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Long-Time Asymptotics for the Integrable Nonlocal Focusing Nonlinear Schrödinger Equation for a Family of Step-Like Initial Data

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Abstract

We study the Cauchy problem for the integrable nonlocal focusing nonlinear Schrödinger (NNLS) equation \(iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0 \) with the step-like initial data close, in a certain sense, to the “shifted step function” \(\chi _R(x)=AH(x-R)\), where H(x) is the Heaviside step function, and \(A>0\) and \(R>0\) are arbitrary constants. Our main aim is to study the large-t behavior of the solution of this problem. We show that for \(R\in \left( \frac{(2n-1)\pi }{2A},\frac{(2n+1)\pi }{2A}\right) \), \(n=1,2,\ldots \), the (x, t) plane splits into \(4n+2\) sectors exhibiting different asymptotic behavior. Namely, there are \(2n+1\) sectors, where the solution decays to 0, whereas in the other \(2n+1\) sectors (alternating with the sectors with decay), the solution approaches (different) constants along each ray \(x/t=const\). Our main technical tool is the representation of the solution of the Cauchy problem in terms of the solution of an associated matrix Riemann–Hilbert problem and its subsequent asymptotic analysis following the ideas of the nonlinear steepest descent method.

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Acknowledgements

We would like to thank the anonymous referee for valuable comments which allowed us to improve the manuscript.

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Authors and Affiliations

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

    Yan Rybalko & Dmitry Shepelsky

  2. V.Karazin Kharkiv National University, Kharkiv, Ukraine

    Dmitry Shepelsky

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  1. Yan Rybalko
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Correspondence to Yan Rybalko.

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Communicated by H.-T. Yau.

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Rybalko, Y., Shepelsky, D. Long-Time Asymptotics for the Integrable Nonlocal Focusing Nonlinear Schrödinger Equation for a Family of Step-Like Initial Data. Commun. Math. Phys. 382, 87–121 (2021). https://doi.org/10.1007/s00220-021-03941-2

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