Attractors of nonlinear Hamiltonian partial differential equations

A. I. Komech ; E. A. Kopylova

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 75 (2020) no. 1, pp. 1-87

Résumé

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.

Détail

Keywords: Hamiltonian equations, nonlinear partial differential equations, wave equation, Maxwell equations, limiting amplitude principle, limiting absorption principle, attractor, steady states, stationary orbits, adiabatic effective dynamics, symmetry group, Schrödinger equation, wave-particle duality.
Mots-clés : Klein–Gordon equation, soliton, Lie group, quantum transitions

Bibliographie

[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. | Zbl

[3] R. K. Adair, E. C. Fowler, Strange particles, Intersci. Publ. John Wiley Sons, New York–London, 1963, viii+151 pp. | MR | Zbl

[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. | DOI | MR | Zbl

[5] S. Agmon, “Spectral properties of Schrödinger operators and scattering theory”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2:2 (1975), 151–218 | MR | Zbl

[6] S. Albeverio, R. Figari, “Quantum fields and point interactions”, Rend. Mat. Appl. (7), 39:2 (2018), 161–180 | MR | Zbl

[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 | DOI | MR | Zbl

[8] A. V. Babin, M. I. Vishik, Attractors of evolution equations, Stud. Math. Appl., 25, North-Holland Publishing Co., Amsterdam, 1992, x+532 pp. | MR | MR | Zbl | Zbl

[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 Poincaré, 14:6 (2013), 1573–1597 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[11] D. Bambusi, L. Galgani, “Some rigorous results on the Pauli–Fierz model of classical electrodynamics”, Ann. Inst. H. Poincaré Phys. Théor., 58:2 (1993), 155–171 | MR | Zbl

[12] V. E. Barnes et al., “Observation of a hyperon with strangeness minus three”, Phys. Rev. Lett., 12:8 (1964), 204–206 | DOI

[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 | DOI | MR | Zbl

[14] M. Beceanu, M. Goldberg, “Schrödinger dispersive estimates for a scaling-critical class of potentials”, Comm. Math. Phys., 314:2 (2012), 471–481 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[16] A. Bensoussan, C. Iliine, A. Komech, “Breathers for a relativistic nonlinear wave equation”, Arch. Ration. Mech. Anal., 165:4 (2002), 317–345 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[19] F. A. Berezin, L. D. Faddeev, “A remark on Schrödinger's equation with a singular potential”, Soviet Math. Dokl., 2 (1961), 372–375 | MR | Zbl

[20] N. Bohr, “On the constitution of atoms and molecules. I”, Philos. Mag. (6), 26:151 (1913), 1–25 ; “II”:153, 476–502 ; “III”:155, 857–875 | DOI | Zbl | DOI | DOI

[21] N. Bohr, “Discussions with Einstein on epistemological problems in atomic physics”, Albert Einstein: philosopher-scientist, The Library of Living Philosophers, 7, The Library of Living Philosophers, Inc., Evaston, IL, 1949, 201–241 | DOI

[22] N. Boussaid, “Stable directions for small nonlinear Dirac standing waves”, Comm. Math. Phys., 268:3 (2006), 757–817 | DOI | MR | Zbl

[23] N. Boussaid, S. Cuccagna, “On stability of standing waves of nonlinear Dirac equations”, Comm. Partial Differential Equations, 37:6 (2012), 1001–1056 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[25] V. S. Buslaev, G. S. Perel'man, “Scattering for the nonlinear Schrödinger equation: states close to a soliton”, St. Petersburg Math. J., 4:6 (1993), 1111–1142 | MR | Zbl

[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 | DOI | MR | Zbl

[27] V. S. Buslaev, C. Sulem, “On asymptotic stability of solitary waves for nonlinear Schrödinger equations”, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20:3 (2003), 419–475 | DOI | MR | Zbl

[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. | MR | Zbl

[29] G. M. Coclite, V. Georgiev, “Solitary waves for Maxwell–Schrödinger equations”, Electron. J. Differential Equations, 2004 (2004), 94, 31 pp. (electronic) | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[32] F. H. J. Cornish, “Classical radiation theory and point charges”, Proc. Phys. Soc., 86:3 (1965), 427–442 | DOI | MR

