Attractors of nonlinear Hamiltonian partial differential equations

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Abstract

This is a survey of the theory of attractors of nonlinear Hamiltonian partial differential equations since its appearance in 1990. Included are results on global attraction to stationary states, to solitons, and to stationary orbits, together with results on adiabatic effective dynamics of solitons and their asymptotic stability, and also results on numerical simulation. The results obtained are generalized in the formulation of a new general conjecture on attractors of $G$-invariant nonlinear Hamiltonian partial differential equations. This conjecture suggests a novel dynamical interpretation of basic quantum phenomena: Bohr transitions between quantum stationary states, de Broglie's wave-particle duality, and Born's probabilistic interpretation.Bibliography: 212 titles.

About the authors

Aleksandr Il'ich Komech

Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute)

Email: alexander.komech@univie.ac.at
Doctor of physico-mathematical sciences, Professor

Elena Andreevna Kopylova

Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute)

Email: elena.kopylova@univie.ac.at

References

  1. M. Abraham, “Prinzipien der Dynamik des Elektrons”, Phys. Z., 4 (1902), 57–63
  2. M. Abraham, Theorie der Elektrizität, v. 2, Elektromagnetische Theorie der Strahlung, B. G. Teubner, Leipzig, 1905, 404 pp.
  3. R. K. Adair, E. C. Fowler, Strange particles, Intersci. Publ. John Wiley & Sons, New York–London, 1963, viii+151 pp.
  4. R. Adami, D. Noja, C. Ortoleva, “Orbital and asymptotic stability for standing waves of a nonlinear Schrödinger equation with concentrated nonlinearity in dimension three”, J. Math. Phys., 54:1 (2013), 013501, 33 pp.
  5. S. Agmon, “Spectral properties of Schrödinger operators and scattering theory”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2:2 (1975), 151–218
  6. S. Albeverio, R. Figari, “Quantum fields and point interactions”, Rend. Mat. Appl. (7), 39:2 (2018), 161–180
  7. L. Andersson, P. Blue, “Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior”, J. Hyperbolic Differ. Equ., 12:4 (2015), 689–743
  8. А. В. Бабин, М. И. Вишик, Аттракторы эволюционных уравнений, Наука, M., 1989, 296 с.
  9. V. Bach, T. Chen, J. Faupin, J. Fröhlich, I. M. Sigal, “Effective dynamics of an electron coupled to an external potential in non-relativistic QED”, Ann. Henri Poincare, 14:6 (2013), 1573–1597
  10. D. Bambusi, S. Cuccagna, “On dispersion of small energy solutions to the nonlinear Klein Gordon equation with a potential”, Amer. J. Math., 133:5 (2011), 1421–1468
  11. D. Bambusi, L. Galgani, “Some rigorous results on the Pauli–Fierz model of classical electrodynamics”, Ann. Inst. H. Poincare Phys. Theor., 58:2 (1993), 155–171
  12. V. E. Barnes et al., “Observation of a hyperon with strangeness minus three”, Phys. Rev. Lett., 12:8 (1964), 204–206
  13. M. Beals, W. Strauss, “$L^p$ estimates for the wave equation with a potential”, Comm. Partial Differential Equations, 18:7-8 (1993), 1365–1397
  14. M. Beceanu, M. Goldberg, “Schrödinger dispersive estimates for a scaling-critical class of potentials”, Comm. Math. Phys., 314:2 (2012), 471–481
  15. M. Beceanu, M. Goldberg, “Strichartz estimates and maximal operators for the wave equation in $mathbb{R}^3$”, J. Funct. Anal., 266:3 (2014), 1476–1510
  16. A. Bensoussan, C. Iliine, A. Komech, “Breathers for a relativistic nonlinear wave equation”, Arch. Ration. Mech. Anal., 165:4 (2002), 317–345
  17. H. Berestycki, P.-L. Lions, “Nonlinear scalar field equations. I. Existence of a ground state”, Arch. Ration. Mech. Anal., 82:4 (1983), 313–345
  18. H. Berestycki, P.-L. Lions, “Nonlinear scalar field equations. II. Existence of infinitely many solutions”, Arch. Ration. Mech. Anal., 82:4 (1983), 347–375
  19. Ф. А. Березин, Л. Д. Фаддеев, “Замечание об уравнении Шредингера с сингулярным потенциалом”, Докл. АН СССР, 137:5 (1961), 1011–1014
  20. N. Bohr, “On the constitution of atoms and molecules. I”, Philos. Mag. (6), 26:151 (1913), 1–25
  21. Н. Бор, “Дискуссии с Эйнштейном о проблемах теории познания в атомной физике”, УФН, 66 (1958), 571–598
  22. N. Boussaid, “Stable directions for small nonlinear Dirac standing waves”, Comm. Math. Phys., 268:3 (2006), 757–817
  23. N. Boussaid, S. Cuccagna, “On stability of standing waves of nonlinear Dirac equations”, Comm. Partial Differential Equations, 37:6 (2012), 1001–1056
  24. V. S. Buslaev, A. I. Komech, E. A. Kopylova, D. Stuart, “On asymptotic stability of solitary waves in Schrödinger equation coupled to nonlinear oscillator”, Comm. Partial Differential Equations, 33:4 (2008), 669–705
  25. В. С. Буслаев, Г. С. Перельман, “Рассеяние для нелинейного уравнения Шрeдингера: состояния, близкие к солитону”, Алгебра и анализ, 4:6 (1992), 63–102
  26. V. S. Buslaev, G. S. Perelman, “On the stability of solitary waves for nonlinear Schrödinger equations”, Nonlinear evolution equations, Amer. Math. Soc. Transl. Ser. 2, 164, Adv. Math. Sci., 22, Amer. Math. Soc., Providence, RI, 1995, 75–98
  27. V. S. Buslaev, C. Sulem, “On asymptotic stability of solitary waves for nonlinear Schrödinger equations”, Ann. Inst. H. Poincare Anal. Non Lineaire, 20:3 (2003), 419–475
  28. V. V. Chepyzhov, M. I. Vishik, Attractors for equations of mathematical physics, Amer. Math. Soc. Colloq. Publ., 49, Amer. Math. Soc., Providence, RI, 2002, xii+363 pp.
