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The Focusing NLS Equation with Step-Like Oscillating Background: Scenarios of Long-Time Asymptotics

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Abstract

We consider the Cauchy problem for the focusing nonlinear Schrödinger equation with initial data approaching two different plane waves \(A_j\mathrm {e}^{\mathrm {i}\phi _j}\mathrm {e}^{-2\mathrm {i}B_jx}\), \(j=1,2\) as \(x\rightarrow \pm \infty \). Using Riemann–Hilbert techniques and Deift–Zhou steepest descent arguments, we study the long-time asymptotics of the solution. We detect that each of the cases \(B_1<B_2\), \(B_1>B_2\), and \(B_1=B_2\) deserves a separate analysis. Focusing mainly on the first case, the so-called shock case, we show that there is a wide range of possible asymptotic scenarios. We also propose a method for rigorously establishing the existence of certain higher-genus asymptotic sectors.

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Acknowledgements

The authors are grateful to the two referees whose comments and suggestions have improved the manuscript. J. Lenells acknowledges support from the Göran Gustafsson Foundation, the Ruth and Nils-Erik Stenbäck Foundation, the Swedish Research Council, Grant No. 2015-05430, and the European Research Council, Grant Agreement No. 682537.

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Authors and Affiliations

  1. Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université de Paris, 75205, Paris Cedex 13, France

    Anne Boutet de Monvel

  2. Department of Mathematics, KTH Royal Institute of Technology, 100 44, Stockholm, Sweden

    Jonatan Lenells

  3. B. Verkin Institute for Low Temperature Physics and Engineering, 47 Nauky Avenue, 61103, Kharkiv, Ukraine

    Dmitry Shepelsky

Authors
  1. Anne Boutet de Monvel
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  2. Jonatan Lenells
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  3. Dmitry Shepelsky
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Correspondence to Anne Boutet de Monvel.

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Communicated by H.-T. Yau.

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Boutet de Monvel, A., Lenells, J. & Shepelsky, D. The Focusing NLS Equation with Step-Like Oscillating Background: Scenarios of Long-Time Asymptotics. Commun. Math. Phys. 383, 893–952 (2021). https://doi.org/10.1007/s00220-021-03946-x

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