Free Access
Issue
Math. Model. Nat. Phenom.
Volume 7, Number 2, 2012
Solitary waves
Page(s) 95 - 112
DOI https://doi.org/10.1051/mmnp/20127209
Published online 29 February 2012
  1. M. Ablowitz, B. Prinar, A. Trubatch. Discrete and Continuous Nonlinear Schrödinger Systems. London Mathematical Society Lecture Note Series, No. 302, Cambridge University Press, 2004. [Google Scholar]
  2. Yu.V. Bludov, V.V. Konotop, N. Akhmediev. Matter rogue waves. Phys. Rev. A, 80 (2009), No. 3, 033610. [CrossRef] [Google Scholar]
  3. J.L. Bona, Y.A. Li. Decay and analyticity of solitary waves. J. Math. Pures Appl. 76 (1997), No. 5, 377-430. [CrossRef] [MathSciNet] [Google Scholar]
  4. J. Bourgain. Periodic Nonlinear Schrödinger Equation and Invariant Measures. Comm. Math. Phys. 166 (1994), No. 1, 1–26. [CrossRef] [MathSciNet] [Google Scholar]
  5. J. Bourgain. Invariant measures for NLS in Infinite Volume. Comm. Math. Phys. 210 (2000), No. 3, 605–620. [CrossRef] [MathSciNet] [Google Scholar]
  6. J. Bourgain. Invariant measures for the 2D-defocusing nonlinear Schrödinger equation. Comm. Math. Phys. 176 (1996), No. 2, 421–445. [CrossRef] [MathSciNet] [Google Scholar]
  7. J. Bourgain. On nonlinear Schrödinger equations, Les relations entre les mathématiques et la physique théorique, 11-21, Inst. Hautes Études Sci., Bures-sur-Yvette, 1998. [Google Scholar]
  8. D. Brydges, G. Slade. Statistical Mechanics of the 2-Dimensional Focusing Nonlinear Schrödinger Equation, Comm. Math. Phys. 182 (1996), No. 2, 485–504. [CrossRef] [MathSciNet] [Google Scholar]
  9. R. Burioni, D. Cassi, P. Sodano, A. Trombettoni, A. Vezzani. Soliton propagation on chains with simple nonlocal defects, Phys. D 216 (2006), No. 1, 71–76. [CrossRef] [MathSciNet] [Google Scholar]
  10. Y. Burlakov. The phase space of the cubic Schroedinger equation : A numerical study. Thesis (Ph.D.)–University of California, Berkeley. Preprint 740 (1998). [Google Scholar]
  11. N. Burq, N. Tzvetkov. Random data Cauchy theory for supercritical wave equations I : local theory. Invent. Math. 173 (2008), No. 3, 449–475. [CrossRef] [MathSciNet] [Google Scholar]
  12. N. Burq, N. Tzvetkov. Random data Cauchy theory for supercritical wave equations II : a global existence result. Invent. Math. 173 (2008), No. 3 477–496. [CrossRef] [MathSciNet] [Google Scholar]
  13. L. Caffarelli, S. Salsa, L. Silvestre. Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian. Invent. Math. 171 2008, No. 2, 425–461. [CrossRef] [MathSciNet] [Google Scholar]
  14. T. Cazenave. Semilinear Schrödinger equations. Courant Lecture Notes, No. 10, American Mathematical Society and Courant Institute of Mathematical Sciences, New York, 2003. [Google Scholar]
  15. T. Cazenave, P.-L. Lions. Orbital stability of standing waves for some nonlinear Schrodinger equations. Comm. Math. Phys. 85 (1982), no. 4, 549–561. [CrossRef] [MathSciNet] [Google Scholar]
  16. S. Chatterjee, K. Kirkpatrick. Probabilistic methods for discrete nonlinear Schrödinger equations, to appear in Comm. Pure Appl. Math. [Google Scholar]
  17. M. Christ, J. Colliander, T. Tao. Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations. Amer. J. Math. 125 (2003), no. 6, 1235–1293. [CrossRef] [MathSciNet] [Google Scholar]
  18. J. Colliander, T. Oh. Almost sure well-posedness of the periodic cubic nonlinear Schrödinger equation below L2, arXiv :0904.2820. [Google Scholar]
  19. R. Cont, P. Tankov. Financial modelling with jump processes. Chapman & Hall/CRC Financial Mathematics Series, Boca Raton, FL, 2004. [Google Scholar]
  20. L. Erdős, B. Schlein, H.-T. Yau. Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems. Invent. Math. 167 (2007), No. 3 515–614. [CrossRef] [MathSciNet] [Google Scholar]