  29. G. M. Coclite, V. Georgiev, “Solitary waves for Maxwell–Schrödinger equations”, Electron. J. Differential Equations, 2004 (2004), 94, 31 pp. (electronic)
  30. A. Comech, “On global attraction to solitary waves. Klein–Gordon equation with mean field interaction at several points”, J. Differential Equations, 252:10 (2012), 5390–5413
  31. A. Comech, “Weak attractor of the Klein–Gordon field in discrete space-time interacting with a nonlinear oscillator”, Discrete Contin. Dyn. Syst., 33:7 (2013), 2711–2755
  32. F. H. J. Cornish, “Classical radiation theory and point charges”, Proc. Phys. Soc., 86:3 (1965), 427–442
  33. S. Cuccagna, “Stabilization of solutions to nonlinear Schrödinger equations”, Comm. Pure Appl. Math., 54:9 (2001), 1110–1145
  34. S. Cuccagna, “The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states”, Comm. Math. Phys., 305:2 (2011), 279–331
  35. S. Cuccagna, T. Mizumachi, “On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations”, Comm. Math. Phys., 284:1 (2008), 51–77
  36. M. Dafermos, I. Rodnianski, “A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds”, Invent. Math., 185:3 (2011), 467–559
  37. P. D'Ancona, “Kato smoothing and Strichartz estimates for wave equations with magnetic potentials”, Comm. Math. Phys., 335:1 (2015), 1–16
  38. P. D'Ancona, L. Fanelli, L. Vega, N. Visciglia, “Endpoint Strichartz estimates for the magnetic Schrödinger equation”, J. Funct. Anal., 258:10 (2010), 3227–3240
  39. S. Demoulini, D. Stuart, “Adiabatic limit and the slow motion of vortices in a Chern–Simons–Schrödinger system”, Comm. Math. Phys., 290:2 (2009), 597–632
  40. P. A. M. Dirac, “Classical theory of radiating electrons”, Proc. Roy. Soc. London Ser. A, 167:929 (1938), 148–169
  41. R. Donninger, W. Schlag, A. Soffer, “On pointwise decay of linear waves on a Schwarzschild black hole background”, Comm. Math. Phys., 309:1 (2012), 51–86
  42. T. Duyckaerts, C. Kenig, F. Merle, “Profiles of bounded radial solutions of the focusing, energy-critical wave equation”, Geom. Funct. Anal., 22:3 (2012), 639–698
  43. T. Duyckaerts, C. Kenig, F. Merle, “Scattering for radial, bounded solutions of focusing supercritical wave equations”, Int. Math. Res. Not. IMRN, 2014:1 (2014), 224–258
  44. T. Duyckaerts, C. Kenig, F. Merle, “Concentration-compactness and universal profiles for the non-radial energy critical wave equation”, Nonlinear Anal., 138 (2016), 44–82
  45. А. В. Дымов, “Диссипативные эффекты в одной линейной лагранжевой системе с бесконечным числом степеней свободы”, Изв. РАН. Сер. матем., 76:6 (2012), 45–80
  46. W. Eckhaus, A. van Harten, The inverse scattering transformation and the theory of solitons. An introduction, North-Holland Math. Stud., 50, North-Holland Publishing Co., Amsterdam–New York, 1981, xi+222 pp.
