TYPICAL DROPPING ASYMPTOTICS OF QUASICLASSICAL APPROXIMATIONS TO SOLUTIONS OF THE NONLINEAR SCHRÖDINGER EQUATION-Reference-Cited by-同舟云学术

TYPICAL DROPPING ASYMPTOTICS OF QUASICLASSICAL APPROXIMATIONS TO SOLUTIONS OF THE NONLINEAR SCHRÖDINGER EQUATION

Published:2024-12-15 Issue:5 Volume:60 Page:
ISSN:0374-0641
Container-title:Дифференциальные уравнения
language:
Short-container-title:Differencialʹnye uravneniâ

Author:

Melikhov S. N.¹²,Suleimanov B. I.³,Shavlukov A. M.³

Affiliation:

1. Southern Federal University

2. Southern Mathematical Institute of VSC of RAS

3. Institute of Mathematics with Computing Centre — Subdivision of the Ufa Federal Research Centre of RAS

Abstract

Formal asymptotics are substantiated that describe typical dropping cusp singularity of quasiclassical approximations to solutions of two cases of the integrable nonlinear Schr¨odinger equation −𝑖𝜀Ψ′𝑡=𝜀2Ψ′′𝑥𝑥±2|Ψ|2Ψ, where 𝜀 is a small parameter. The substantiation uses the ideology and facts of the mathematical catastrophe theory and the part of the theorem of Yu.F. Korobeinik, concerning analytical as ℎ → 0 solutions 𝐺(ℎ, 𝑢) of the mixed type linear equation ℎ𝐺′′ℎℎ = 𝐺′′𝑢𝑢 to which the hodograph images of both cases of the systems of equations of these quasiclassical approximations are equivalent.

Publisher

The Russian Academy of Sciences

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