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].

Using the matrix Riemann-Hilbert factorization approach for nonlinear evolution systems which take the form of Lax-pair isospectral deformations and whose corresponding Lax operators contain both discrete and continuous spectra, the leading-order asymptotics as t → ±∞ of the solution to the Cauchy problem for the modified nonlinear Schrödinger equation,, which is a model for nonlinear pulse propagation in optical fibers in the subpicosecond time scale, are obtained: also derived are analogous results for two gauge-equivalent nonlinear evolution equations; in particular, the derivative nonlinear Schrödinger equation, i∂ t q+∂ 2 x q−i∂ x (|q| 2 q) = 0. As an application of these asymptotic results, explicit expressions for position and phase shifts of solitons in the presence of the continuous spectrum are calculated.

For Lax-pair isospectral deformations whose associated spectrum, for given initial data, consists of the disjoint union of a finitely denumerable discrete spectrum (solitons) and a continuous spectrum (continuum), the matrix Riemann-Hilbert problem approach is used to derive the leading-order asymptotics as |t| → ∞ (x/t ∼ O(1)) of solutions (u = u(x, t)) to the Cauchy problem for the defocusing non-linear Schrödinger equation 2π). The D f NLSE dark soliton position shifts in the presence of the continuum are also obtained.36, 37]. The complete details of the asymptotic analysis that constitutes stage (1) of the two-step asymptotic paradigm above, which is quite technical and whose results are essential in order to obtain those of the present paper, can be found in [38]: this paper addresses stage (2) of the above programme via the matrix Riemann-Hilbert problem (RHP) approach [7,12,39,40,41,42,43,44,45,46,47]. It is important to note that, to the best of the author's knowledge as at the time of the presents, the first to obtain asymptotics of solutions to the Cauchy problem for the D f NLSE with finite-density initial data in the solitonless sector were Its and Ustinov [48,49].This paper is organized as follows. In Section 2, the necessary facts from the direct and inverse scattering analysis for the D f NLSE with finite-density initial data are given, the (matrix) RHP analysed asymptotically as |t| → ∞ (x/t ∼ O(1)) is stated, and the results of this paper are summarised in Theorems 2.2.1-2.2.4 (and Corollaries 2.2.1 and 2.2.2). In Section 3, an augmented RHP, which is equivalent to the original one stated in Section 2, is formulated, and it is shown that, as t → +∞, modulo exponentially small terms, the solution of the augmented RHP converges to the solution of an explicitly solvable, model RHP. In Section 4, the model RHP is solved asymptotically as t → +∞, from which the asymptotics of u(x, t) andx ±∞ (|u(x ′ , t)| 2 − 1) dx ′ are derived, and, in Appendix A, the-analogous-asymptotic analysis is succinctly reworked for the case when t → −∞. In Appendices B and C, respectively, formulae which are necessary in order to obtain the remaining asymptotic results of this paper are presented, and a panoramic view of the matrix RH theory in the L 2 -Sobolev space is given [44,45,46,50].

Using the matrix Riemann-Hilbert factorisation approach for non-linear evolution systems which take the form of Lax-pair isospectral deformations, the higher order asymptotics as t → ±∞ (x/t ∼ O(1)) of the solution to the Cauchy problem for the modified non-linear Schrödinger equation,is a model for non-linear pulse propagation in optical fibres in the subpicosecond time scale, are obtained: also derived are analogous results for two gauge-equivalent nonlinear evolution equations; in particular, the derivative non-linear Schrödinger equation, i∂ t q+∂ 2 x q−i∂ x (|q| 2 q) = 0.

It is shown how to calculate asymptotics of integrals over the positive semi-axis of two functions related to the Degenerate Third Painlevé Equation (dP3). As an example, the corresponding results for the meromorphic solution of the dP3 vanishing at the origin are presented.