  47. И. Е. Егорова, Е. А. Копылова, В. А. Марченко, Г. Тешль, “Об уточнении дисперсионных оценок для одномерных уравнений Шрeдингера и Клейна–Гордона”, УМН, 71:3(429) (2016), 3–26
  48. I. Egorova, E. A. Kopylova, G. Teschl, “Dispersion estimates for one-dimensional discrete Schrödinger and wave equations”, J. Spectr. Theory, 5:4 (2015), 663–696
  49. A. Einstein, “Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?”, Ann. der Phys. (4), 18 (1905), 639–641
  50. M. B. Erdoğan, M. Goldberg, W. R. Green, “Dispersive estimates for four dimensional Schrödinger and wave equations with obstructions at zero energy”, Comm. Partial Differential Equations, 39:10 (2014), 1936–1964
  51. M. J. Esteban, V. Georgiev, E. Sere, “Stationary solutions of the Maxwell–Dirac and the Klein–Gordon–Dirac equations”, Calc. Var. Partial Differential Equations, 4:3 (1996), 265–281
  52. Р. Фейнман, З. Лейтон, М. Сэндс, Фейнмановские лекции по физике, т. 5, 6, 7, Мир, М., 1977, 302 с., 349 с., 288 с.
  53. C. Foias, O. Manley, R. Rosa, R. Temam, Navier–Stokes equations and turbulence, Encyclopedia Math. Appl., 83, Cambridge Univ. Press, Cambridge, 2001, xiv+347 pp.
  54. J. Fröhlich, Z. Gang, “Emission of Cherenkov radiation as a mechanism for Hamiltonian friction”, Adv. Math., 264 (2014), 183–235
  55. J. Fröhlich, S. Gustafson, B. L. G. Jonsson, I. M. Sigal, “Solitary wave dynamics in an external potential”, Comm. Math. Phys., 250:3 (2004), 613–642
  56. J. Fröhlich, T.-P. Tsai, H.-T. Yau, “On the point-particle (Newtonian) limit of the non-linear Hartree equation”, Comm. Math. Phys., 225:2 (2002), 223–274
  57. G. I. Gaudry, “Quasimeasures and operators commuting with convolution”, Pacific J. Math., 18:3 (1966), 461–476
  58. M. Gell-Mann, “Symmetries of baryons and mesons”, Phys. Rev. (2), 125:3 (1962), 1067–1084
  59. H.-P. Gittel, J. Kijowski, E. Zeidler, “The relativistic dynamics of the combined particle-field system in renormalized classical electrodynamics”, Comm. Math. Phys., 198:3 (1998), 711–736
  60. M. Goldberg, W. R. Green, “Dispersive estimates for higher dimensional Schrödinger operators with threshold eigenvalues. I. The odd dimensional case”, J. Funct. Anal., 269:3 (2015), 633–682
  61. M. Goldberg, W. R. Green, “Dispersive estimates for higher dimensional Schrödinger operators with threshold eigenvalues. II. The even dimensional case”, J. Spectr. Theory, 7:1 (2017), 33–86
  62. M. Grillakis, J. Shatah, W. Strauss, “Stability theory of solitary waves in the presence of symmetry. I”, J. Funct. Anal., 74:1 (1987), 160–197
  63. M. Grillakis, J. Shatah, W. Strauss, “Stability theory of solitary waves in the presence of symmetry. II”, J. Funct. Anal., 94:2 (1990), 308–348
  64. J. K. Hale, Asymptotic behavior of dissipative systems, Math. Surveys Monogr., 25, Amer. Math. Soc., Providence, RI, 1988, x+198 pp.
  65. F. Halzen, A. D. Martin, Quarks and leptons: an introductory course in modern particle physics, John Wiley & Sons, Inc., New York, 1984, xvi+396 pp.
  66. T. Harada, H. Maeda, “Stability criterion for self-similar solutions with a scalar field and those with a stiff fluid in general relativity”, Classical Quantum Gravity, 21:2 (2004), 371–389
  67. A. Haraux, Systèmes dynamiques dissipatifs et applications, Rech. Math. Appl., 17, Masson, Paris, 1991, xii+132 pp.
  68. W. Heisenberg, “Der derzeitige Stand der nichtlinearen Spinortheorie der Elementarteilchen”, Acta Phys. Austriaca, 14 (1961), 328–339
  69. В. Гейзенберг, Введение в единую полевую теорию элементарных частиц, Мир, М., 1968, 240 с.
  70. Д. Хенри, Геометрическая теория полулинейных параболических уравнений, Мир, М., 1985, 376 с.
  71. E. Hopf, “Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen”, Math. Nachr., 4 (1951), 213–231
  72. L. Hörmander, The analysis of linear partial differential operators, v. I, Grundlehren Math. Wiss., 256, Distribution theory and Fourier analysis, 2nd ed., Springer-Verlag, Berlin, 1990, xii+440 pp.
