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The Derivative Nonlinear Schrödinger Equation with Zero/Nonzero Boundary Conditions: Inverse Scattering Transforms and N-Double-Pole Solutions

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Abstract

In this paper, we report a rigorous theory of the inverse scattering transforms (ISTs) for the derivative nonlinear Schrödinger (DNLS) equation with both zero boundary conditions (ZBCs) and nonzero boundary conditions (NZBCs) at infinity and double zeros of analytical scattering coefficients. The scattering theories for both ZBCs and NZBCs are addressed. The direct scattering problem establishes the analyticity, symmetries, and asymptotic behaviors of the Jost solutions and scattering matrix, and properties of discrete spectra. The inverse scattering problems are formulated and solved with the aid of the matrix Riemann–Hilbert problems, and the reconstruction formulae, trace formulae and theta conditions are also posed. In particular, the IST with NZBCs at infinity is proposed by a suitable uniformization variable, which allows the scattering problem to be solved on a standard complex plane instead of a two-sheeted Riemann surface. The reflectionless potentials with double poles for the ZBCs and NZBCs are both carried out explicitly by means of determinants. Some representative semi-rational bright–bright soliton, dark–bright soliton, and breather–breather solutions are examined in detail. These results and idea can also be extended to other types of DNLS equations such as the Chen–Lee–Liu-type DNLS equation, Gerdjikov–Ivanov-type DNLS equation, and Kundu-type DNLS equation and will be useful to further explore and apply the related nonlinear wave phenomena.

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Acknowledgements

The authors would like to thank the referees for their valuable comments and suggestions. This work was supported by the NSFC under Grants Nos. 11925108 and 11731014, and CAS Interdisciplinary Innovation Team.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Bei**g, 100084, China

    Guoqiang Zhang

  2. Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Bei**g, 100190, China

    Zhenya Yan

  3. School of Mathematical Sciences, University of Chinese Academy of Sciences, Bei**g, 100049, China

    Zhenya Yan

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  1. Guoqiang Zhang
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  2. Zhenya Yan
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Correspondence to Zhenya Yan.

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Communicated by Arnd Scheel.

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The paper was originally announced on 6 December 2018 [ar**v:1812.02387].

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Zhang, G., Yan, Z. The Derivative Nonlinear Schrödinger Equation with Zero/Nonzero Boundary Conditions: Inverse Scattering Transforms and N-Double-Pole Solutions. J Nonlinear Sci 30, 3089–3127 (2020). https://doi.org/10.1007/s00332-020-09645-6

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  • DOI: https://doi.org/10.1007/s00332-020-09645-6

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