It is well known that the Cauchy problem for the wave equation in one dimension can be easily solved.

*J.-P. Dias* and

*M. Figueira* [Ric. Mat. 35, 309–316 (1986;

Zbl 0658.35076)] have shown, that the squared argument of a solution of the so called nonlinear Dirac equation

${\partial}_{t}u+\alpha {\partial}_{x}u=i{\left|u\right|}^{2}u$,

$\alpha $ being a two-by-two Hermitian traceless matrix with

${\alpha}^{2}=1$, compare

*W. E. Thirring* [Ann. Phys. 3, 91–112 (1958;

Zbl 0078.44303)], fulfills the one-dimensional wave equation

${\partial}_{t}^{2}{\left|u\right|}^{2}-{\partial}_{x}^{2}{\left|u\right|}^{2}=0$. The authors of the present paper construct the explicit solutions

$u=({u}_{1},{u}_{2})$ for the cases

$\alpha ={\sigma}_{1},{\sigma}_{2},{\sigma}_{3}$,

${\sigma}_{i}$ being the Pauli matrices. Explicit solutions for the general case,

$\alpha =\mathbf{e}\xb7\sigma $,

$\mathbf{e}$ being a unit vector of

${\mathbb{R}}^{3}$, are not considered as well as actions of SU(2) on solutions. Instead, explicit solutions for two related Cauchy problems with

$\alpha ={\sigma}_{1},{\sigma}_{2},{\sigma}_{3}$, the Dirac-Klein-Gordon system of equations,

${\partial}_{t}u+\alpha {\partial}_{x}u=i\phi u$,

${\partial}_{t}^{2}\phi -{\partial}_{x}^{2}\phi ={u}^{+}\alpha u$, and the Dirac equation with matrix potential

${\partial}_{t}u+\alpha {\partial}_{x}u=iVu$,

$V\left(x\right)$ being a two Hermitian by two matrix commuting with

$\alpha $ are constructed. Eventually, the decay of the solutions for

$t\to \infty $ is considered. This is a nice and well formulated paper.