  73. L. Houllevigue, L'evolution des sciences, Librairie A. Collin, Paris, 1908, xi+287 pp.
  74. В. М. Имайкин, “Солитонные асимптотики для систем типа ‘поле-частица’ ”, УМН, 68:2(410) (2013), 33–90
  75. V. Imaykin, A. Komech, P. A. Markowich, “Scattering of solitons of the Klein–Gordon equation coupled to a classical particle”, J. Math. Phys., 44:3 (2003), 1202–1217
  76. V. Imaykin, A. Komech, N. Mauser, “Soliton-type asymptotics for the coupled Maxwell–Lorentz equations”, Ann. Henri Poincare, 5:6 (2004), 1117–1135
  77. V. Imaykin, A. Komech, H. Spohn, “Soliton-type asymptotic and scattering for a charge coupled to the Maxwell field”, Russ. J. Math. Phys., 9:4 (2002), 428–436
  78. V. Imaykin, A. Komech, H. Spohn, “Scattering theory for a particle coupled to a scalar field”, Discrete Contin. Dyn. Syst., 10:1-2 (2004), 387–396
  79. V. Imaykin, A. Komech, H. Spohn, “Rotating charge coupled to the Maxwell field: scattering theory and adiabatic limit”, Monatsh. Math., 142:1-2 (2004), 143–156
  80. V. Imaykin, A. Komech, H. Spohn, “Scattering asymptotics for a charged particle coupled to the Maxwell field”, J. Math. Phys., 52:4 (2011), 042701, 33 pp.
  81. V. Imaykin, A. Komech, B. Vainberg, “On scattering of solitons for the Klein–Gordon equation coupled to a particle”, Comm. Math. Phys., 268:2 (2006), 321–367
  82. V. Imaykin, A. Komech, B. Vainberg, “Scattering of solitons for coupled wave-particle equations”, J. Math. Anal. Appl., 389:2 (2012), 713–740
  83. J. D. Jackson, Classical electrodynamics, 3rd ed., John Wiley & Sons, Inc., New York, 1999, xxi+808 pp.
  84. A. Jensen, T. Kato, “Spectral properties of Schrödinger operators and time-decay of the wave functions”, Duke Math. J., 46:3 (1979), 583–611
  85. K. Jörgens, “Das Anfangswertproblem im Grossen für eine Klasse nichtlinearer Wellengleichungen”, Math. Z., 77 (1961), 295–308
  86. J.-L. Journe, A. Soffer, C. D. Sogge, “Decay estimates for Schrödinger operators”, Comm. Pure Appl. Math., 44:5 (1991), 573–604
  87. C. Kenig, A. Lawrie, B. Liu, W. Schlag, “Stable soliton resolution for exterior wave maps in all equivariance classes”, Adv. Math., 285 (2015), 235–300
  88. C. E. Kenig, A. Lawrie, W. Schlag, “Relaxation of wave maps exterior to a ball to harmonic maps for all data”, Geom. Funct. Anal., 24:2 (2014), 610–647
  89. C. E. Kenig, F. Merle, “Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case”, Invent. Math., 166:3 (2006), 645–675
  90. C. E. Kenig, F. Merle, “Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation”, Acta Math., 201:2 (2008), 147–212
  91. C. E. Kenig, F. Merle, “Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications”, Amer. J. Math., 133:4 (2011), 1029–1065
  92. А. А. Комеч, А. И. Комеч, “Вариант теоремы Титчмарша о свертке для распределений на окружности”, Функц. анализ и его прил., 47:1 (2013), 26–32
  93. А. И. Комеч, “О стабилизации взаимодействия струны с нелинейным осциллятором”, Вестн. Моск. ун-та. Сер. 1. Матем., мех., 1991, № 6, 35–41
  94. А. И. Комеч, “Линейные уравнения в частных производных с постоянными коэффициентами”, Дифференциальные уравнения с частными производными – 2, Итоги науки и техн. Сер. Соврем. пробл. матем. Фундам. направления, 31, ВИНИТИ, М., 1988, 127–261
  95. A. I. Komech, “On stabilization of string-nonlinear oscillator interaction”, J. Math. Anal. Appl., 196:1 (1995), 384–409
  96. A. I. Komech, “On the stabilization of string-oscillator interaction”, Russ. J. Math. Phys., 3:2 (1995), 227–247
  97. A. Komech, “On transitions to stationary states in one-dimensional nonlinear wave equations”, Arch. Ration. Mech. Anal., 149:3 (1999), 213–228
  98. А. И. Комеч, “Аттракторы нелинейных гамильтоновых одномерных волновых уравнений”, УМН, 55:1(331) (2000), 45–98
  99. A. I. Komech, “On attractor of a singular nonlinear $mathrm U(1)$-invariant Klein–Gordon equation”, Progress in analysis (Berlin, 2001), v. I, II, World Sci. Publ., River Edge, NJ, 2003, 599–611
  100. A. Komech, Quantum mechanics: genesis and achievements, Springer, Dordrecht, 2013, xviii+285 pp.
  101. A. Komech, “Attractors of Hamilton nonlinear PDEs”, Discrete Contin. Dyn. Syst., 36:11 (2016), 6201–6256
  102. A. Komech, “Quantum jumps and attractors of Maxwell–Schrödinger equations”, Nonlinearity (to appear)
  103. A. I. Komech, A. A. Komech, “On the global attraction to solitary waves for the Klein–Gordon equation coupled to a nonlinear oscillator”, C. R. Math. Acad. Sci. Paris, 343:2 (2006), 111–114
  104. A. Komech, A. Komech, “Global attractor for a nonlinear oscillator coupled to the Klein–Gordon field”, Arch. Ration. Mech. Anal., 185:1 (2007), 105–142
  105. A. I. Komech, A. A. Komech, “Global attraction to solitary waves in models based on the Klein–Gordon equation”, SIGMA, 4 (2008), 010, 23 pp.
