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arXiv · 2212.00064

Non-flat conformal blow-up profiles for the 1D critical nonlinear Schr\"odinger equation

Yvan Martel·Ivan Naumkin
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Abstract

For the critical one-dimensional nonlinear Schr\"odinger equation, we construct blow-up solutions that concentrate a soliton at the origin at the conformal blow-up rate, with a non-flat blow-up profile. More precisely, we obtain a blow-up profile that equals $|x|+i\kappa x^2$ near the origin, where $\kappa$ is a universal real constant. Such profile differs from the flat profiles obtained in the same context by Bourgain and Wang [Construction of blowup solutions for the nonlinear Schr\"odinger equation with critical nonlinearity. Ann. Sc. Norm. Super. Pisa Cl. Sci. 25 (1997)].

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Yvan Martel, Ivan Naumkin. 2022-11-30. Non-flat conformal blow-up profiles for the 1D critical nonlinear Schr\"odinger equation. https://doi.org/10.2140/tunis.2023.5.505

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