  21. L. Erdős, B. Schlein, H.-T. Yau. Derivation of the Gross-Pitaevskii Equation for the Dynamics of Bose-Einstein Condensate. Ann. Math. (2) 172 (2010), no. 1, 291–370. [CrossRef] [Google Scholar]
  22. P. Felmer, A. Quaas, J. Tan. Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian. Preprint : http://www.capde.cl/publication/abstract/frac-FQTreview1.pdf [Google Scholar]
  23. S. Flach, K. Kladko, R.S. MacKay. Energy thresholds for discrete breathers in one-, two- and three-dimensional lattices, Phys. Rev. Lett. 78 (1997), 1207–1210. [CrossRef] [Google Scholar]
  24. Yu. Gaididei, S. Mingaleev, P. Christiansen, K. Rasmussen. Effect of nonlocal dispersion on self-trapping excitations. Phys. Rev. E 55 (1997), No. 5, 6141–6150. [CrossRef] [Google Scholar]
  25. Yu. Gaididei, S. Mingaleev, P. Christiansen, K. Rasmussen. Effect of nonlocal dispersion on self-interacting excitations. Phys. Lett. A 222 (1996), 152-156. [CrossRef] [Google Scholar]
  26. J. Ginibre, G. Velo. On a class of nonlinear Schrodinger equations. I. The Cauchy problem, general case. J. Funct. Anal. 32 (1979), no. 1, 1–32. [CrossRef] [MathSciNet] [Google Scholar]
  27. Z. Guo, Y. Wang. Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrdinger and wave equation. arXiv :1007.4299. [Google Scholar]
  28. J. Holmer, R. Platte, S. Roudenko. Blow-up criteria for the 3D cubic nonlinear Schrodinger equation, Nonlinearity 23 (2010), No. 4, 977–1030. [CrossRef] [MathSciNet] [Google Scholar]
  29. R. Jordan, C. Josserand. Statistical equilibrium states for the nonlinear Schr"odinger equation. Nonlinear waves : computation and theory (Athens, GA,1999). Math. Comput. Simulation 55 (2001), No. 4-6, 433–447. [Google Scholar]
  30. R. Jordan, B. Turkington. Statistical equilibrium theories for the nonlinear Schr"odinger equation, Advances in wave interaction and turbulence (South Hadley, MA, 2000), 27–39, Contemp. Math. 283, Amer. Math. Soc., Providence, RI, 2001. [Google Scholar]
  31. T. Kato. On nonlinear Schrödinger equations. t Ann. Inst. H. Poincaré Phys. Theor. 46 (1987), no. 1, 113–129. [Google Scholar]
  32. C. Kenig, Y. Martel, L Robbiano. Local well-posedness and blow up in the energy space for a class of L2 critical dispersion generalized Benjamin-Ono equations. Preprint arXiv :1006.0122 [Google Scholar]
  33. P. Kevrekidis, D. Pelinovsky, A. Stefanov. Asymptotic stability of small bound states in the discrete nonlinear Schrödinger equation. SIAM J. Math. Anal. 41 (2009), No. 5, 2010–2030. [CrossRef] [MathSciNet] [Google Scholar]
  34. K. Kirkpatrick, E. Lenzmann, G. Staffilani. On the continuum limit for discrete NLS with long-range lattice interactions, arXiv :1108.6136v1. [Google Scholar]
  35. K. Kirkpatrick, B. Schlein, G. Staffilani. Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics. Amer. J. Math. 133 (2011), no. 1, 91–130. [CrossRef] [MathSciNet] [Google Scholar]
  36. A. Komech, E. Kopylova, M. Kunze. Dispersive estimates for 1D discrete Schrödinger and Klein-Gordon equations. Appl. Anal. 85 (2006), no. 12, 1487–1508. [CrossRef] [MathSciNet] [Google Scholar]
  37. O. A. Ladyzhenskaya. The Boundary Value Problems of Mathematical Physics. Applied Mathematical Sciences, No. 49. Springer-Verlag New York, 1985. [Google Scholar]
  38. N. Laskin. Fractional Schrödinger equation. Phys. Rev. E (3) 66 (2002), no. 5, 056108, 7 pp. [Google Scholar]
  39. J. Lebowitz, H. Rose, E. Speer. Statistical Mechanics of the Nonlinear Schrodinger equation, J. Stat. Phys. 50 (1988), No. 3-4, 657–687. [CrossRef] [Google Scholar]
  40. R. S. MacKay, S. Aubry. Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators, Nonlinearity 7 (1994), No. 6, 1623–1643. [CrossRef] [MathSciNet] [Google Scholar]
  41. H. P. McKean, K. L. Vaninsky. Brownian motion with restoring drift : the petit and micro-canonical ensembles. Comm. Math. Phys. 160 (1994), no. 3, 615–630. [CrossRef] [MathSciNet] [Google Scholar]
