The degenerate third Painlevé equation,and a ∈ C, and the associated tau-function are studied via the Isomonodromy Deformation Method. Connection formulae for asymptotics of the general as τ → ±0 and ±i0 solution and general regular as τ → ±∞ and ±i∞ solution are obtained.

Using the matrix Riemann-Hilbert factorisation approach for non-linear evolution equations (NLEEs) integrable in the sense of the inverse scattering method, we obtain, in the solitonless sector, the leading-order asymptotics as t → ±∞ of the solution to the Cauchy initial-value problem for the modified non-linear Schrödinger equation, i∂ t u+ 1 2 ∂ 2x u+|u| 2 u+is∂ x (|u| 2 u) = 0, s ∈ R >0 : also obtained are analogous results for two gauge-equivalent NLEEs; in particular, the derivative non-linear Schrödinger equation, i∂ t q+∂ 2 x q−i∂ x (|q| 2 q) = 0.

The degenerate third Painlevé equation, u ′′ (τ ) = (u ′ , and a ∈ C, is studied via the Isomonodromy Deformation Method. Asymptotics of general regular and singular solutions as τ → ±∞ and τ → ±i∞ are derived and parametrized in terms of the monodromy data of the associated 2 × 2 linear auxiliary problem introduced in [1]. Using these results, three-real-parameter families of solutions that have infinite sequences of zeroes and poles that are asymptotically located along the real and imaginary axes are distinguished: asymptotics of these zeroes and poles are also obtained.

Let Λ R denote the linear space over R spanned by z k , k ∈ Z. Define the real inner product (with varying exponential weights)Define the even degree and odd degree monic orthogonal Laurent polynomials: π π π 2n (z) := (ξ (2n) n ) −1 φ 2n (z) and π π π 2n+1 (z) := (ξ (2n+1) −n−1 ) −1 φ 2n+1 (z). Asymptotics in the double-scaling limit as N, n → ∞ such that N/n = 1+o(1) of π π π 2n+1 (z) (in the entire complex plane), ξ (2n+1) −n−1 , φ 2n+1 (z) (in the entire complex plane), and Hankel determinant ratios associated with the real-valued, bi-infinite, strong moment sequencek∈Z are obtained by formulating the odd degree monic orthogonal Laurent polynomial problem as a matrix Riemann-Hilbert problem on R, and then extracting the largen behaviour by applying the non-linear steepest-descent method introduced in [1] and further developed in [2,3].