Steen-Ermakov-Pinney equation and integrable nonlinear deformation of the one-dimensional Dirac equation (Q723523)

The paper considers the linear one-dimensional Dirac equation \[ \frac{df}{dx} = \Lambda f , \quad \Lambda = \begin{pmatrix} \lambda & q_1 (x) \\ q_2 (x) & - \lambda \end{pmatrix} \] with \(\lambda \in \mathbb C\), \(q_i \in C^\infty (\mathbb R, \mathbb C^2)\), \(f \in L_\infty (\mathbb R , \mathbb C^2) \); and the nonlinear Dirac equation \[ \frac{d g}{d x} = \Lambda, g + (\delta g), \] with \(\delta g\) a nonlinear term which can depend on \(g\) itself. It is shown that for \[ \begin{aligned} \delta g_1 & =(\alpha / g_1) \;q_2 \;\exp \left[ D_x^{-1} [ q_1 (g_1/g_2) + q_2 (g_2/g_1) ] \right], \\ \delta g_2 & = (\alpha / g_2) \;q_1 \;\exp \left[ D_x^{-1} [ q_1 (g_1/g_2) + q_2 (g_2/g_1) ] \right], \end{aligned} \] with \(\alpha \in \mathbb C\) an arbitrary constant, solutions to the nonlinear Dirac equation are obtained from the fundamental solution matrix \(F\) for the linear one as \[ \begin{pmatrix} g_1 \\ g_2 \end{pmatrix} = \begin{pmatrix} (c_{12} f_{11}^2 - c_{21} f_{11} f_{12} - c_{21} f_{12}^2 + c_{22} f_{12} f_{11} )^{1/2} \\ (c_{12} f_{21}^2 - c_{11} f_{21} f_{22} + c_{22} f_{22} f_{21} - c_{21} f_{22}^2 )^{1/2} \end{pmatrix} \] The proof is based on a generalization of the Steen-Ermakov-Pinney approach.