[33] S. Cuccagna, “Stabilization of solutions to nonlinear Schrödinger equations”, Comm. Pure Appl. Math., 54:9 (2001), 1110–1145 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[37] P. D'Ancona, “Kato smoothing and Strichartz estimates for wave equations with magnetic potentials”, Comm. Math. Phys., 335:1 (2015), 1–16 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[40] P. A. M. Dirac, “Classical theory of radiating electrons”, Proc. Roy. Soc. London Ser. A, 167:929 (1938), 148–169 | DOI | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[45] A. V. Dymov, “Dissipative effects in a linear Lagrangian system with infinitely many degrees of freedom”, Izv. Math., 76:6 (2012), 1116–1149 | DOI | DOI | MR | Zbl

[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. | MR | Zbl

[47] I. E. Egorova, E. A. Kopylova, V. A. Marchenko, G. Teschl, “Dispersion estimates for one-dimensional Schrödinger and Klein–Gordon equations revisited”, Russian Math. Surveys, 71:3 (2016), 391–415 | DOI | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[49] A. Einstein, “Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?”, Ann. der Phys. (4), 18 (1905), 639–641 | DOI

[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 | DOI | MR | Zbl

[51] M. J. Esteban, V. Georgiev, E. Séré, “Stationary solutions of the Maxwell–Dirac and the Klein–Gordon–Dirac equations”, Calc. Var. Partial Differential Equations, 4:3 (1996), 265–281 | DOI | MR | Zbl

[52] R. P. Feynman, R. B. Leighton, M. Sands, The Feynman lectures on physics, v. 2, Mainly electromagnetism and matter, Addison-Wesley Publishing Co., Inc., Reading, MA–London, 1964, xii+569 pp. | MR | MR | MR | MR | Zbl

[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. | DOI | MR | Zbl

[54] J. Fröhlich, Z. Gang, “Emission of Cherenkov radiation as a mechanism for Hamiltonian friction”, Adv. Math., 264 (2014), 183–235 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[57] G. I. Gaudry, “Quasimeasures and operators commuting with convolution”, Pacific J. Math., 18:3 (1966), 461–476 | DOI | MR | Zbl

[58] M. Gell-Mann, “Symmetries of baryons and mesons”, Phys. Rev. (2), 125:3 (1962), 1067–1084 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[64] J. K. Hale, Asymptotic behavior of dissipative systems, Math. Surveys Monogr., 25, Amer. Math. Soc., Providence, RI, 1988, x+198 pp. | MR | Zbl

[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 | DOI | MR | Zbl

[67] A. Haraux, Systèmes dynamiques dissipatifs et applications, Rech. Math. Appl., 17, Masson, Paris, 1991, xii+132 pp. | MR | Zbl

[68] W. Heisenberg, “Der derzeitige Stand der nichtlinearen Spinortheorie der Elementarteilchen”, Acta Phys. Austriaca, 14 (1961), 328–339 | MR | Zbl

[69] W. Heisenberg, Introduction to the unified field theory of elementary particles, Intersci. Publ. John Wiley Sons, London–New York–Sydney, 1966, ix+177 pp. | Zbl

[70] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Math., 840, Springer-Verlag, Berlin–New York, 1981, iv+348 pp. | DOI | MR | MR | Zbl | Zbl

[71] E. Hopf, “Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen”, Math. Nachr., 4 (1951), 213–231 | DOI | MR | Zbl

[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. ; L. Khermander, Analiz lineinykh differentsialnykh operatorov s chastnymi proizvodnymi, v. 1, Teoriya raspredelenii i analiz Fure, Mir, M., 1986, 462 pp. | DOI | MR | Zbl | MR | Zbl

[73] L. Houllevigue, L'évolution des sciences, Librairie A. Collin, Paris, 1908, xi+287 pp.

