Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
  1. Home
  2. Books
  3. Dispersive Partial Differential Equations
  • Cited by 23
Publisher:
Cambridge University Press
Online publication date:
April 2016
Print publication year:
2016
Online ISBN:
9781316563267
DOI:
https://doi.org/10.1017/CBO9781316563267
Subjects:
Mathematics (general), Mathematics, Differential and Integral Equations, Dynamical Systems and Control Theory, Abstract Analysis
Series:
London Mathematical Society Student Texts (86)
Information

Book description

The area of nonlinear dispersive partial differential equations (PDEs) is a fast developing field which has become exceedingly technical in recent years. With this book, the authors provide a self-contained and accessible introduction for graduate or advanced undergraduate students in mathematics, engineering, and the physical sciences. Both classical and modern methods used in the field are described in detail, concentrating on the model cases that simplify the presentation without compromising the deep technical aspects of the theory, thus allowing students to learn the material in a short period of time. This book is appropriate both for self-study by students with a background in analysis, and for teaching a semester-long introductory graduate course in nonlinear dispersive PDEs. Copious exercises are included, and applications of the theory are also presented to connect dispersive PDEs with the more general areas of dynamical systems and mathematical physics.

Reviews

'The exercises in each chapter, while not at all trivial, tremendously enhance one’s understanding of the material. The focus on periodic boundary conditions sets this book apart from related ones in the area, and yet the authors do a nice job discussing related well-posedness results and estimates on the real line as well. While certainly not meant for students without any analysis background, a thorough work-through of this book certainly brings one to the frontier of the theory of nonlinear dispersive PDEs.'

Eric Stachura Source: MAA Reviews

'The book is a manual for beginning graduate students in the field of the general theory of nonlinear partial differential equations. The material is presented in the rigorous mathematical style, providing proofs of formal theorems, rather than less strict considerations which may be often encountered in physics literature.'

