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Dispersive Asymptotics for Linear and Integrable Equations by the \(\overline {\partial }\) Steepest Descent Method

  • Momar Dieng
  • Kenneth D. T.-R. McLaughlin
  • Peter D. MillerEmail author
Chapter
Part of the Fields Institute Communications book series (FIC, volume 83)

Abstract

We present a new and relatively elementary method for studying the solution of the initial-value problem for dispersive linear and integrable equations in the large-t limit, based on a generalization of steepest descent techniques for Riemann-Hilbert problems to the setting of \({\overline {\partial }}\)-problems. Expanding upon prior work (Dieng and McLaughlin, Long-time asymptotics for the NLS equation via \({\overline {\partial }}\) methods, arXiv:0805.2807, 2008) of the first two authors, we develop the method in detail for the linear and defocusing nonlinear Schrödinger equations, and show how in the case of the latter it gives sharper asymptotics than previously known under essentially minimal regularity assumptions on initial data.

Notes

Acknowledgements

The first two authors were supported in part by NSF grants DMS-0451495, DMS-0800979, and the second author was supported by NSF Grant DMS-1733967. The third author was supported in part by NSF grant DMS-1812625.

References

  1. 1.
    M. Borghese, R. Jenkins, and K. D. T.-R. McLaughlin, “Long time asymptotic behavior of the focusing nonlinear Schrödinger equation,” Ann. Inst. H. Poincaré Anal. Non Linéaire35, no. 4, 887–920, 2018.Google Scholar
  2. 2.
    S. Cuccagna and R. Jenkins, “On the asymptotic stability of N-soliton solutions of the defocusing nonlinear Schrödinger equation,” Commun. Math. Phys.343, 921–969, 2016.CrossRefGoogle Scholar
  3. 3.
    P. Deift, A. Its, and X. Zhou, “Long-time asymptotic for integrable nonlinear wave equations,” in A. S. Fokas and V. E. Zakharov, editors, Important Developments in Soliton Theory 1980–1990, 181–204, Springer-Verlag, Berlin, 1993.CrossRefGoogle Scholar
  4. 4.
    P. Deift and X. Zhou, “A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the mKdV equation,” Ann. Math.137, 295–368, 1993.MathSciNetCrossRefGoogle Scholar
  5. 5.
    P. Deift and X. Zhou, “Long-time asymptotics for integrable systems. Higher order theory,” Comm. Math. Phys.165, 175–191, 1994.MathSciNetCrossRefGoogle Scholar
  6. 6.
    P. Deift and X. Zhou, Long-time behavior of the non-focusing nonlinear Schrödinger equation — A case study, volume 5 of New Series: Lectures in Math. Sci., University of Tokyo, 1994.Google Scholar
  7. 7.
    P. Deift and X. Zhou, “Perturbation theory for infinite-dimensional integrable systems on the line. A case study,” Acta Math.188, no. 2, 163–262, 2002.Google Scholar
  8. 8.
    P. Deift and X. Zhou, “Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space,” Comm. Pure Appl. Math.56, 1029–1077, 2003.MathSciNetCrossRefGoogle Scholar
  9. 9.
    M. Dieng and K. D. T.-R. McLaughlin, “Long-time asymptotics for the NLS equation via \({\overline {\partial }}\) methods,” arXiv:0805.2807, 2008.Google Scholar
  10. 10.
    A. R. Its, “Asymptotic behavior of the solutions to the nonlinear Schrödinger equation, and isomonodromic deformations of systems of linear differential equations,” Dokl. Akad. Nauk SSSR261, 14–18, 1981. (In Russian.)Google Scholar
  11. 11.
    R. Jenkins, J. Liu, P. Perry, and C. Sulem, “Soliton resolution for the derivative nonlinear Schrödinger equation,” Commun. Math. Phys., doi.org/10.1007/s00220-018-3138-4, 2018.Google Scholar
  12. 12.
    J. Liu, P. A. Perry, and C. Sulem, “Long-time behavior of solutions to the derivative nonlinear Schrödinger equation for soliton-free initial data,” Ann. Inst. H. Poincaré Anal. Non Linéaire35, no. 1, 217–265, 2018.Google Scholar
  13. 13.
    K. D. T.-R. McLaughlin and P. D. Miller, “The \(\overline {\partial }\) steepest descent method and the asymptotic behavior of polynomials orthogonal on the unit circle with fixed and exponentially varying nonanalytic weights,” Intern. Math. Res. Papers2006, Article ID 48673, 1–77, 2006.Google Scholar
  14. 14.
    K. D. T.-R. McLaughlin and P. D. Miller, “The \(\overline {\partial }\) steepest descent method for orthogonal polynomials on the real line with varying weights,” Intern. Math. Res. Notices2008, Article ID rnn075, 1–66, 2008.Google Scholar
  15. 15.
    P. D. Miller and Z.-Y. Qin, “Initial-boundary value problems for the defocusing nonlinear Schrödinger equation in the semiclassical limit,” Stud. Appl. Math.134, no. 3, 276–362, 2015.Google Scholar
  16. 16.
    N. I. Muskhelishvili, Singular Integral Equations, Boundary Problems of Function Theory and Their Application to Mathematical Physics, Second edition, Dover Publications, New York, 1992.Google Scholar
  17. 17.
    P. A. Perry, “Inverse scattering and global well-posedness in one and two space dimensions,” in P. D. Miller, P. Perry, and J.-C. Saut, editors, Nonlinear Dispersive Partial Differential Equations and Inverse Scattering, Fields Institute Communications, volume 83, 161–252, Springer, New York, 2019. https://doi.org/10.1007/978-1-4939-9806-7_4.CrossRefGoogle Scholar
  18. 18.
    F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, and B. V. Saunders, eds., NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.0.17, 2017.
  19. 19.
    H. Segur and M. J. Ablowitz, “Asymptotic solutions and conservation laws for the nonlinear Schrödinger equation,” J. Math. Phys.17, 710–713 (part I) and 714–716 (part II), 1976.Google Scholar
  20. 20.
    V. E. Zakharov and S. V. Manakov, “Asymptotic behavior of nonlinear wave systems integrated by the inverse method,” Sov. Phys. JETP44, 106–112, 1976.Google Scholar
  21. 21.
    X. Zhou, “The L 2-Sobolev space bijectivity of the scattering and inverse-scattering transforms,” Comm. Pure Appl. Math.51, 697–731, 1989.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Momar Dieng
    • 1
  • Kenneth D. T.-R. McLaughlin
    • 1
    • 2
  • Peter D. Miller
    • 3
    Email author
  1. 1.Department of MathematicsUniversity of ArizonaTucsonUSA
  2. 2.Department of MathematicsColorado State UniversityFort CollinsUSA
  3. 3.Department of MathematicsUniversity of Michigan–Ann ArborAnn ArborUSA

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