[74] V. M. Imaykin, “Soliton asymptotics for systems of ‘field-particle’ type”, Russian Math. Surveys, 68:2 (2013), 227–281 | DOI | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[76] V. Imaykin, A. Komech, N. Mauser, “Soliton-type asymptotics for the coupled Maxwell–Lorentz equations”, Ann. Henri Poincaré, 5:6 (2004), 1117–1135 | DOI | MR | Zbl

[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 | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[82] V. Imaykin, A. Komech, B. Vainberg, “Scattering of solitons for coupled wave-particle equations”, J. Math. Anal. Appl., 389:2 (2012), 713–740 | DOI | MR | Zbl

[83] J. D. Jackson, Classical electrodynamics, 3rd ed., John Wiley Sons, Inc., New York, 1999, xxi+808 pp. ; Dzh. Dzhekson, Klassicheskaya elektrodinamika, Mir, M., 1965, 704 pp. | MR | Zbl

[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 | DOI | MR | Zbl

[85] K. Jörgens, “Das Anfangswertproblem im Großen für eine Klasse nichtlinearer Wellengleichungen”, Math. Z., 77 (1961), 295–308 | DOI | MR | Zbl

[86] J.-L. Journé, A. Soffer, C. D. Sogge, “Decay estimates for Schrödinger operators”, Comm. Pure Appl. Math., 44:5 (1991), 573–604 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[92] A. A. Komech, A. I. Komech, “On the Titchmarsh convolution theorem for distributions on the circle”, Funct. Anal. Appl., 47:1 (2013), 21–26 | DOI | DOI | MR | Zbl

[93] A. I. Komech, “On the stabilization of interaction of a string with a nonlinear oscillator”, Moscow Univ. Math. Bull., 46:6 (1991), 34–39 | MR | Zbl

[94] A. I. Komech, “Linear partial differential equations with constant coefficients”, Partial differential equations II, Encyclopaedia Math. Sci., 31, Springer, Berlin, 1994, 121–255 | DOI | MR | MR | Zbl | Zbl

[95] A. I. Komech, “On stabilization of string-nonlinear oscillator interaction”, J. Math. Anal. Appl., 196:1 (1995), 384–409 | DOI | MR | Zbl

[96] A. I. Komech, “On the stabilization of string-oscillator interaction”, Russ. J. Math. Phys., 3:2 (1995), 227–247 | MR | Zbl

[97] A. Komech, “On transitions to stationary states in one-dimensional nonlinear wave equations”, Arch. Ration. Mech. Anal., 149:3 (1999), 213–228 | DOI | MR | Zbl

[98] A. I. Komech, “Attractors of non-linear Hamiltonian one-dimensional wave equations”, Russian Math. Surveys, 55:1 (2000), 43–92 | DOI | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[100] A. Komech, Quantum mechanics: genesis and achievements, Springer, Dordrecht, 2013, xviii+285 pp. | DOI | MR | Zbl

[101] A. Komech, “Attractors of Hamilton nonlinear PDEs”, Discrete Contin. Dyn. Syst., 36:11 (2016), 6201–6256 | DOI | MR | Zbl

[102] A. Komech, “Quantum jumps and attractors of Maxwell–Schrödinger equations”, Nonlinearity (to appear); 2019, 14 pp., arXiv: 1907.04297

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[106] A. Komech, A. Komech, “Global attraction to solitary waves for Klein–Gordon equation with mean field interaction”, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26:3 (2009), 855–868 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[110] A. I. Komech, E. A. Kopylova, “Weighted energy decay for 1D Klein–Gordon equation”, Comm. Partial Differential Equations, 35:2 (2010), 353–374 | DOI | MR | Zbl

[111] A. I. Komech, E. A. Kopylova, “Weighted energy decay for 3D Klein–Gordon equation”, J. Differential Equations, 248:3 (2010), 501–520 | DOI | MR | Zbl

[112] A. Komech, E. Kopylova, Dispersion decay and scattering theory, John Wiley Sons, Inc., Hoboken, NJ, 2012, xxvi+175 pp. | DOI | MR | Zbl

[113] A. I. Komech, E. A. Kopylova, “Dispersion decay for the magnetic Schrödinger equation”, J. Funct. Anal., 264:3 (2013), 735–751 | DOI | MR | Zbl

