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We consider real-valued solutions , , of the second Painlevé equation which are parameterized in terms of the monodromy data of the associated Flaschka–Newell system of rational differential equations. Our analysis describes the transition, as , between the oscillatory power-like decay asymptotics for (Ablowitz–Segur) to the power-like growth behavior for (Hastings–McLeod) and from the latter to the singular oscillatory power-like growth for (Kapaev). It is shown that the transition asymptotics are of Boutroux type; that is, they are expressed in terms of Jacobi elliptic functions. As applications of our results we obtain asymptotics for the Airy kernel determinant in a double scaling limit , , as well as asymptotics for the spectrum of .
The simplest case of the Langlands functoriality principle asserts the existence of the symmetric powers of a cuspidal representation of over the adèles of , where is a number field. In 1978, Gelbart and Jacquet proved the existence of . After this, progress was slow, eventually leading, through the work of Kim and Shahidi, to the existence of and . In this series of articles we revisit this problem using recent progress in the deformation theory of modular Galois representations. As a consequence, our methods apply only to classical modular forms on a totally real number field; the present article proves the existence, in this “classical” case, of and .