On the Stokes geometry of perturbed tangential Pearcey systems (Q824245)

The Pearcey system \(\mathcal{M}\) with a large parameter \(\eta\) is given by \[ \Bigl(4\eta^{-3}\frac{\partial^3}{\partial x_1^3}+2x_2\eta^{-1}\frac{\partial} {\partial x_1} + x_1 \Bigr)\psi(x,\eta)=0, \quad \Bigl(\eta^{-1}\frac{\partial}{\partial x_2}-\eta^{-2}\frac{\partial^2} {\partial x_1^2} \Bigr)\psi(x,\eta)=0. \] For a hyperplane \(Y(c):\) \(x_2=c(x_1-1)\) \((c\not=0)\) (respectively, \(Y_2:\) \(x_1=A_1\) \((\not=0)\)), the tangential system \(\mathcal{N}(c)\) (respectively, \(\mathcal{N}_2\)) of \(\mathcal{M}\) admits a non-hereditary turning point (NHTP), which does not originate from a turning point of \(\mathcal{M}\), and is a consequence of the relation between characteristic varieties of \(\mathcal{M}\) and \(\mathcal{N}(c)\) (respectively, \(\mathcal{N}_2\)). Let \(\tilde{\mathcal{N}}(c)\), \(\tilde{\mathcal{N}}_2\) be perturbed systems given by adding lower-order terms with their characteristic varieties kept intact. This paper investigates the Stokes geometry of \(\tilde{\mathcal{N}}(c)\) and \(\tilde{\mathcal{N}}_2\). For \(\tilde{\mathcal{N}}(1)\), as a concrete example, bicharacteristics emanating from the NHTP are computed, and the global figure is examined. Viewing the resulting global structure, the authors define a new kind of virtual turning point of \(\tilde{\mathcal{N}}(1)\) and show the existence of infinitely many virtual turning points in the Stokes geometry of \(\tilde {\mathcal{N}}(1)\). In the Stokes geometry of \(\mathcal{N}_2 \) and a slight perturbation of it, the activity of Stokes curves is studied by the use of the exact steepest descent method near a special point at which three active Stokes curves meet forming a linearly ordered crossing point.