[114] A. Komech, E. Kopylova, “On eigenfunction expansion of solutions to the Hamilton equations”, J. Stat. Phys., 154:1-2 (2014), 503–521 | DOI | MR | Zbl

[115] A. I. Komech, E. A. Kopylova, “Weighted energy decay for magnetic Klein–Gordon equation”, Appl. Anal., 94:2 (2015), 218–232 | DOI | MR | Zbl

[116] A. Komech, E. Kopylova, “On the eigenfunction expansion for Hamilton operators”, J. Spectr. Theory, 5:2 (2015), 331–361 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | MR | Zbl

[124] A. I. Komech, A. E. Merzon, “Scattering in the nonlinear Lamb system”, Phys. Lett. A, 373:11 (2009), 1005–1010 | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[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. | DOI | Zbl

[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 | DOI | MR | Zbl

[129] A. Komech, H. Spohn, “Long-time asymptotics for the coupled Maxwell–Lorentz equations”, Comm. Partial Differential Equations, 25:3-4 (2000), 559–584 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[131] E. A. Kopylova, “Dispersion estimates for discrete Schrödinger and Klein–Gordon equations”, St. Petersburg Math. J., 21:5 (2010), 743–760 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[134] E. A. Kopylova, “Dispersive estimates for the Schrödinger and Klein–Gordon equations”, Russian Math. Surveys, 65:1 (2010), 95–142 | DOI | DOI | MR | Zbl

[135] E. A. Kopylova, “Asymptotic stability of solitons for nonlinear hyperbolic equations”, Russian Math. Surveys, 68:2 (2013), 283–334 | DOI | DOI | MR | Zbl

[136] E. Kopylova, “On global attraction to solitary waves for the Klein–Gordon equation with concentrated nonlinearity”, Nonlinearity, 30:11 (2017), 4191–4207 | DOI | MR | Zbl

[137] E. Kopylova, “On global attraction to stationary states for wave equation with concentrated nonlinearity”, J. Dynam. Differential Equations, 30:1 (2018), 107–116 | DOI | MR | Zbl

[138] E. Kopylova, “On dispersion decay for 3D Klein–Gordon equation”, Discrete Contin. Dyn. Syst., 38:11 (2018), 5765–5780 | DOI | MR | Zbl

[139] E. A. Kopylova, A. I. Komech, “Long time decay for 2D Klein–Gordon equation”, J. Funct. Anal., 259:2 (2010), 477–502 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[143] E. Kopylova, A. Komech, “Global attractor for 1D Dirac field coupled to nonlinear oscillator”, Comm. Math. Phys., Publ. online 2019, 1–31 | DOI

[144] E. Kopylova, G. Teschl, “Dispersion estimates for one-dimensional discrete Dirac equations”, J. Math. Anal. Appl., 434:1 (2016), 191–208 | DOI | MR | Zbl

[145] V. V. Kozlov, “Kinetics of collisionless continuous medium”, Regul. Chaotic Dyn., 6:3 (2001), 235–251 | DOI | MR | Zbl

[146] V. V. Kozlov, O. G. Smolyanov, “Wigner function and diffusion in a collision-free medium of quantum particles”, Theory Probab. Appl., 51:1 (2007), 168–181 | DOI | DOI | MR | Zbl

[147] V. V. Kozlov, D. V. Treshchev, “Weak convergence of solutions of the Liouville equation for nonlinear Hamiltonian systems”, Theor. Math. Phys., 134:3 (2003), 339–350 | DOI | DOI | MR | Zbl

[148] V. V. Kozlov, D. V. Treshchev, “Evolution of measures in the phase space of nonlinear Hamiltonian systems”, Theor. Math. Phys., 136:3 (2003), 1325–1335 | DOI | DOI | MR | Zbl

[149] M. G. Kreĭn, G. K. Langer, “The spectral function of a selfadjoint operator in a space with indefinite metric”, Soviet Math. Dokl., 4 (1963), 1236–1239 | MR | Zbl

