Abstract
In this paper we are concerned with the asymptotics of solutions of the second Painleve equation (PII), which involves a parameter α. First, we review the asymptotics of solutions of PII in the special case when α = 0, including the Hastings-McLeod solution, which is probably the most studied solution of PII. Then we discuss the asymptotics of PII in the general case when α 6= 0. We show numerically the existence of monotonic increasing and monotonic decreasing solutions for PII with α = n ∈ Z, which exist on the whole real line and are generalizations of the Hastings-McLeod solution. We formulate a conjecture about these generalized Hastings-McLeod solutions.
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