The nonlinear Schrödinger equation on the half-line

Fokas, Athanassios; Himonas, A.; Mantzavinos, Dionyssios

doi:10.1090/tran/6734

The nonlinear Schrödinger equation on the half-line

Authors: Athanassios S. Fokas, A. Alexandrou Himonas and Dionyssios Mantzavinos
Journal: Trans. Amer. Math. Soc. 369 (2017), 681-709
MSC (2010): Primary 35Q55, 35G16, 35G31
DOI: https://doi.org/10.1090/tran/6734
Published electronically: March 1, 2016
MathSciNet review: 3557790
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The initial-boundary value problem (ibvp) for the cubic nonlinear Schrödinger (NLS) equation on the half-line with data in Sobolev spaces is analysed via the formula obtained through the unified transform method and a contraction mapping approach. First, the linear Schrödinger (LS) ibvp with initial and boundary data in Sobolev spaces is solved and the basic space and time estimates of the solution are derived. Then, the forced LS ibvp is solved for data in Sobolev spaces on the half-line $[0, \infty )$ for the spatial variable and on an interval $[0, T]$, $0<T<\infty$, for the temporal variable by decomposing it into a free ibvp and a forced ibvp with zero data, and its solution is estimated appropriately. Furthermore, using these estimates, well-posedness of the NLS ibvp with data $(u(x,0), u(0,t))$ in $H_x^s(0,\infty )\times H_t^{(2s+1)/4}(0,T)$, $s>1/2$, is established via a contraction mapping argument. In addition, this work places Fokas’ unified transform method for evolution equations into the broader Sobolev spaces framework.

References

Jerry L. Bona, S. M. Sun, and Bing-Yu Zhang, A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane, Trans. Amer. Math. Soc. 354 (2002), no. 2, 427–490. MR 1862556, DOI https://doi.org/10.1090/S0002-9947-01-02885-9

J. L. Bona, S. M. Sun and B.-Y. Zhang, Nonhomogeneous boundary value problems of one-dimensional nonlinear Schrödinger equations. arXiv:1503.00065v1 (2015).

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), no. 2, 107–156. MR 1209299, DOI https://doi.org/10.1007/BF01896020

J. Bourgain, Global solutions of nonlinear Schrödinger equations, American Mathematical Society Colloquium Publications, vol. 46, American Mathematical Society, Providence, RI, 1999. MR 1691575

Robert W. Boyd, Nonlinear optics, 3rd ed., Elsevier/Academic Press, Amsterdam, 2008. MR 2475397

Robert Carroll and Qiyue Bu, Solution of the forced nonlinear Schrödinger (NLS) equation using PDE techniques, Appl. Anal. 41 (1991), no. 1-4, 33–51. MR 1103847, DOI https://doi.org/10.1080/00036819108840015

Thierry Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. MR 2002047

Thierry Cazenave and Fred B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal. 14 (1990), no. 10, 807–836. MR 1055532, DOI https://doi.org/10.1016/0362-546X%2890%2990023-A

A. Chabchoub, N. P. Hoffmann, and N. Akhmediev, Rogue wave observation in a water wave tank, Phys. Rev. Lett. 106 (2011), 204502.

J. E. Colliander and C. E. Kenig, The generalized Korteweg-de Vries equation on the half line, Comm. Partial Differential Equations 27 (2002), no. 11-12, 2187–2266. MR 1944029, DOI https://doi.org/10.1081/PDE-120016157

Walter Craig, Thomas Kappeler, and Walter Strauss, Microlocal dispersive smoothing for the Schrödinger equation, Comm. Pure Appl. Math. 48 (1995), no. 8, 769–860. MR 1361016, DOI https://doi.org/10.1002/cpa.3160480802

W. Craig, C. Sulem, and P.-L. Sulem, Nonlinear modulation of gravity waves: a rigorous approach, Nonlinearity 5 (1992), no. 2, 497–522. MR 1158383

Bernard Deconinck, Thomas Trogdon, and Vishal Vasan, The method of Fokas for solving linear partial differential equations, SIAM Rev. 56 (2014), no. 1, 159–186. MR 3246302, DOI https://doi.org/10.1137/110821871

Charles L. Epstein and John Schotland, The bad truth about Laplace’s transform, SIAM Rev. 50 (2008), no. 3, 504–520. MR 2429447, DOI https://doi.org/10.1137/060657273