[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 | DOI | MR | Zbl

[151] J. Krieger, W. Schlag, Concentration compactness for critical wave maps, EMS Monogr. Math., Eur. Math. Soc., Zürich, 2012, vi+484 pp. | DOI | MR | Zbl

[152] M. Kunze, H. Spohn, “Adiabatic limit for the Maxwell–Lorentz equations”, Ann. Henri Poincaré, 1:4 (2000), 625–653 | DOI | MR | Zbl

[153] O. A. Ladyzhenskaya, “O printsipe predelnoi amplitudy”, UMN, 12:3(75) (1957), 161–164 | MR | Zbl

[154] G. L. Lamb, Jr., Elements of soliton theory, Pure Appl. Math., John Wiley Sons, Inc., New York, 1980, xiii+289 pp. | MR | Zbl

[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 | DOI | MR | Zbl

[156] L. Landau, “On the problem of turbulence”, Dokl. AN SSSR, 44 (1944), 311–314 | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[159] B. Ya. Levin, Lectures on entire functions, Transl. Math. Monogr., 150, Amer. Math. Soc., Providence, RI, 1996, xvi+248 pp. | MR | Zbl

[160] L. Lewin, Advanced theory of waveguides, Iliffe and Sons, Ltd., London, 1951, 192 pp.

[161] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris; Gauthier-Villars, Paris, 1969, xx+554 pp. | MR | MR | Zbl | Zbl

[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 | DOI | MR | Zbl

[163] L. Lusternik, L. Schnirelmann, Méthodes topologiques dans les problèmes variationnels, Actualités Sci. Indust., 188, Hermann, Paris, 1934, 51 pp. | Zbl | Zbl

[164] L. Lyusternik, L. Shnirelman, “Topologicheskie metody v variatsionnykh zadachakh i ikh prilozheniya k differentsialnoi geometrii poverkhnostei”, UMN, 2:1(17) (1947), 166–217 | MR

[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 | MR | Zbl

[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 | DOI | MR | Zbl

[167] Y. Martel, F. Merle, “Asymptotic stability of solitons of the subcritical gKdV equations revisited”, Nonlinearity, 18:1 (2005), 55–80 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[169] M. Merkli, I. M. Sigal, “A time-dependent theory of quantum resonances”, Comm. Math. Phys., 201:3 (1999), 549–576 | DOI | MR | Zbl

[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 | 3.0.CO;2-7 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[171] C. S. Morawetz, “The limiting amplitude principle”, Comm. Pure Appl. Math., 15:3 (1962), 349–361 | DOI | MR | Zbl

[172] C. S. Morawetz, “Time decay for the nonlinear Klein–Gordon equations”, Proc. Roy. Soc. London Ser. A, 306 (1968), 291–296 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[174] K. Nakanishi, W. Schlag, Invariant manifolds and dispersive Hamiltonian evolution equations, Zur. Lect. Adv. Math., Eur. Math. Soc., Zürich, 2011, vi+253 pp. | DOI | MR | Zbl

[175] Y. Ne'eman, “Unified interactions in the unitary gauge theory”, Nuclear Phys., 30 (1962), 347–349 | DOI | MR

[176] D. Noja, A. Posilicano, “Wave equations with concentrated nonlinearities”, J. Phys. A, 38:22 (2005), 5011–5022 | DOI | MR | Zbl

[177] R. L. Pego, M. I. Weinstein, “Asymptotic stability of solitary waves”, Comm. Math. Phys., 164:2 (1994), 305–349 | DOI | MR | Zbl

[178] G. Perelman, “Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations”, Comm. Partial Differential Equations, 29:7-8 (2004), 1051–1095 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[180] M. Reed, B. Simon, Methods of modern mathematical physics, v. III, Scattering theory, Academic Press [Harcourt Brace Jovanovich, Publishers], New York–London, 1979, xv+463 pp. | MR | MR | Zbl | Zbl

[181] M. Reed, B. Simon, Methods of modern mathematical physics, v. IV, Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York–London, 1978, xv+396 pp. | MR | MR | Zbl | Zbl

