium. Their <pressure> gradient balances the remaining centrifugal and magnetic curvature forces. At $v=0$ there is no propagation; the condition is just $f(R,z)g(R,z)=h(R)$, giving additional steady profiles. These cases exhaust the local possibilities under the stated ansatz and domain qualifications.
There is a closure distinction in the wording. <Incompressible flow> is the formal infinite-sound-speed limit, not the simultaneous imposition of finite-$\gamma$ adiabatic compression and an independently fixed density; see https://arxiv.org/pdf/1604.03835[Ogilvie's discussion of simplified fluid models, section 2.8]. If the cover-sheet <pressure> equation is nevertheless imposed in addition, then axisymmetry and purely azimuthal motion give $p_t=0$. In addition to $(FG)_z=0$, the necessary and sufficient local <pressure>-gradient conditions, with stationary $\Phi$, are
$$
\boxed{\partial_t\partial_z(b^2)=0,\qquad
\partial_t\left[\frac12\partial_R(b^2)-\frac{4FG}{R}\right]=0.}
$$
They follow by requiring the gradients $p_z/\rho=-\Phi_z-\tfrac12(b^2)_z$ and $p_R/\rho=4FG/R-\Phi_R-\tfrac12(b^2+v^2)_R$ to be time independent; the remaining spatially uniform <pressure> variation is removed by $C(t)$.
For a single traveling profile with $v\ne0$, these extra conditions require its square to be affine in its traveling argument, $f^2=a(R)(z+vt)+b_0(R)$ or $g^2=a(R)(z-vt)+b_0(R)$, with $v(R)a(R)$ constant across a connected radial region. Smooth bounded profiles on the entire axial line then have $a=0$ and are steady. For two nonzero exponential profiles the extra conditions force $\kappa=0$. Thus arbitrary finite-amplitude traveling pulses are valid for the intended incompressible <pressure> model, but not generally for the overconstrained finite-$\gamma$ closure. The force compatibility derived above makes the distinction explicit.
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