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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 t u+α x u=i|u| 2 u, α being a two-by-two Hermitian traceless matrix with α 2 =1, compare W. E. Thirring [Ann. Phys. 3, 91–112 (1958; Zbl 0078.44303)], fulfills the one-dimensional wave equation t 2 |u| 2 - x 2 |u| 2 =0. The authors of the present paper construct the explicit solutions u=(u 1 ,u 2 ) for the cases α=σ 1 ,σ 2 ,σ 3 , σ i being the Pauli matrices. Explicit solutions for the general case, α=𝐞·σ, 𝐞 being a unit vector of 3 , are not considered as well as actions of SU(2) on solutions. Instead, explicit solutions for two related Cauchy problems with α=σ 1 ,σ 2 ,σ 3 , the Dirac-Klein-Gordon system of equations, t u+α x u=iφu, t 2 φ- x 2 φ=u + αu, and the Dirac equation with matrix potential t u+α x u=iVu, V(x) being a two Hermitian by two matrix commuting with α are constructed. Eventually, the decay of the solutions for t is considered. This is a nice and well formulated paper.
MSC:
35C05Solutions of PDE in closed form
35L45First order hyperbolic systems, initial value problems
81Q05Closed and approximate solutions to quantum-mechanical equations
References:
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[3]3. Dias, J.P., Figueira, M.: Time decay for the solutions of a nonlinear Dirac equation in one space dimension. Ricerche Mat. 35, 309–316 (1986)
[4]4. Fang, Y.F.: On the Dirac-Klein-Gordon equation in one space dimension. Differential Integral Equations 17, 1321–1346 (2004)
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[7]7. Reed, M.: Abstract non-linear wave equations. Lecture Notes in Mathematics 507 (1976)
[8]8. Strauss, W.A.: Nonlinear invariant wave equations. Invariant wave equations (Proc. ”Ettore Majorana” Internat. School of Math. Phys., Erice, 1977), Lecture Notes in Phys. 73, 197–249 (1978)
[9]9. Thirring, W.E.: A soluble relativistic field theory. Ann. Physics 3, 91–112 (1958)