[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 | DOI | MR | Zbl

[183] I. Rodnianski, W. Schlag, A. Soffer, Asymptotic stability of $N$-soliton states of NLS, 2003, 70 pp., arXiv: math/0309114

[184] I. Rodnianski, W. Schlag, A. Soffer, “Dispersive analysis of charge transfer models”, Comm. Pure Appl. Math., 58:2 (2005), 149–216 | DOI | MR | Zbl

[185] W. Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., 1973, xiii+397 pp. | MR | MR | Zbl

[186] E. Schrödinger, “Quantisierung als Eigenwertproblem. I”, Ann. Phys., 79(384):4 (1926), 361–376 ; “II”, 79(384):6 (1926), 489–527 ; “III”, 80(385):13 (1926), 437–490 ; “IV”, 81(386):18 (1926), 109–139 | DOI | DOI | DOI | DOI | Zbl | Zbl | Zbl | Zbl

[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 | MR

[188] I. Segal, “Dispersion for non-linear relativistic equations. II”, Ann. Sci. École Norm. Sup. (4), 1:4 (1968), 459–497 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[190] A. Soffer, “Soliton dynamics and scattering”, International congress of mathematicians, v. III, Eur. Math. Soc., Zürich, 2006, 459–471 | MR | Zbl

[191] A. Soffer, M. I. Weinstein, “Multichannel nonlinear scattering for nonintegrable equations”, Comm. Math. Phys., 133:1 (1990), 119–146 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[193] A. Soffer, M. I. Weinstein, “Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations”, Invent. Math., 136:1 (1999), 9–74 | DOI | MR | Zbl

[194] A. Soffer, M. I. Weinstein, “Selection of the ground state for nonlinear Schrödinger equations”, Rev. Math. Phys., 16:8 (2004), 977–1071 | DOI | MR | Zbl

[195] H. Spohn, Dynamics of charged particles and their radiation field, Cambridge Univ. Press, Cambridge, 2004, xvi+360 pp. | DOI | MR | Zbl

[196] W. A. Strauss, “Decay and asymptotics for $\square u=F(u)$”, J. Funct. Anal., 2:4 (1968), 409–457 | DOI | MR | Zbl

[197] W. A. Strauss, “Existence of solitary waves in higher dimensions”, Comm. Math. Phys., 55:2 (1977), 149–162 | DOI | MR | Zbl

[198] W. A. Strauss, “Nonlinear scattering theory at low energy”, J. Funct. Anal., 41:1 (1981), 110–133 | DOI | MR | Zbl

[199] W. A. Strauss, “Nonlinear scattering theory at low energy: sequel”, J. Funct. Anal., 43:3 (1981), 281–293 | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[201] D. Tataru, “Local decay of waves on asymptotically flat stationary space-times”, Amer. J. Math., 135:2 (2013), 361–401 | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[203] E. C. Titchmarsh, “The zeros of certain integral functions”, Proc. London Math. Soc. (2), 25:1 (1926), 283–302 | DOI | MR | Zbl

[204] D. Treschev, “Oscillator and thermostat”, Discrete Contin. Dyn. Syst., 28:4 (2010), 1693–1712 | DOI | MR | Zbl

[205] T.-P. Tsai, “Asymptotic dynamics of nonlinear Schrödinger equations with many bound states”, J. Differential Equations, 192:1 (2003), 225–282 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[208] M. I. Weinstein, “Modulational stability of ground states of nonlinear Schrödinger equations”, SIAM J. Math. Anal., 16:3 (1985), 472–491 | DOI | MR | Zbl

[209] D. R. Yafaev, “On a zero-range interaction of a quantum particle with the vacuum”, J. Phys. A, 25:4 (1992), 963–978 | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[211] K. Yajima, “Dispersive estimates for Schrödinger equations with threshold resonance and eigenvalue”, Comm. Math. Phys., 259:2 (2005), 475–509 | DOI | MR | Zbl

[212] Ya. B. Zel'dovich, “Scattering by a singular potential in perturbation theory and in the momentum representation”, Soviet Physics. JETP, 11:3 (1960), 594–597 | MR