A. S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London Ser. A 453 (1997), no. 1962, 1411–1443. MR 1469927, DOI https://doi.org/10.1098/rspa.1997.0077

Athanassios S. Fokas, A unified approach to boundary value problems, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 78, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. MR 2451953

Athanasios S. Fokas and A. Alexandrou Himonas, Well-posedness of an integrable generalization of the nonlinear Schrödinger equation on the circle, Lett. Math. Phys. 96 (2011), no. 1-3, 169–189. MR 2788909, DOI https://doi.org/10.1007/s11005-011-0488-7

A. S. Fokas and A. R. Its, The nonlinear Schrödinger equation on the interval, J. Phys. A 37 (2004), no. 23, 6091–6114. MR 2074625, DOI https://doi.org/10.1088/0305-4470/37/23/009

A. S. Fokas, A. R. Its, and L.-Y. Sung, The nonlinear Schrödinger equation on the half-line, Nonlinearity 18 (2005), no. 4, 1771–1822. MR 2150354, DOI https://doi.org/10.1088/0951-7715/18/4/019

A. S. Fokas and E. A. Spence, Synthesis, as opposed to separation, of variables, SIAM Rev. 54 (2012), no. 2, 291–324. MR 2916309, DOI https://doi.org/10.1137/100809647

A. S. Fokas and L. Y. Sung, Initial-boundary value problems for linear dispersive evolution equations on the half-line. Unpublished.

C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett. 19 (1967), no. 19, 1095–1097.

Jean-Michel Ghidaglia and Jean-Claude Saut, Nonelliptic Schrödinger equations, J. Nonlinear Sci. 3 (1993), no. 2, 169–195. MR 1220173, DOI https://doi.org/10.1007/BF02429863

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. II. Scattering theory, general case, J. Functional Analysis 32 (1979), no. 1, 33–71. MR 533219, DOI https://doi.org/10.1016/0022-1236%2879%2990077-6

L. Grafakos, Personal communication. 12pt

H. Hasimoto and H. Ono, Nonlinear modulation of gravity waves, J. Phys. Soc. Japan 33 (1972), 805–811.

Justin Holmer, The initial-boundary-value problem for the 1D nonlinear Schrödinger equation on the half-line, Differential Integral Equations 18 (2005), no. 6, 647–668. MR 2136703

Justin Holmer, The initial-boundary value problem for the Korteweg-de Vries equation, Comm. Partial Differential Equations 31 (2006), no. 7-9, 1151–1190. MR 2254610, DOI https://doi.org/10.1080/03605300600718503

David Jerison and Carlos E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal. 130 (1995), no. 1, 161–219. MR 1331981, DOI https://doi.org/10.1006/jfan.1995.1067

Tosio Kato, On nonlinear Schrödinger equations. II. $H^s$-solutions and unconditional well-posedness, J. Anal. Math. 67 (1995), 281–306. MR 1383498, DOI https://doi.org/10.1007/BF02787794

Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), no. 1, 33–69. MR 1101221, DOI https://doi.org/10.1512/iumj.1991.40.40003

David Lannes, The water waves problem, Mathematical Surveys and Monographs, vol. 188, American Mathematical Society, Providence, RI, 2013. Mathematical analysis and asymptotics. MR 3060183

Felipe Linares and Gustavo Ponce, Introduction to nonlinear dispersive equations, Universitext, Springer, New York, 2009. MR 2492151

J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. II, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 182. MR 0350178

Alan C. Newell and Jerome V. Moloney, Nonlinear optics, Advanced Topics in the Interdisciplinary Mathematical Sciences, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1992. MR 1163192

D. H. Peregrine, Water waves, nonlinear Schrödinger equations and their solutions, J. Austral. Math. Soc. Ser. B 25 (1983), no. 1, 16–43. MR 702914, DOI https://doi.org/10.1017/S0334270000003891

E. G. Titchmarsh, Introduction to the theory of Fourier integrals, Clarendon Press, 1937.

Yoshio Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac. 30 (1987), no. 1, 115–125. MR 915266

V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Applied Mech. Tech. Phys. 9 (1968), 190–194.

V. E. Zakharov and S. V. Manakov, On the complete integrability of a nonlinear Schrödinger equation, J. Theor. and Math. Phys. 19 (1974), 551–559.

V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Ž. Èksper. Teoret. Fiz. 61 (1971), no. 1, 118–134 (Russian, with English summary); English transl., Soviet Physics JETP 34 (1972), no. 1, 62–69. MR 0406174