Boris A. Malomed Source: Zentralblatt MATH

Refine List

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

Contents

Contents
References
References
[1] A., Babin, A. A., Ilyin, and E. S., Titi, On the regularization mechanism for the periodic Korteweg–de Vries equation, Comm. Pure Appl. Math. 64 (2011), no. 5, 591–648.
[2] A., Babin, A., Mahalov, and B., Nicolaenko, Regularity and integrability of 3D Euler and Navier–Stokes equations for rotating fluids, Asymptot. Anal. 15 (1997), no.2, 103–150.
[3] J. M., Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst. 10 (2004), no. 1–2, 31–52.
[4] M., Beals, Self-spreading and strength of singularities of solutions to semilinear wave equations, Ann. of Math. 118 (1983), 187–214.
[5] H., Berestycki and P.-L., Lions, Existence d'ondes solitaires dans des problemes nonlinéaires du type Klein–Gordon, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 7, A395–A398.
[6] M. V., Berry, Quantum fractals in boxes, J. Phys. A: Math. Gen. 29 (1996), 6617–6629.
[7] M. V., Berry and S., Klein, Integer, fractional and fractal Talbot effects, J. Mod. Optics 43 (1996), 2139–2164.
[8] M. V., Berry and Z. V., Lewis, On the Weierstrass–Mandelbrot fractal function, Proc. Roy. Soc. London A 370 (1980), 459–484.
[9] M. V., Berry, I., Marzoli, and W., Schleich, Quantum carpets, carpets of light, Physics World 14 (2001), no.6, 39–44.
[10] H., Biagioni and F., Linares, Ill-posedness for the Derivative Schrödinger and Generalized Benjamin-Ono equations, Trans. Amer. Math. Soc. 353 (2001), 3649–3659.
[11] B., Birnir, C. E., Kenig, G., Ponce, N., Svanstedt, and L., Vega, On the ill-posedness of the IVP for the generalized Korteweg–de Vries and nonlinear Schrödinger equations, J. London Math. Soc. 53 (1996), pp. 551–559.
[12] E., Bombieri and J., Pila, The number of integral points on arcs and ovals, Duke Math. J. 59 (1989), 337–357.
[13] J. L., Bona and R., Smith, The initial-value problem for the Korteweg–de Vries equation, Philos. Trans. Roy. Soc. London Ser. A 278 (1975), no. 1287, 555–601.
[14] J. L., Bona and N., Tzvetkov, Sharp wellposedness results for the BBM equation, Discrete Contin. Dyn. Syst. 23 (2009), no.4, 1241–1252.
[15] J., Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part I: Schrödinger equations, GAFA 3 (1993), 209–262.
[16] J., Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part II: The KdV equation, GAFA 3 (1993), 209–262.
[17] J., Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, Internat. Math. Res. Notices (1996), no. 6, 277–304.
[18] J., Bourgain, Periodic Korteweg–de Vries equation with measures as initial data, Sel. Math. New ser. 3 (1997), 115–159.
[19] J., Bourgain, Refinement of Strichartz’ inequality and applications to 2D–NLS with critical nonlinearity, Internat. Math. Res. Notices (1998), no. 5, 253–283.
[20] J., Bourgain, Global Solutions of Nonlinear Schrödinger Equations, AMS Colloquium Publications 46. American Mathematical Society, Providence, RI, 1999.
[21] J., Bourgain, A remark on normal forms and the “I-method” for periodic NLS, J. Anal. Math. 94 (2004), 125–157.
[22] J., Bourgain and C., Demeter, The proof of the l2 decoupling conjecture, preprint 2014, http://arxiv.org/abs/1403.5335.
[23] N., Burq, P., Gérard, and N., Tzvetkov, An instability property of the nonlinear Schrödinger equation on Sd, Math. Res. Lett. 9 (2002), 323–335.
[24] N., Burq, P., Gérard, and N., Tzvetkov, Global solutions for the nonlinear Schrödinger equation on three-dimensional compact manifolds, Mathematical Aspects of Nonlinear Dispersive Equations, 111–129, Ann. of Math. Stud., 163, Princeton Univ. Press, Princeton, NJ, 2007.
[25] M., Cabral and R., Rosa, Chaos for a damped and forced KdV equation, Physica D 192 (2004), 265–278.
[26] L., Carleson, Some analytical problems related to statistical mechanics, Euclidean Harmonic Analysis, 5–45, Lecture Notes in Math. 779 Springer- Verlag, Berlin and New York, 1979.