  106. A. Komech, A. Komech, “Global attraction to solitary waves for Klein–Gordon equation with mean field interaction”, Ann. Inst. H. Poincare Anal. Non Lineaire, 26:3 (2009), 855–868
  107. A. Komech, A. Komech, “On global attraction to solitary waves for the Klein–Gordon field coupled to several nonlinear oscillators”, J. Math. Pures Appl. (9), 93:1 (2010), 91–111
  108. A. Komech, A. Komech, “Global attraction to solitary waves for a nonlinear Dirac equation with mean field interaction”, SIAM J. Math. Anal., 42:6 (2010), 2944–2964
  109. A. Komech, E. Kopylova, “Scattering of solitons for the Schrödinger equation coupled to a particle”, Russ. J. Math. Phys., 13:2 (2006), 158–187
  110. A. I. Komech, E. A. Kopylova, “Weighted energy decay for 1D Klein–Gordon equation”, Comm. Partial Differential Equations, 35:2 (2010), 353–374
  111. A. I. Komech, E. A. Kopylova, “Weighted energy decay for 3D Klein–Gordon equation”, J. Differential Equations, 248:3 (2010), 501–520
  112. A. Komech, E. Kopylova, Dispersion decay and scattering theory, John Wiley & Sons, Inc., Hoboken, NJ, 2012, xxvi+175 pp.
  113. A. I. Komech, E. A. Kopylova, “Dispersion decay for the magnetic Schrödinger equation”, J. Funct. Anal., 264:3 (2013), 735–751
  114. A. Komech, E. Kopylova, “On eigenfunction expansion of solutions to the Hamilton equations”, J. Stat. Phys., 154:1-2 (2014), 503–521
  115. A. I. Komech, E. A. Kopylova, “Weighted energy decay for magnetic Klein–Gordon equation”, Appl. Anal., 94:2 (2015), 218–232
  116. A. Komech, E. Kopylova, “On the eigenfunction expansion for Hamilton operators”, J. Spectr. Theory, 5:2 (2015), 331–361
  117. A. I. Komech, E. A. Kopylova, S. A. Kopylov, “On nonlinear wave equations with parabolic potentials”, J. Spectr. Theory, 3:4 (2013), 485–503
  118. A. I. Komech, E. A. Kopylova, M. Kunze, “Dispersive estimates for 1D discrete Schrödinger and Klein–Gordon equations”, Appl. Anal., 85:12 (2006), 1487–1508
  119. A. I. Komech, E. A. Kopylova, H. Spohn, “Scattering of solitons for Dirac equation coupled to a particle”, J. Math. Anal. Appl., 383:2 (2011), 265–290
  120. A. Komech, E. Kopylova, D. Stuart, “On asymptotic stability of solitons in a nonlinear Schrödinger equation”, Commun. Pure Appl. Anal., 11:3 (2012), 1063–1079
  121. A. I. Komech, E. A. Kopylova, B. R. Vainberg, “On dispersive properties of discrete 2D Schrödinger and Klein–Gordon equations”, J. Funct. Anal., 254:8 (2008), 2227–2254
  122. A. Komech, M. Kunze, H. Spohn, “Effective dynamics for a mechanical particle coupled to a wave field”, Comm. Math. Phys., 203:1 (1999), 1–19
  123. A. I. Komech, N. J. Mauser, A. P. Vinnichenko, “Attraction to solitons in relativistic nonlinear wave equations”, Russ. J. Math. Phys., 11:3 (2004), 289–307
  124. A. I. Komech, A. E. Merzon, “Scattering in the nonlinear Lamb system”, Phys. Lett. A, 373:11 (2009), 1005–1010
  125. A. I. Komech, A. E. Merzon, “On asymptotic completeness for scattering in the nonlinear Lamb system”, J. Math. Phys., 50:2 (2009), 023514, 10 pp.
  126. A. I. Komech, A. E. Merzon, “On asymptotic completeness of scattering in the nonlinear Lamb system. II”, J. Math. Phys., 54:1 (2013), 012702, 9 pp.
  127. A. Komech, A. Merzon, Stationary diffraction by wedges. Method of automorphic functions on complex characteristics, Lecture Notes in Math., 2249, Springer, Cham, 2019, xi+165 pp.
