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arXiv · 2311.09845

Typical dropping asymptotics of quasiclassical approximations to solutions of the nonlinear Schr\"odinger equation

Sergej Melikhov·Bulat Suleimanov·Azamat Shavlukov
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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 $-i\varepsilon\Psi'_{t}=\varepsilon^2\Psi''_{xx}\pm2|\Psi| ^2\Psi$, where $\varepsilon$ 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 $h\to 0$ 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.

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Sergej Melikhov, Bulat Suleimanov, Azamat Shavlukov. 2023-11-16. Typical dropping asymptotics of quasiclassical approximations to solutions of the nonlinear Schr\"odinger equation. https://arxiv.org/abs/2311.09845

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