[27] F., Catoire and W.-M., Wang, Bounds on Sobolev norms for the defocusing nonlinear Schrödinger equation on general flat tori, Commun. Pure Appl. Anal. 9 (2010), no. 2, 483–491.
[28] T., Cazenave, An Introduction to Nonlinear Schrödinger Equations, Textos de Métodos Matemáticos, 22, Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro, 1989.
[29] T., Cazenave, Blow Up and Scattering in the Nonlinear Schrödinger Equation, Textos de Métodos Matemáticos, 30, Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro, 1994.
[30] T., Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics 10. New York University, Courant Institute of Mathematical Sciences, NY; American Mathematical Society, Providence, RI, 2003.
[31] T., Cazenave and F. B., Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case. Nonlinear Semigroups, Partial Differential Equations and Attractors (Washington, DC, 1987), 18–29, Lecture Notes in Math., 1394, Springer, Berlin, 1989.
[32] N., Chang, J., Shatah, and K., Uhlenbeck, Schrödinger maps, Comm. Pure Appl. Math. 53 (2000), 590–602.
[33] G., Chen and P. J., Olver, Numerical simulation of nonlinear dispersive quantization, Discrete Contin. Dyn. Syst. 34 (2014), no. 3, 991–1008.
[34] G., Chen and P. J., Olver, Dispersion of discontinuous periodic waves, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469 (2013), no. 2149, 20120407, 21 pp.
[35] Y., Cho, G., Hwang, S., Kwon, and S., Lee, well-posedness and ill-posedness for the cubic fractional Schrödinger equations, Discrete Contin. Dyn. Syst. 35 (2015), no. 7, 2863–2880.
[36] V., Chousionis, M.B., Erdoğan, and N., Tzirakis, Fractal solutions of linear and nonlinear dispersive partial differential equations, Proc. Lond. Math. Soc. 110 (2015), no. 3, 543–564.
[37] M., Christ, Power series solution of a nonlinear Schrödinger equation in Mathematical Aspects of Nonlinear Dispersive Equations, 131–155, Ann. of Math. Stud., 163, Princeton Univ. Press, Princeton, NJ, 2007.
[38] M., Christ, J., Colliander, and T., Tao, Asymptotics, frequency modulation, low regularity ill-posedness for canonical defocusing equations, Amer. J. Math. 125 (2003), 1235–1293.
[39] M., Christ, J., Colliander, and T., Tao, Instability of the periodic nonlinear Schrödinger equation, arxiv:math/0311227v1.
[40] J., Chung, Z., Guo, S., Kwon, and T., Oh, Normal form approach to global wellposedness of the quadratic derivative Schrödinger equation on the circle, Preprint 2015, http://arxiv.org/abs/1509.08139.
[41] J., Colliander, M., Keel, G., Staffilani, H., Takaoka, and T., Tao, Global wellposedness for Schrödinger equation with derivative, SIAM J. Math. Anal. 33 (2001), 649–669.
[42] J., Colliander, M., Keel, G., Staffilani, H., Takaoka, T., Tao, Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation, Math. Research Letters, 9 (2002), 659–682.
[43] J., Colliander, M., Keel, G., Staffilani, H., Takaoka, and T., Tao, Sharp global well-posedness for KdV and modified KdV on R and T, J. Amer. Math. Soc. 16 (2003), 705–749.
[44] J., Colliander, M., Keel, G., Staffilani, H., Takaoka, and T., Tao, Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on R3, Comm. Pure Appl. Math. 57 (2004), no. 8, 987–1014.
[45] J., Colliander, S., Kwon, and T., Oh, A remark on normal forms and the “upsidedown” I-method for periodic NLS: growth of higher Sobolev norms, J. Anal. Math. 118 (2012), no. 1, 55–82.
[46] J., Colliander, G., Staffilani, and H., Takaoka, Global wellposedness for KdV below L2, Math. Res. Lett. 6 (1999), no. 5–6, 755–778.
[47] E., Compaan, Smoothing and global attractors for the Majda–Biello system on the torus, Differential Integral Equations 29 (2016), no. 3–4, 269–308.
