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Interaction with an obstacle in the 2D focusing nonlinear Schrödinger equation

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We present a numerical study of solutions to the 2d cubic and quintic focusing nonlinear Schrödinger equation in the exterior of a smooth, compact and strictly convex obstacle (a disk) with Dirichlet boundary condition. We first investigate the effect of the obstacle on the behavior of solutions traveling towards the obstacle at different angles and with different velocities directions. We introduce a new concept of weak and strong interactions of the solutions with the obstacle. Next, we study the existence of blow-up solutions depending on the type of the interaction and show how the presence of the obstacle changes the overall behavior of solutions (e.g., from blow-up to global existence), especially in the strong interaction case, as well as how it affects the shape of solutions compared to their initial data (e.g., splitting into transmitted and reflected parts). We also investigate the influence of the size of the obstacle on the eventual existence of blow-up solutions in the strong interaction case in terms of the transmitted and the reflected parts of the mass. Moreover, we show that the sharp threshold for global existence vs. finite time blow-up solutions in the mass critical case in the presence of the obstacle is the same as the one given by Weinstein for NLS in the whole Euclidean space \(\mathbb {R}^d\). Finally, we construct new wall-type initial data that blows up in finite time after a strong interaction with an obstacle and having a very distinct dynamics compared with all other blow-up scenarios and dynamics for the NLS in the whole Euclidean space \(\mathbb {R}^d\).

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Acknowledgements

All authors would like to thank Thomas Duyckaerts (Paris-13) for fruitful discussions on this problem. Most of the research on this project was done while O.L. was visiting the Department of Mathematics and Statistics at Florida International University, Miami, FL, during his PhD training. He thanks the department for its hospitality and support. The initial numerical investigations started when K.Y. visited T. Duyckaerts at IHP and LAGA, Paris-13.

Funding

Research of K.Y was partially supported by the Fundamental Research Funds for the Central Universities Project No.2023CDJXY-042 (PI: Yang), and partially supported by the NSF grant DMS-1927258 (PI: Roudenko). Research of S.R. was partially supported by the NSF grant DMS-1927258, and part of O.L.’s research visit to FIU was funded by the same grant DMS-1927258 (PI: Roudenko).

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Authors and Affiliations

  1. LAGA - Institut Galilée, Université Sorbonne Paris Nord, UMR 7539, Paris, France

    Oussama Landoulsi

  2. Department of Mathematics & Statistics, Florida International University, Miami, FL, 33199, USA

    Svetlana Roudenko

  3. Key Laboratory of Nonlinear Analysis and its Applications (Ministry of Education), and College of Mathematics & Statistics, Chongqing University, Chongqing, 401331, China

    Kai Yang

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  1. Oussama Landoulsi
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  3. Kai Yang
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Correspondence to Kai Yang.

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Communicated by: Jon Wilkening

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Landoulsi, O., Roudenko, S. & Yang, K. Interaction with an obstacle in the 2D focusing nonlinear Schrödinger equation. Adv Comput Math 49, 71 (2023). https://doi.org/10.1007/s10444-023-10055-x

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