Inverse Scattering in 60 Minutes
Journées équations aux dérivées partielles (2016), Talk no. 8, 17 p.

This lecture reports on joint work with Robert Jenkins, Jiaqi Liu, and Catherine Sulem. We illustrate the strengths of the inverse scattering method for addressing large-time behavior of completely integrable dispersive PDE’s by proving global well-posedness and determining large-time asymptotic behavior for the Derivative Nonlinear Schrödinger equation (DNLS) for soliton-free initial data. Our work uses techniques from the work of Deift and Zhou on the defocussing NLS together with further developments due to Dieng and McLaughlin.

Published online:
DOI: 10.5802/jedp.649
Perry, Peter 1

1 Department of Mathematics University of Kentucky Lexington Kentucky 40506-0027, USA
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     title = {Inverse {Scattering} in 60 {Minutes}},
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Perry, Peter. Inverse Scattering in 60 Minutes. Journées équations aux dérivées partielles (2016), Talk no. 8, 17 p. doi : 10.5802/jedp.649.

[1] Ablowitz, M. J.; Kaup, D. J.; Newell, A. C.; Segur, H. The inverse scattering transform-Fourier analysis for nonlinear problems. Studies in Appl. Math. 53 (1974), no. 4, 249–315.

[2] Borghese, M., Jenkins, R., McLaughlin, K. T.-R. Long-time asymptotic behavior of the focusing nonlinear Schrödinger equation. Preprint, . | arXiv

[3] Cuccagna, S., Jenkins, J. On asymptotic stability of N-solitons of the defocusing nonlinear Schrödinger equation. Preprint, , to appear in Comm. Math. Phys. | arXiv

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

[5] Deift, P. A.; Its, A. R.; Zhou, X. Long-time asymptotics for integrable nonlinear wave equations. Important developments in soliton theory, 181–204, Springer Ser. Nonlinear Dynam., Springer, Berlin, 1993.

[6] Deift, P., Zhou, X. A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation. Ann. of Math. (2) 137 (1993), 295–368.

[7] Deift, P. A.; Zhou, X. Long-time asymptotics for integrable systems. Higher order theory. Comm. Math. Phys., 165 (1994), no. 1, 175–191.

[8] Deift, P., Zhou, X. Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space. Dedicated to the memory of Jürgen K. Moser. Comm. Pure Appl. Math., 56 (2003), 1029–1077.

[9] Dieng, M., McLaughlin, K D.-T. Long-time asymptotics for the NLS equation via dbar methods. 2008. | arXiv

[10] Do, Y. (2011). A nonlinear stationary phase method for oscillatory Hilbert-Riemann problem. Intern. Math. Res. Not., 12 (2011), 2650–2765.

[11] Gardner, C. S.; Greene, J. M.; Kruskal, M. D.; Miura, R. M. Method for solving the Korteweg-de Vries equation. Phys. Rev. Letters 19 (1967), 1095–1097.

[12] Gel’fand, I. M., and Levitan, B. M., On the determination of a differential equation from its spectral function. Amer. Math. Soc. Transl., Ser. 2, 1 (1955), pp. 253–304.

[13] Hayashi, N. The initial value problem for the derivative nonlinear Schrödinger equation in the energy space. Nonlinear Anal. 20 (1993), no. 7, 823–833.

[14] Hayashi, N. Naumkin, P. Uchida, H. Large time behavior of solutions for derivative cubic nonlinear Schrödinger equations. Publ. Res. Inst. Math. Sci. 35 (1999), no. 3, 501–513.

[15] Its, A. R. Asymptotic behavior of the solutions to the nonlinear Schrödinger equation, and isomonodromic deformations of systems of linear differential equations. (Russian) Dokl. Akad. Nauk SSSR 261 (1981), no. 1, 14–18. English translation in Soviet Math. Dokl. 24 (1982), no. 3, 452–456.

[16] Jenkins, R.; Liu, J.; Perry, P.; Sulem, C. Global well-posedness for the derivative nonlinear Schrödinger equation in the presence of solitons. In preparation.

[17] Kaup, D. J., Newell, A. C. An exact solution for a derivative nonlinear Schrödinger equation. J. Mathematical Phys. 19 (1978), 798–801.

[18] Kitaev, A. V.; Vartanian, A. H. Leading-order temporal asymptotics of the modified nonlinear Schrödinger equation: solitonless sector. Inverse Problems 13 (1997), no. 5, 1311–1339.

[19] Korteweg, D. J.; de Vries, G. On the change of form of long waves advancing in a rectangular canal, and a new type of long stationary waves. Phil. Mag. 39 (1895), 422–443.

[20] Lee, J.-H. Analytic properties of Zakharov-Shabat inverse scattering problem with a polynomial spectral dependence of degree 1 in the potential. Thesis (Ph.D.), 1983, Yale University.

[21] Lee, J.-H. (1989) Global solvability of the derivative nonlinear Schrödinger equation. Trans. Amer. Math. Soc. 314:107–118.

[22] Liu, J., Perry, P., Sulem, C. Global existence for the derivative nonlinear Schrödinger equation by the method of inverse scattering. Comm. Partial Differential Equations 41 (2016), no. 11, 1692–1760.

[23] Liu, J., Perry, P., Sulem, C. Long-Time Asymptotic Behavior of Solutions to the Derivative Nonlinear Schrödinger Equation for Soliton-Free Initial Data. Preprint, , submitted to Ann. Inst. Henri Poincaré C - Analyse non-linéaire. | arXiv

[24] 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. IMRP Int. Math. Res. Pap. (2006), Art. ID 48673, 1–77.

[25] Pelinovsky, D.E., Shimabukuro, Y. Existence of global solutions to the derivative NLS equation with the inverse scattering transform method. | arXiv

[26] Tovbis, Alexander; Venakides, Stephanos. The eigenvalue problem for the focusing nonlinear Schrödinger equation: new solvable cases. Phys. D. 146 (2000), no. 1-4, 150–164.

[27] Varzugin, G. G. Asymptotics of oscillatory Riemann-Hilbert problems. J. Math. Phys. 37 (1996), no. 11, 5869–5892.

[28] Whittaker, E.T., Watson, G.G. A course in modern analysis, Cambridge Univ. Press, 1915.

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

[30] Zakharov, V.E., Manakov, S.V. Asymptotic behavior of nonlinear wave systems integrated by the inverse scattering method. Soviet Physics JETP 44 (1976), no. 1, 106–112; translated from Z. Eksper. Teoret. Fiz. 71 (1976), no. 1, 203–215.

[31] Zakharov, V. E.; Shabat, A. B. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Soviet Physics JETP 34 (1972), no. 1, 62–69; translated from Ž. Èksper. Teoret. Fiz. 61 (1971), no. 1, 118–134.

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