[48] B. E., Dahlberg and C. E., Kenig, A note on the almost everywhere behavior of solutions to the Schrödinger equation, Harmonic Analysis (Minneapolis, Minn., 1981), 205–209, Lecture Notes in Math. 908, Springer, Berlin-New York, 1982.
[49] A., Deliu and B., Jawerth, Geometrical dimension versus smoothness, Constr. Approx. 8 (1992), 211–222.
[50] S., Demirbaş, Local well-posedness for 2-D Schrödinger equation on irrational tori and bounds on Sobolev norms, Preprint 2013, to appear in Communications in Pure and Applied Analysis, http://arxiv.org/abs/1307.0051.
[51] S., Demirbaş, M. B., Erdoğan, and N., Tzirakis, Existence and uniqueness theory for the fractional Schrödinger equation on the torus, http://arxiv.org/ abs/1312.5249.
[52] B., Dodson, Global well-posedness and scattering for the defocusing, L2 critical, nonlinear Schrödinger equation when d = 1, http://arxiv.org/abs/1010.0040.
[53] B., Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, preprint 2011, http://arxiv.org/abs/1104.1114.
[54] P. F., Embid and A. J., Majda, Averaging over fast gravity waves for geophysical flows with arbitrary potential vorticity, Comm. Partial Diff. Eqs. 21 (1996), 619– 658.
[55] M. B., Erdoğan, J., Marzuola, K., Newhall, and N., Tzirakis, Global Attractors for Forced and Weakly Damped Zakharov System on the Torus, SIAM J. Appl. Dyn. Syst. 14 (2015), 1978–1990.
[56] M. B., Erdoğan and N., Tzirakis, Global smoothing for the periodic KdV evolution, Internat. Math. Res. Notices (2013), 4589–4614.
[57] M. B., Erdoğan and N., Tzirakis, Long time dynamics for forced and weakly damped KdV on the torus, Commun. Pure Appl. Anal. 12 (2013), 2669–2684.
[58] M. B., Erdoğan and N., Tzirakis, Smoothing and global attractors for the Zakharov system on the torus, Analysis & PDE 6 (2013), no. 3, 723–750.
[59] M. B., Erdoğan and N., Tzirakis, Talbot effect for the cubic nonlinear Schrödinger equation on the torus, Math. Res. Lett. 20 (2013), 1081–1090.
[60] M. B., Erdoğan and N., Tzirakis, Regularity properties of the cubic nonlinear Schrödinger equation on the half line, http://arxiv.org/abs/1509.03546.
[61] G. B., Folland, Real analysis. Modern Techniques and Their Applications, Pure and Applied Mathematics, John Wiley & Sons, Inc., NY, 1984.
[62] G., Fonseca, F., Linares, and G., Ponce, Global well-posedness for the modified Korteweg–de Vries equation, Comm. PDE 24 (1999), 683–705.
[63] G., Fonseca, F., Linares, and G., Ponce, Global existence for the critical generalized KdV equation, Proc. Amer. Math. Soc. 131 (2003), 1847–1855.
[64] P., Germain, N., Masmudi, and J., Shatah, Global solutions for the gravity water waves equation in dimension 3, C. R. Math. Acad. Sci. Paris 347 (2009), 897– 902.
[65] J. M., Ghidaglia, Weakly damped forced Korteweg–de Vries equations behave as a finite dimensional dynamical system in the long time, J. Diff. Eqs. 74 (1988), 369–390.
[66] J. M., Ghidaglia, A note on the strong convergence towards attractors of damped forced KdV equations, J. Diff. Eqs. 110 (1994), 356–359.
[67] J., Ginibre and G., Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Annal. Non linéaire 2 (1985), 309–327.
[68] R. T., Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 18 (1977), no. 9, 1794–1797.
[69] O., Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg– de Vries equations, Discrete Contin. Dyn. Syst. 6 (2000), 625–644.
[70] O., Goubet, Asymptotic smoothing effect for a weakly damped nonlinear Schrödinger equation in T2, J. Diff. Eqs. 165 (2000), no. 1, 96–122.
[71] D. J., Griffiths, Introduction to Quantum Mechanics, Prentice Hall, 1995.
[72] Z., Guo, Global well-posedness of Korteweg–de Vries equation in H3/4(R), J. Math. Pures Appl. 91 (2009), no. 6, 583–597.
[73] Z., Guo, Local well-posedness for dispersion generalized Benjamin–Ono equations in Sobolev spaces, J. Diff. Eqs. 252 (2012), 2053–2084.
[74] N. J., Hitchin, G. B., Segal, and R. S., Ward, Integrable Systems: Twistors, Loop Groups, and Riemann Surfaces, Oxford University Press, 1999.