  128. A. Komech, H. Spohn, “Soliton-like asymptotics for a classical particle interacting with a scalar wave field”, Nonlinear Anal., 33:1 (1998), 13–24
  129. A. Komech, H. Spohn, “Long-time asymptotics for the coupled Maxwell–Lorentz equations”, Comm. Partial Differential Equations, 25:3-4 (2000), 559–584
  130. A. Komech, H. Spohn, M. Kunze, “Long-time asymptotics for a classical particle interacting with a scalar wave field”, Comm. Partial Differential Equations, 22:1-2 (1997), 307–335
  131. Е. А. Копылова, “Дисперсионные оценки для дискретных уравнений Шредингера и Клейна–Гордона”, Алгебра и анализ, 21:5 (2009), 87–113
  132. E. A. Kopylova, “On asymptotic stability of solitary waves in the discrete Schrödinger equation coupled to a nonlinear oscillator”, Nonlinear Anal., 71:7-8 (2009), 3031–3046
  133. E. A. Kopylova, “On asymptotic stability of solitary waves in discrete Klein–Gordon equation coupled to a nonlinear oscillator”, Appl. Anal., 89:9 (2010), 1467–1492
  134. Е. А. Копылова, “Дисперсионные оценки для уравнений Шрeдингера и Клейна–Гордона”, УМН, 65:1(391) (2010), 97–144
  135. Е. А. Копылова, “Асимптотическая устойчивость солитонов для нелинейных гиперболических уравнений”, УМН, 68:2(410) (2013), 91–144
  136. E. Kopylova, “On global attraction to solitary waves for the Klein–Gordon equation with concentrated nonlinearity”, Nonlinearity, 30:11 (2017), 4191–4207
  137. E. Kopylova, “On global attraction to stationary states for wave equation with concentrated nonlinearity”, J. Dynam. Differential Equations, 30:1 (2018), 107–116
  138. E. Kopylova, “On dispersion decay for 3D Klein–Gordon equation”, Discrete Contin. Dyn. Syst., 38:11 (2018), 5765–5780
  139. E. A. Kopylova, A. I. Komech, “Long time decay for 2D Klein–Gordon equation”, J. Funct. Anal., 259:2 (2010), 477–502
  140. E. A. Kopylova, A. I. Komech, “On asymptotic stability of moving kink for relativistic Ginzburg–Landau equation”, Comm. Math. Phys., 302:1 (2011), 225–252
  141. E. Kopylova, A. I. Komech, “On asymptotic stability of kink for relativistic Ginzburg–Landau equations”, Arch. Ration. Mech. Anal., 202:1 (2011), 213–245
  142. E. Kopylova, A. Komech, “On global attractor of 3D Klein–Gordon equation with several concentrated nonlinearities”, Dyn. Partial Differ. Equ., 16:2 (2019), 105–124
  143. E. Kopylova, A. Komech, “Global attractor for 1D Dirac field coupled to nonlinear oscillator”, Comm. Math. Phys., Publ. online 2019, 1–31
  144. E. Kopylova, G. Teschl, “Dispersion estimates for one-dimensional discrete Dirac equations”, J. Math. Anal. Appl., 434:1 (2016), 191–208
  145. V. V. Kozlov, “Kinetics of collisionless continuous medium”, Regul. Chaotic Dyn., 6:3 (2001), 235–251
  146. В. В. Козлов, О. Г. Смолянов, “Функция Вигнера и диффузия в бесстолкновительной среде, состоящей из квантовых частиц”, Теория вероятн. и ее примен., 51:1 (2006), 109–125
  147. В. В. Козлов, Д. В. Трещeв, “Слабая сходимость решений уравнения Лиувилля для нелинейных гамильтоновых систем”, ТМФ, 134:3 (2003), 388–400
  148. В. В. Козлов, Д. В. Трещeв, “Эволюция мер в фазовом пространстве нелинейных гамильтоновых систем”, ТМФ, 136:3 (2003), 496–506
  149. М. Г. Крейн, Г. К. Лангер, “О спектральной функции самосопряженного оператора в пространстве с индефинитной метрикой”, Докл. АН СССР, 152:1 (1963), 39–42
  150. J. Krieger, K. Nakanishi, W. Schlag, “Center-stable manifold of the ground state in the energy space for the critical wave equation”, Math. Ann., 361:1-2 (2015), 1–50
  151. J. Krieger, W. Schlag, Concentration compactness for critical wave maps, EMS Monogr. Math., Eur. Math. Soc., Zürich, 2012, vi+484 pp.
  152. M. Kunze, H. Spohn, “Adiabatic limit for the Maxwell–Lorentz equations”, Ann. Henri Poincare, 1:4 (2000), 625–653
  153. О. А. Ладыженская, “О принципе предельной амплитуды”, УМН, 12:3(75) (1957), 161–164
  154. G. L. Lamb, Jr., Elements of soliton theory, Pure Appl. Math., John Wiley & Sons, Inc., New York, 1980, xiii+289 pp.
