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The explicit solutions to the nonlinear Dirac equation and Dirac-Klein-Gordon equation. (English) Zbl 1146.35021
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 ${\partial }_{t}u+\alpha {\partial }_{x}u=i{|u|}^{2}u$, $\alpha$ 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 ${\partial }_{t}^{2}{|u|}^{2}-{\partial }_{x}^{2}{|u|}^{2}=0$. The authors of the present paper construct the explicit solutions $u=\left({u}_{1},{u}_{2}\right)$ for the cases $\alpha ={\sigma }_{1},{\sigma }_{2},{\sigma }_{3}$, ${\sigma }_{i}$ being the Pauli matrices. Explicit solutions for the general case, $\alpha =𝐞·\sigma$, $𝐞$ being a unit vector of ${ℝ}^{3}$, 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.
##### MSC:
 35C05 Solutions of PDE in closed form 35L45 First order hyperbolic systems, initial value problems 81Q05 Closed and approximate solutions to quantum-mechanical equations
##### References:
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