[75] Y., Hu and X., Li, Discrete Fourier restriction associated with KdV equations, Anal. PDE 6 (2013), no. 4, 859–892.
[76] T. W., Hungerford, Algebra, Holt, Rinehart & Winston, Inc., New York– Montreal, Que.–London, 1974.
[77] A. D., Ionescu and C. E., Kenig, Global well-posedness of the Benjamin–Ono equation in low-regularity spaces, J. Amer. Math. Soc. 20 (2007), 753-798.
[78] L., Kapitanski and I., Rodnianski, Does a quantum particle knows the time?, in: Emerging Applications of Number Theory, D., Hejhal, J., Friedman, M. C., Gutzwiller and A. M., Odlyzko, eds., IMA Volumes in Mathematics and Its Applications, vol. 109, Springer Verlag, New York, 1999, pp. 355–371.
[79] T., Kappeler, B., Schaad, and P., Topalov, Qualitative features of periodic solutions of KdV, Comm. PDE 38 (2013), 1626–1673.
[80] T., Kappeler, B., Schaad, and P., Topalov, Scattering-like phenomena of the periodic defocusing NLS equation, http://arxiv.org/abs/1505.07394.
[81] T., Kappeler and P., Topalov, Global wellposedness of KdV in H-1(T,R), Duke Math. J. 135, no. 2 (2006), 327–360.
[82] T., Kato, On the Korteweg–de Vries equation, Manuscripta Math. 28 (1979), 89–99.
[83] T., Kato and G., Ponce, Commutator estimates and the Euler and Navier–Stokes equations, Commun. Pure Appl. Math. 41 (1988), 891–907.
[84] Y., Katznelson, An Introduction to Harmonic Analysis. Third edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004.
[85] M., Keel and T., Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955–980.
[86] M., Keel and T., Tao, Local and global well-posedness of wave maps on R1+1 for rough data, Internat. Math. Res. Notices 1998, no. 21, 1117–1156.
[87] C. E., Kenig, The Cauchy problem for the quasilinear Schrödinger equation, http://arxiv.org/abs/1309.3291.
[88] C. E., Kenig, G., Ponce, and L., Vega, Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993), 527–620.
[89] C. E., Kenig, G., Ponce, and L., Vega, Small solutions to nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 10 (1993), no. 3, 255–288.
[90] C. E., Kenig, G., Ponce, and L., Vega, On the Zakharov and Zakharov–Schulman systems, J. Funct. Anal. 127 (1995), no. 1, 204–234.
[91] C. E., Kenig, G., Ponce, and L., Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9 (1996), no. 2, 573–603.
[92] C. E., Kenig, G., Ponce, and L., Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J. 106 (2001), 617–633.
[93] S., Keraani and A., Vargas, A smoothing property for the L2-critical NLS equations and an application to blow up theory, Ann. Inst. H. Poincaré Annal. Non linéaire 26 (2009), 745–762.
[94] A. Y., Khinchin, Continued Fractions, Translated from the 3rd Russian edition of 1961, The University of Chicago Press, 1964.
[95] N., Kishimoto, Remark on the periodic mass critical nonlinear Schrödinger equation, Proc. Amer. Math. Soc. 142 (2014), 2649–2660.
[96] S., Klainerman and M., Machedon, space-time estimates for null forms and a local existence theorem, Comm. Pure Appl. Math. 46 (1993), 1221–1268.
[97] H., Koch and N., Tzvetkov, Nonlinear wave interactions for the Benjamin-Ono equation, Internat. Math. Res. Notices 2005, no. 30, 1833–1847.
[98] L., Kuipers and H., Niederreiter, Uniform Distribution of Sequences, Pure and Applied Mathematics. Wiley-Interscience, New York-London-Sydney, 1974.
[99] S. B., Kuksin, Analysis of Hamiltonian PDEs, Oxford University Press, 2000.
[100] S., Kwon and T., Oh, On unconditional well-posedness of modified KdV, Internat. Math. Res. Notices 2012, no. 15, 3509–3534.
[101] M. K., Kwong, Uniqueness of positive solutions of Δu - u + up = 0 in Rn, Arch. Rational Mech. Anal. 105 (1989), no. 3, 243–266.
[102] P., Lévy, Théorie de l'Addition des Variables Aléatoires, Paris, 1937.