  42. H. P. McKean, K. L. Vaninsky. Action-angle variables for the cubic Schrödinger equation. Comm. Pure Appl. Math. 50 (1997), no. 6, 489–562. [CrossRef] [MathSciNet] [Google Scholar]
  43. H. P. McKean, K. L. Vaninsky. Cubic Schrödinger : the petit canonical ensemble in action-angle variables. Comm. Pure Appl. Math. 50 (1997), no. 7, 593–622. [CrossRef] [MathSciNet] [Google Scholar]
  44. B. Malomed, M. I. Weinstein. Soliton dynamics in the discrete nonlinear Schrödinger equation. Phys. Lett. A 220 (1996), 91–96. [CrossRef] [Google Scholar]
  45. M. Maris. On the existence, regularity and decay of solitary waves to a generalized Benjamin-Ono equation, Nonlinear Anal. 51 (2002), No. 6, 1073–1085. [CrossRef] [MathSciNet] [Google Scholar]
  46. S. Mingaleev, P. Christiansen, Yu. Gaididei, M. Johannson, K. Rasmussen. Models for Energy and Charge Transport and Storage in Biomolecules. J. Biol. Phys. 25 (1999), 41-63. [CrossRef] [PubMed] [Google Scholar]
  47. L. Molinet. On ill-posedness for the one-dimensional periodic cubic Schrödinger equation. Math. Res. Lett. 16 (2009), no. 1, 111-120. [MathSciNet] [Google Scholar]
  48. A. Nahmod, T. Oh, L. Rey-Bellet, G. Staffilani. Invariant weighted Wiener measures adn almost sure global well-posedness for the periodic derivative NLS. arXiv :1007.1502. [Google Scholar]
  49. T. Oh, C. Sulem. On the one-dimensional cubic nonlinear Schrodinger equation below L2, arXiv :1007.2073. [Google Scholar]
  50. D. Pelinovsky, P. Kevrekidis. Stability of discrete dark solitons in nonlinear Schrödinger lattices. J. Phys. A 41 (2008), 185–206. [Google Scholar]
  51. D. Pelinovsky, A. Sakovich. Internal modes of discrete solitons near the anti-continuum limit of the dNLS equation. Physica D 240 (2011), 265–281. [CrossRef] [MathSciNet] [Google Scholar]
  52. D. Pelinovsky, A. Stefanov. On the spectral theory and dispersive estimates for a discrete Schrödinger equation in one dimension. J. Math. Phys. 49, (2008), no. 11,113501, 17 pp. [Google Scholar]
  53. B. Rider. On the ∞-volume limit of focussing cubic Schrödinger equation. Comm. Pure Appl. Math. 55 (2002), No. 10, 1231–1248. [CrossRef] [MathSciNet] [Google Scholar]
  54. B. Rider. Fluctuations in the thermodynamic limit of focussing cubic Schrödinger. J. Stat. Phys. 113 (2003), 575–594. [CrossRef] [Google Scholar]
  55. B. Rumpf. Simple statistical explanation for the localization of energy in nonlinear lattices with two conserved quantities. Phys. Rev. E 69 (2004), 016618. [CrossRef] [Google Scholar]
  56. S. Samko, A. Kilbas, O. Marichev. Fractional Integrals and Derivatives : Theory and Applications. Gordon and Breach Science Publishers, Amsterdam, 1993. [Google Scholar]
  57. Y. Sire, E. Valdinoci. Fractional Laplacian phase transitions and boundary reactions : a geometric inequality and a symmetry result. J. Funct. Anal. 256 (2009), no. 6, 1842–1864. [CrossRef] [MathSciNet] [Google Scholar]
  58. A. Stefanov, P. Kevrekidis. Asymptotic behaviour of small solutions for the discrete nonlinear Schrodinger and Klein-Gordon equations. Nonlinearity 18 (2005), No. 4, 1841-1857. [CrossRef] [MathSciNet] [Google Scholar]
  59. C. Sulem, P.L. Sulem. The nonlinear Schrödinger equation : self-focusing and wave collapse. Springer, 1999. [Google Scholar]
  60. N. Tzvetkov. Invariant measures for the defocusing NLS. Ann. Inst. Fourier (Grenoble) 58 (2008), no. 7, 2543–2604. [CrossRef] [MathSciNet] [Google Scholar]
  61. M. I. Weinstein. Excitation Thresholds for Nonlinear Localized Modes on Lattices. Nonlinearity 12 (1999), No. 3, 673–691. [CrossRef] [MathSciNet] [Google Scholar]
  62. V. E. Zakharov. Stability of periodic waves of finite amplitude on a surface of deep fluid. J. Appl. Mech. Tech. Phys. 2 (1968), 190–198. [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.