Long-time behavior of solutions to the derivative nonlinear Schrödinger equation for soliton-free initial data
Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 1, pp. 217-265.

On établit le comportement au temps long des solutions de l'équation de Schrödinger nonlinéraire avec dérivée dans des espaces de Sobolev à poids, sous l'hypothèse que les conditions initiales ne supportent pas de solitons. Notre approche utilise l'inverse scattering et la méthode de la plus grande pente (“steepest descent”) nonlinéaire de Deift et Zhou revisitée par Dieng et McLaughlin.

The large-time behavior of solutions to the derivative nonlinear Schrödinger equation is established for initial conditions in some weighted Sobolev spaces under the assumption that the initial conditions do not support solitons. Our approach uses the inverse scattering setting and the nonlinear steepest descent method of Deift and Zhou as recast by Dieng and McLaughlin.

DOI : 10.1016/j.anihpc.2017.04.002
Mots-clés : Riemann–Hilbert problem, Inverse scattering method, Nonlinear steepest descent method 
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     title = {Long-time behavior of solutions to the derivative nonlinear {Schr\"odinger} equation for soliton-free initial data},
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Liu, Jiaqi; Perry, Peter A.; Sulem, Catherine. Long-time behavior of solutions to the derivative nonlinear Schrödinger equation for soliton-free initial data. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 1, pp. 217-265. doi : 10.1016/j.anihpc.2017.04.002. http://www.numdam.org/articles/10.1016/j.anihpc.2017.04.002/

[1] Liu, J.; Perry, P.; Sulem, C. Global existence for the derivative nonlinear Schrodinger equation by the method of inverse scattering (ArXiv e-prints) | arXiv | MR | Zbl

[2] Fan, E. Darboux transformation and soliton-like solutions for the Gerdjikov–Ivanov equation, J. Phys. A, Volume 33 (2000) no. 39, pp. 6925–6933 | DOI | MR | Zbl

[3] Kaup, D.J.; Newell, A.C. An exact solution for a derivative nonlinear Schrödinger equation, J. Math. Phys., Volume 19 (1978) no. 4, pp. 798–801 | MR | Zbl

[4] Lee, J.-H. Analtyic Properties of Zakharov–Shabat Inverse Scattering Problem with a Polynomial Spectral Dependence of Degree 1 in the Potential, Yale University, 1983 http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:8329296 ProQuest LLC, Ann Arbor, MI, thesis (Ph.D.) | MR

[5] Pelinovsky, D.E.; Shimabukuro, Y. Existence of global solutions to the derivative NLS equation with the inverse scattering transform method, Int. Math. Res. Not. (2017) | DOI | MR | Zbl

[6] Hayashi, N.; Naumkin, P.I.; Uchida, H. Large time behavior of solutions for derivative cubic nonlinear Schrödinger equations, Publ. Res. Inst. Math. Sci., Volume 35 (1999) no. 3, pp. 501–513 | DOI | MR | Zbl

[7] Zakharov, V.E.; Manakov, S.V. Asymptotic behavior of non-linear wave systems integrated by the inverse scattering method, Zh. Èksp. Teor. Fiz., Volume 71 (1976) no. 1, pp. 203–215 | MR

[8] Deift, P.; Zhou, X. A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation, Ann. Math., Volume 137 (1993) no. 2, pp. 295–368 | DOI | MR | Zbl

[9] Deift, P.A.; Its, A.R.; Zhou, X. Long-time asymptotics for integrable nonlinear wave equations, Important Developments in Soliton Theory, Springer Ser. Nonlinear Dynam., Springer, Berlin, 1993, pp. 181–204 | DOI | MR | Zbl

[10] Deift, P.; Zhou, X. Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space, Commun. Pure Appl. Math., Volume 56 (2003) no. 8, pp. 1029–1077 (dedicated to the memory of Jürgen K. Moser) | DOI | MR | Zbl

[11] Varzugin, G.G. Asymptotics of oscillatory Riemann–Hilbert problems, J. Math. Phys., Volume 37 (1996) no. 11, pp. 5869–5892 | DOI | MR | Zbl

[12] Do, Y. A nonlinear stationary phase method for oscillatory Riemann–Hilbert problems, Int. Math. Res. Not., Volume 2011 (2011) no. 12, pp. 2650–2765 | DOI | MR | Zbl

[13] Dieng, M.; McLaughlin, K.D.T. Long-time asymptotics for the NLS equation via dbar methods (ArXiv e-prints) | arXiv

[14] McLaughlin, K.T.-R.; Miller, P.D. The ∂‾ steepest descent method and the asymptotic behavior of polynomials orthogonal on the unit circle with fixed and exponentially varying nonanalytic weights, Int. Math. Res. Pap. (2006), pp. 1–77 | MR | Zbl

[15] Cuccagna, S.; Jenkins, R. On asymptotic stability of N-solitons of the defocusing nonlinear Schrodinger equation (ArXiv e-prints) | arXiv | DOI | Zbl

[16] Borghese, M.; Jenkins, R.; McLaughlin, K.D.T.-R. Long time asymptotic behavior of the focusing nonlinear Schrodinger equation (ArXiv e-prints) | arXiv | Numdam | MR | Zbl

[17] Kitaev, A.V.; Vartanian, A.H. Leading-order temporal asymptotics of the modified nonlinear Schrödinger equation: solitonless sector, Inverse Probl., Volume 13 (1997) no. 5, pp. 1311–1339 | DOI | MR | Zbl

[18] Xu, J.; Fan, E. Long-time asymptotic for the derivative nonlinear Schrödinger equation with decaying initial value | arXiv | DOI

[19] Deift, P.A.; Zhou, X. Long-time asymptotics for integrable systems. Higher order theory, Commun. Math. Phys., Volume 165 (1994) no. 1, pp. 175–191 http://projecteuclid.org/euclid.cmp/1104271038 | DOI | MR | Zbl

[20] Its, A.R. Asymptotic behavior of the solutions to the nonlinear Schrödinger equation, and isomonodromic deformations of systems of linear differential equations, Dokl. Akad. Nauk SSSR, Volume 261 (1981) no. 1, pp. 14–18 | MR | Zbl

[21] Digital, N.I.S.T. Library of mathematical functions, release 1.0.11 of 2016-06-08 http://dlmf.nist.gov/ (online companion to [24] )

[22] Cuccagna, S.; Pelinovsky, D.E. The asymptotic stability of solitons in the cubic NLS equation on the line, Appl. Anal., Volume 93 (2014) no. 4, pp. 791–822 | DOI | MR | Zbl

[23] Whittaker, E.T.; Watson, G.N. A Course of Modern Analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996 an introduction to the general theory of infinite processes and of analytic functions, with an account of the principal transcendental functions, reprint of the fourth (1927) edition | DOI | JFM | MR

[24] Olver, F.W.J.; Lozier, D.W.; Boisvert, R.F.; Clark, C.W. NIST Handbook of Mathematical Functions, Cambridge University Press, New York, NY, 2010 (print companion to [21] ) | MR | Zbl

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