[103] Y., Li, Y., Wu, and G., Xu, Global well-posedness for the mass-critical nonlinear Schrödinger equation on T, J. Diff. Eqs. 250 (2011), no. 6, 2715–2736.
[104] F., Linares, D., Pilod, and J.-C., Saut, Dispersive perturbations of Burgers and hyperbolic equations I: Local theory, SIAM J. Math. Anal. 46 (2014), no. 2, 1505- 1537.
[105] F., Linares and G., Ponce, Introduction to Nonlinear Dispersive Equations, UTX, Springer–Verlag, 2009.
[106] L., Molinet, Global well-posedness in the energy space for the Benjamin–Ono equation on the circle, Math. Ann. 337 (2007), 353–383.
[107] L., Molinet, On ill-posedness for the one-dimensional periodic cubic Schrödinger equation, Math. Res. Lett. 16 (2009), 111-120.
[108] L., Molinet, Sharp ill-posedness result for the periodic Benjamin–Ono equation, J. Funct. Anal. 257, (2009), 3488-3516.
[109] L., Molinet, A note on ill-posedness for the KdV equation, Differential Integral Equations 24 (2011), 759–765.
[110] L., Molinet, Sharp ill-posedness results for the KdV and mKdV equations on the torus, Adv. Math. 230 (2012), 1895–1930.
[111] L., Molinet, Global well-posedness in L2 for the periodic Benjamin-Ono equation, Amer. J. Math. 130 (2008), no. 3, 635-683.
[112] L., Molinet, J. C., Saut, and N., Tzvetkov, Ill-posedness issues for the Benjamin– Ono and related equations, Siam J. Math. Anal. 33 (2001), 982–988.
[113] H. L., Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, CBMS Regional conference series in Mathematics Volume 84, 1990.
[114] S J., Montgomery-Smith, Time decay for the bounded mean oscillation of solutions of the Schrödinger and wave equation, Duke Math. J. 19 (1998), 393– 408.
[115] A., Moyua and L., Vega, Bounds for the maximal function associated to periodic solutions of one-dimensional dispersive equations, Bull. Lond. Math. Soc. 40 (2008), no. 1, 117–128.
[116] C., Muscalu and W., Schlag, Classical and Multilinear Harmonic Analysis. Vol. I. Cambridge Studies in Advanced Mathematics, 137, Cambridge University Press, Cambridge, 2013.
[117] L., Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa 13 (1959), 115–162.
[118] T., Ogawa and Y., Tsutsumi, Blow-up of H1 solutions for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity, Proc. Amer. Math. Soc. 111 (1991), no. 2, 487–496.
[119] S., Oh, Resonant phase–shift and global smoothing of the periodic Korteweg–de Vries equation in low regularity settings, Adv. Diff. Eqs. 18 (2013), 633–662.
[120] P. J., Olver, Dispersive quantization, Amer. Math. Monthly 117 (2010), no. 7, 599–610.
[121] K. I., Oskolkov, A class of I. M. Vinogradov's series and its applications in harmonic analysis, in: Progress in Approximation Theory (Tampa, FL, 1990), Springer Ser. Comput. Math. 19, Springer, New York, 1992, pp. 353–402.
[122] H., Pecher, Global well-posedness below energy space for the 1-dimensional Zakharov system, Internat. Math. Res. Notices 2001, no. 19, 1027–1056.
[123] J., Pöschel and E., Trubowitz, Inverse Spectral Theory, Pure Appl. Math. 130, Academic Press, Boston, 1987.
[124] L., Rayleigh, On copying diffraction-gratings, and on some phenomena connected therewith, Philos. Mag. 11 (1881), 196–205.
[125] I., Rodnianski, Fractal solutions of the Schrödinger equation, Contemp. Math. 255 (2000), 181–187.
[126] J. C., Saut and R., Temam, Remarks on the Korteweg–de Vries equation, Israel J. Math. 24 (1976), 78–87.
[127] J., Shatah, Normal forms and quadratic nonlinear Klein–Gordon equations, Comm. Pure Appl. Math. 38 (1985), 685–696.
[128] V., Sohinger, Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on S1, Differential and Integral Equations 24 (2011), no. 7–8, 653–718.
[129] V., Sohinger, Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on R, Indiana Math. J. 60 (2011), no. 5, 1487– 1516.
[130] G., Staffilani, On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations, Duke Math. J. 86 (1997), no. 1, 109–142.