  155. H. Lamb, “On a peculiarity of the wave-system due to the free vibrations of a nucleus in an extended medium”, Proc. London Math. Soc., 32 (1900), 208–211
  156. L. Landau, “On the problem of turbulence”, Докл. АН СССР, 44 (1944), 311–314
  157. H. Langer, “Spectral functions of definitizable operators in Krein spaces”, Functional analysis (Dubrovnik, 1981), Lecture Notes in Math., 948, Springer, Berlin–New York, 1982, 1–46
  158. P. D. Lax, C. S. Morawetz, R. S. Phillips, “Exponential decay of solutions of the wave equation in the exterior of a star-shaped obstacle”, Comm. Pure Appl. Math., 16:4 (1963), 477–486
  159. B. Ya. Levin, Lectures on entire functions, Transl. Math. Monogr., 150, Amer. Math. Soc., Providence, RI, 1996, xvi+248 pp.
  160. L. Lewin, Advanced theory of waveguides, Iliffe and Sons, Ltd., London, 1951, 192 pp.
  161. Ж.-Л. Лионс, Некоторые методы решения нелинейных краевых задач, Мир, М., 1972, 587 с.
  162. E. Long, D. Stuart, “Effective dynamics for solitons in the nonlinear Klein–Gordon–Maxwell system and the Lorentz force law”, Rev. Math. Phys., 21:4 (2009), 459–510
  163. Л. А. Люстерник, Л. Г. Шнирельман, Топологические методы в вариационных задачах, МГУ, М., 1930, 68 с.
  164. Л. Люстерник, Л. Шнирельман, “Топологические методы в вариационных задачах и их приложения к дифференциальной геометрии поверхностей”, УМН, 2:1(17) (1947), 166–217
  165. B. Marshall, W. Strauss, S. Wainger, “$L^{p}-L^{q}$ estimates for the Klein–Gordon equation”, J. Math. Pures Appl. (9), 59:4 (1980), 417–440
  166. Y. Martel, “Asymptotic $N$-soliton-like solutions of the subcritical and critical generalized Korteweg–de Vries equations”, Amer. J. Math., 127:5 (2005), 1103–1140
  167. Y. Martel, F. Merle, “Asymptotic stability of solitons of the subcritical gKdV equations revisited”, Nonlinearity, 18:1 (2005), 55–80
  168. Y. Martel, F. Merle, T.-P. Tsai, “Stability and asymptotic stability in the energy space of the sum of $N$ solitons for subcritical gKdV equations”, Comm. Math. Phys., 231:2 (2002), 347–373
  169. M. Merkli, I. M. Sigal, “A time-dependent theory of quantum resonances”, Comm. Math. Phys., 201:3 (1999), 549–576
  170. J. R. Miller, M. I. Weinstein, “Asymptotic stability of solitary waves for the regularized long-wave equation”, Comm. Pure Appl. Math., 49:4 (1996), 399–441
  171. C. S. Morawetz, “The limiting amplitude principle”, Comm. Pure Appl. Math., 15:3 (1962), 349–361
  172. C. S. Morawetz, “Time decay for the nonlinear Klein–Gordon equations”, Proc. Roy. Soc. London Ser. A, 306 (1968), 291–296
  173. C. S. Morawetz, W. A. Strauss, “Decay and scattering of solutions of a nonlinear relativistic wave equation”, Comm. Pure Appl. Math., 25:1 (1972), 1–31
  174. K. Nakanishi, W. Schlag, Invariant manifolds and dispersive Hamiltonian evolution equations, Zur. Lect. Adv. Math., Eur. Math. Soc., Zürich, 2011, vi+253 pp.
  175. Y. Ne'eman, “Unified interactions in the unitary gauge theory”, Nuclear Phys., 30 (1962), 347–349
  176. D. Noja, A. Posilicano, “Wave equations with concentrated nonlinearities”, J. Phys. A, 38:22 (2005), 5011–5022
  177. R. L. Pego, M. I. Weinstein, “Asymptotic stability of solitary waves”, Comm. Math. Phys., 164:2 (1994), 305–349
  178. G. Perelman, “Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations”, Comm. Partial Differential Equations, 29:7-8 (2004), 1051–1095
  179. C.-A. Pillet, C. E. Wayne, “Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations”, J. Differential Equations, 141:2 (1997), 310–326
  180. М. Рид, Б. Саймон, Методы современной математической физики, т. 3, Теория рассеяния, Мир, М., 1982, 445 с.
  181. М. Рид, Б. Саймон, Методы современной математической физики, т. 4, Анализ операторов, Мир, М., 1982, 430 с.
