Typical Dropping Asymptotics in the Semiclassical Approximations to Solutions of the Nonlinear Schrödinger Equation

Pub Date : 2024-09-11 DOI:10.1134/s0012266124050045

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

引用次数: 0

Abstract

Formal asymptotics are substantiated that describe a typical dropping cusp singularity in the semiclassical approximations to solutions of two cases of the integrable nonlinear Schrödinger equation \(-i\varepsilon \Psi ^{\prime }_{t} = \varepsilon ^2\Psi ^{\prime \prime }_{xx}\pm 2|\Psi | ^2\Psi \), where \(\varepsilon \) is a small parameter. The substantiation uses the ideas and facts of the mathematical catastrophe theory and the part of Yu.F. Korobeinik’s theorem concerning analytical, as \(h\to 0\), solutions \(G(h,u) \) of the mixed type linear equation \(hG^{\prime \prime }_{hh}=G^{\prime \prime }_{uu}\) to which the hodograph images of both cases of the systems of equations of these semiclassical approximations are equivalent.

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非线性薛定谔方程解的半经典近似中的典型丢弃渐近法

Abstract Formal asymptics are substantiated that describe a typical dropping cusp singularity in the semiclassical approximations to solutions of two cases of the integrable nonlinearSchrödinger equation \(-i\varepsilon \Psi ^{\prime }_{t} = \varepsilon ^2\Psi ^{\prime \prime }_{xx}\pm 2|\Psi | ^2\Psi \)、其中 \(\varepsilon \)是一个小参数。证明使用了数学灾难理论的概念和事实，以及 Yu.F.Korobeinik's storem concerning analytical, as \(h\to 0\), solutions\(G(h,u) \) of the mixed type linear equation\(hG^{\prime \prime }_{hh}=G^{\prime \prime }_{uu}\) to which the hodograph images of the both cases of the systems of equations of these semiclassicalapproximations are equivalent.

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