[131] A., Stefanov, Strichartz estimates for the Schrödinger equation with radial data, Proc. Amer. Math. Soc. 129 (2001), 1395–1401.
[132] E. M., Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956), 482–492.
[133] E. M., Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993.
[134] E. M., Stein and R., Shakarchi, Functional Analysis. Introduction to Further Topics in Analysis. Princeton Lectures in Analysis, 4. Princeton University Press, Princeton, NJ, 2011.
[135] E. M., Stein and G., Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, 32. Princeton University Press, Princeton, NJ, 1971.
[136] R. S., Strichartz, Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), 705–714.
[137] C., Sulem and P. L., Sulem, The Nonlinear Schrödinger Equation, Applied Mathematical Sciences 139. Springer–VerlagNY 1999.
[138] P.L., Sulem, C., Sulem and C., Bardos, On the continuous limit for a system of classical spins, Comm. Math. Phys. 107 (1986), no. 3, 431–454.
[139] H., Takaoka, Y., Tsutsumi, Well-posedness of the Cauchy problem for the modified KdV equation with periodic boundary condition, Internat. Math. Res. Notices 2004, no. 56, 3009–3040.
[140] H. F., Talbot, Facts related to optical science, No. IV, Philo. Mag. 9 (1836), 401–407.
[141] T., Tao, Spherically averaged endpoint Strichartz estimates for the two–dimensional Schrödinger equation, Commun. PDE 25 (2000), 1471–1485.
[142] T., Tao, Global well-posedness of the Benjamin–Ono equation in H1(R), J. Hyperbolic Differ. Equ. 1 (2004), no. 1, 27–49.
[143] T., Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics 106. American Mathematical Society, Providence, RI, 2006.
[144] M., Taylor, The Schrödinger equation on spheres, Pacific J. Math. 209 (2003), 145–155.
[145] M., Taylor, Tidbits in Harmonic Analysis, Lecture Notes, UNC, 1998.
[146] M., Taylor, Tools for PDE. Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Mathematical Surveys and Monographs, 81. American Mathematical Society, Providence, RI, 2000.
[147] R., Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Phyiscs, Applied Mathematical Sciences 68, Springer, 1997.
[148] P., Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 477–478.
[149] H., Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983.
[150] K., Tsugawa, Existence of the global attractor for weakly damped forced KdV equation on Sobolev spaces of negative index, Commun. Pure Appl. Anal. 3 (2004), no. 2, 301–318.
[151] Y., Tsutsumi, L2 solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac. 30 (1987), 115–125.
[152] N., Tzvetkov, Remark on the local ill-posedness for KdV equation, C. R. Acad. Sci. Paris 329 Série I (1999), 1043–1047.
[153] M., Wadati, H., Sanuki, and K., Konno, Relationships among inverse method, Bäcklund transformation and an infinite number of conservation laws, Progr. Theoret. Phys. 53 (1975), 419–436.
[154] T., Wolff, Lectures on Harmonic Analysis, University Lecture Series, 29. American Mathematical Society, Providence, RI, 2003.
[155] J., Yang, Nonlinear waves in integrable and nonintegrable systems, Mathematical Modeling and Computation 16, SIAM, Philadelphia, PA, 2010.
[156] X., Yang, Global attractor for the weakly damped forced KdV equation in Sobolev spaces of low regularity, Nonlinear Differ. Equ. Appl., published online.
[157] V. E., Zakharov and A. B., Shabat, 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.
[158] Y., Zhang, J., Wen, S. N., Zhu, and M., Xiao, Nonlinear Talbot effect, Phys. Rev. Lett. 104, 183901 (2010).
[159] A., Zygmund, On Fourier coefficients and transforms of functions of two variables, Stud. Math. 50 (1974), 189–201.

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Book summary page views

Total views: 0 *
Loading metrics...

* Views captured on Cambridge Core between #date#. This data will be updated every 24 hours.

Usage data cannot currently be displayed.