Effect of a small dispersion on self-focusing in a spatially one-dimensional case

Abstract

Abstract. The effect of the small dispersion on the self-focusing of solutions of the equations of nonlinear geometric optics in one-dimensional case is investigated. In the main order this influence is described by means of the universal special solution of the nonlinear Schrцdinger equation, which is isomonodromic. Analytic and asymptotic properties of this solution are described.

“Quantizations” of isomonodromic Hamilton system $H^{\frac{7}{2}+1}$

“…The leading term in this asymptotics is similar to the integrand in (14). Thus, the similarities between this Fourier integral and a simultaneous solution of the pair of equations (7), (8) are extended to the corresponding IDM equations according the general theory of such isomonodromic analogues of Fourier integrals of special form and the practice of applying them, see [34], [46] and the references in the latter work.…”

Meromorphic Solution of the Degenerate Third Painlevé Equation Vanishing at the Origin

“…Miller and B.I. Suleimanov for discussions of the papers [1,11]. The author is indebted to the referees for their significant contribution to improving the quality of the original version of this paper.…”

“…Suleimanov. In [11] he studied some asymptotics of a function which can be identified as a special solution of the second member of the hierarchy of the third Painlevé equation ( 2 P 3 -function), which generally depends on two variables. When one of these variables vanishes then 2 P 3 -equation reduces to the third Painlevé equations (P 3 ): one of them is equivalent to the well-known similarity reduction of the sin-Gordon equation, and the other one to the solution u(τ ) for the following values of the coefficients:…”

Trans-Series Asymptotics of Solutions to the Degenerate Painlevé III Equation: A Case Study

“…(i) It was shown in [54] that a variant of the DP3E (1.1) appears in the characterisation of the effect of the small dispersion on the self-focusing of solutions of the fundamental equations of non-linear optics in the one-dimensional case, where the main order of the influence of this effect is described via a universal special monodromic solution of the non-linear Schrödinger equation (NLSE); in particular, the author studies the asymptotics of a function that can be identified as a solution (the so-called 'Suleimanov solution') of a slightly modified, yet equivalent, version of the DP3E (1.1) for the parameter values a = i/2 and b = 64k −3 , where k > 0 is a physical variable.…”

“…(ii) In [41], an extensive number-theoretic and asymptotic analysis of the universal special monodromic solution considered in [54] is presented: the author studies a particular meromorphic solution of the DP3E (1.1) that vanishes at the origin; more specifically, it is proved that, for −i2a ∈ Z, the aforementioned solution exists and is unique, and, for the case a−i/2 ∈ Z, this solution exists and is unique provided that u(τ ) = −u(−τ ). The bulk of the analysis presented in [41] focuses on the study of the Taylor expansion coefficients of the solution of the DP3E (1.1) that is holomorphic at τ = 0; in particular, upon invoking the 'normalisation condition' b = a and taking ε = +1, it is shown that, for general values of the parameter a, these coefficients are rational functions of a 2 that possess remarkable number-theoretic properties: en route, novel notions such as super-generating functions and quasi-periodic fences are introduced.…”