  182. I. Rodnianski, W. Schlag, “Time decay for solutions of Schrödinger equations with rough and time-dependent potentials”, Invent. Math., 155:3 (2004), 451–513
  183. I. Rodnianski, W. Schlag, A. Soffer, Asymptotic stability of $N$-soliton states of NLS, 2003, 70 pp.
  184. I. Rodnianski, W. Schlag, A. Soffer, “Dispersive analysis of charge transfer models”, Comm. Pure Appl. Math., 58:2 (2005), 149–216
  185. У. Рудин, Функциональный анализ, Мир, М., 1975, 443 с.
  186. Э. Шредингер, “Квантование как задача о собственных значениях. I, II, III, IV”, Избранные труды по квантовой механике, Наука, М., 1976, 9–20, 21–50, 75–115, 116–138
  187. I. Segal, “Quantization and dispersion for nonlinear relativistic equations”, Mathematical theory of elementary particles (Dedham, MA, 1965), M.I.T. Press, Cambridge, MA, 1966, 79–108
  188. I. Segal, “Dispersion for non-linear relativistic equations. II”, Ann. Sci. Ecole Norm. Sup. (4), 1:4 (1968), 459–497
  189. I. M. Sigal, “Non-linear wave and Schrödinger equations. I. Instability of periodic and quasiperiodic solutions”, Comm. Math. Phys., 153:2 (1993), 297–320
  190. A. Soffer, “Soliton dynamics and scattering”, International congress of mathematicians, v. III, Eur. Math. Soc., Zürich, 2006, 459–471
  191. A. Soffer, M. I. Weinstein, “Multichannel nonlinear scattering for nonintegrable equations”, Comm. Math. Phys., 133:1 (1990), 119–146
  192. A. Soffer, M. I. Weinstein, “Multichannel nonlinear scattering for nonintegrable equations. II. The case of anisotropic potentials and data”, J. Differential Equations, 98:2 (1992), 376–390
  193. A. Soffer, M. I. Weinstein, “Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations”, Invent. Math., 136:1 (1999), 9–74
  194. A. Soffer, M. I. Weinstein, “Selection of the ground state for nonlinear Schrödinger equations”, Rev. Math. Phys., 16:8 (2004), 977–1071
  195. H. Spohn, Dynamics of charged particles and their radiation field, Cambridge Univ. Press, Cambridge, 2004, xvi+360 pp.
  196. W. A. Strauss, “Decay and asymptotics for $square u=F(u)$”, J. Funct. Anal., 2:4 (1968), 409–457
  197. W. A. Strauss, “Existence of solitary waves in higher dimensions”, Comm. Math. Phys., 55:2 (1977), 149–162
  198. W. A. Strauss, “Nonlinear scattering theory at low energy”, J. Funct. Anal., 41:1 (1981), 110–133
  199. W. A. Strauss, “Nonlinear scattering theory at low energy: sequel”, J. Funct. Anal., 43:3 (1981), 281–293
  200. D. Stuart, “Existence and Newtonian limit of nonlinear bound states in the Einstein–Dirac system”, J. Math. Phys., 51:3 (2010), 032501, 13 pp.
  201. D. Tataru, “Local decay of waves on asymptotically flat stationary space-times”, Amer. J. Math., 135:2 (2013), 361–401
  202. R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Appl. Math. Sci., 68, 2nd ed., Springer-Verlag, New York, 1997, xxii+648 pp.
  203. E. C. Titchmarsh, “The zeros of certain integral functions”, Proc. London Math. Soc. (2), 25:1 (1926), 283–302
  204. D. Treschev, “Oscillator and thermostat”, Discrete Contin. Dyn. Syst., 28:4 (2010), 1693–1712
  205. T.-P. Tsai, “Asymptotic dynamics of nonlinear Schrödinger equations with many bound states”, J. Differential Equations, 192:1 (2003), 225–282
  206. T.-P. Tsai, H.-T. Yau, “Classification of asymptotic profiles for nonlinear Schrödinger equations with small initial data”, Adv. Theor. Math. Phys., 6:1 (2002), 107–139
  207. T.-P. Tsai, H.-T. Yau, “Asymptotic dynamics of nonlinear Schrödinger equations: resonance-dominated and dispersion-dominated solutions”, Comm. Pure Appl. Math., 55:2 (2002), 153–216
  208. M. I. Weinstein, “Modulational stability of ground states of nonlinear Schrödinger equations”, SIAM J. Math. Anal., 16:3 (1985), 472–491
  209. D. R. Yafaev, “On a zero-range interaction of a quantum particle with the vacuum”, J. Phys. A, 25:4 (1992), 963–978
  210. D. R. Yafaev, “A point interaction for the discrete Schrödinger operator and generalized Chebyshev polynomials”, J. Math. Phys., 58:6 (2017), 063511, 24 pp.
  211. K. Yajima, “Dispersive estimates for Schrödinger equations with threshold resonance and eigenvalue”, Comm. Math. Phys., 259:2 (2005), 475–509
  212. Я. Б. Зельдович, “Рассеяние сингулярным потенциалом в теории возмущений и в импульсном представлении”, ЖЭТФ, 38:3 (1960), 819–824

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