Abstract:
We consider an extended version of the second Painlevé equation $(\mathrm P_{\mathrm{II}})$, which appears as the simplest member of a recently-derived extended second Painlevé hierarchy. For this third-order system we consider the application of the Ablowitz–Ramani–Segur algorithm, use its auto-Bäcklund transformations (auto-BTs) to construct sequences of rational solutions and solutions defined in terms of Bessel functions, the latter constituting the analogues for the extended $\mathrm P_{\mathrm{II}}$ of the well-known Airy function solutions of $\mathrm P_{\mathrm{II}}$. In addition, we present two new Bäcklund transformations, which extend the Schwarzian $\mathrm P_{\mathrm{II}}$ equation due to Weiss and an auto-BT due to Gambier. Finally, we use the auto-BTs of extended $\mathrm P_{\mathrm{II}}$ also to derive a new third-order discrete system.
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