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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. {\it J.-P. Dias} and {\it 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 $\partial_tu + \alpha\partial_xu = i|u|^2u$, $\alpha$ being a two-by-two Hermitian traceless matrix with $\alpha^2=1$, compare {\it 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 $u=(u_1, u_2)$ for the cases $\alpha = \sigma_1, \sigma_2, \sigma_3$, $\sigma_i$ being the Pauli matrices. Explicit solutions for the general case, $\alpha = {\bold e}\cdot {\pmb \sigma}$, ${\bold e}$ being a unit vector of $\Bbb R^3$, 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, $\partial_tu + \alpha\partial_xu = i\varphi u$, $\partial_t^2\varphi - \partial_x^2\varphi = u^+ \alpha u$, and the Dirac equation with matrix potential $\partial_tu + \alpha\partial_xu = iVu$, $V(x)$ being a two Hermitian by two matrix commuting with $\alpha$ are constructed. Eventually, the decay of the solutions for $t \rightarrow \infty$ is considered. This is a nice and well formulated paper.

35C05Solutions of PDE in closed form
35L45First order hyperbolic systems, initial value problems
81Q05Closed and approximate solutions to quantum-mechanical equations
Full Text: DOI
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