Attractors of non-linear Hamiltonian one-dimensional wave equations

A. I. Komech

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 55 (2000) no. 1, pp. 43-92

Résumé

A theory is constructed for attractors of all finite-energy solutions of conservative one-dimensional wave equations on the whole real line. The attractor of a non-degenerate (that is, generic) equation is the set of all stationary solutions. Each finite-energy solution converges as $t\to\pm\infty$ to this attractor in the Frechet topology determined by local energy seminorms. The attraction is caused by energy dissipation at infinity. Our results provide a mathematical model of Bohr transitions (“quantum jumps”) between stationary states in quantum systems.

Bibliographie

[1] Babin A. V., Vishik M. I., Attraktory evolyutsionnykh uravnenii, Nauka, M., 1989

[2] Balabane M., Cazenave T., Douady A., Merle F., “Existence of excited states for a nonlinear Dirac field”, Comm. Math. Phys., 119 (1988), 153–176 | DOI | MR | Zbl

[3] Berestycki H., Lions P. L., “Nonlinear scalar field equations. I; II”, Arch. Rational Mech. Anal., 82 (1983), 313–345; 347–375 | MR | Zbl

[4] Bohr N., “On the constitution of atoms and molecules”, Phil. Mag., 26 (1913), 1–25

[5] Buslaev V. S., Perel'man G. S., “On nonlinear scattering of states which are close to a soliton”, Astérisque, 208, 1992, 49–63 | MR

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

[7] Buslaev V. S., Perel'man G. S., “On the stability of solitary waves for nonlinear Schrödinger equations”, Amer. Math. Soc. Trans. Ser. 2, 164 (1995), 75–98 | MR | Zbl

[8] Chadam J. M., “Asymptotics for $\square u=m^2u+G(x,t,u,u_x,u_t)$. I; II”, Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. (3), 26 (1972), 33–65; 67–95 | MR | Zbl

[9] Chadam J. M., “On certain global solutions of the Cauchy problem for the (classical) coupled Klein–Gordon–Dirac equation in one and three space dimensions”, Arch. Rational Mech. Anal., 54 (1974), 223–237 | DOI | MR | Zbl

[10] Egorov Yu. E., O stabilizatsii vzaimodeistviya struny s nelineinymi ostsillyatorami, Dis. $\dots$ kand. fiz.-matem. nauk, MGU, M., 1997

[11] Egorov Yu. E., Komech A. I., “Ob attraktorakh i asimptotike reshenii v nelineinoi zadache Lemba”, Vestnik MGU. Ser. 1. Matem., mekh., 1996, no. 5, 80–88

[12] Egorov Yu. E., Komech A. I., “Stabilizatsiya vzaimodeistviya struny s nelineinym ostsillyatorom”, UMN, 50:4 (1995), 152

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

[14] Esteban M. J., Séré E., “Stationary states of the nonlinear Dirac equation: a variational approach”, Comm. Math. Phys., 171:2 (1995), 323–350 | DOI | MR | Zbl

[15] Feinman R., Leiton R., Sends M., Feinmanovskie lektsii po fizike. Elektrodinamika, Mir, M., 1966

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

[17] Ginibre J., Velo G., “Time decay of finite energy solutions of the nonlinear Klein–Gordon and Schrödinger equations”, Ann. Inst. H. Poincaré Phys. Théor., 43 (1985), 399–442 | MR | Zbl

[18] Glassey R. T., Strauss W. A., “Decay of classical Yang–Mills fields”, Comm. Math. Phys., 65 (1979), 1–13 | DOI | MR | Zbl

[19] Glassey R. T., Strauss W. A., “Decay of Yang–Mills field coupled to a scalar field”, Comm. Math. Phys., 67 (1979), 51–67 | DOI | MR | Zbl

[20] Grillakis M., Shatah J., Strauss W. A., “Stability theory of solitary waves in the presence of symmetry. I; II”, J. Funct. Anal., 74 (1987), 160–197 ; 94 (1990), 308–348 | DOI | MR | Zbl | DOI | MR | Zbl

[21] Hale J., Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, 1988 | Zbl

[22] Hörmander L., “On the fully nonlinear Cauchy problem with small data”, IMA Vol. Math. Appl., 30 (1991), 51–81 | Zbl

[23] Dzhekson D., Klassicheskaya elektrodinamika, Mir, M., 1965

[24] Keller J. B., Bonilla L. L., “Irreversibility and nonrecurrence”, J. Statist. Phys., 42:5/6 (1986), 1115–1125 | DOI | MR

[25] Khruslov E. Ya., “Asimptotika resheniya zadachi Koshi dlya uravneniya Kortevega–de Friza s nachalnymi dannymi tipa stupenki”, Matem. sb., 99:2 (1976), 261–281 | MR | Zbl

[26] Klainerman S., “Long-time behavior of solutions to nonlinear evolution equations”, Arch. Rational Mech. Anal., 78 (1982), 73–98 | DOI | MR | Zbl

[27] Komech A. I., “O stabilizatsii vzaimodeistviya struny s nelineinym ostsillyatorom”, UMN, 46:6 (1991), 166–167 | MR

[28] Komech A. I., “Stabilizatsiya vzaimodeistviya struny s nelineinym ostsillyatorom”, Vestnik MGU. Ser. 1. Matem., mekh., 1991, no. 6, 35–41 | MR

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

[30] Komech A. I., “O perekhodakh k statsionarnym sostoyaniyam v nekotorykh beskonechnomernykh gamiltonovykh sistemakh”, Dokl. RAN, 347:3 (1996), 309–311 | MR | Zbl

[31] Komech A. I., “O stabilizatsii vzaimodeistviya struny s konechnym chislom ostsillyatorov”, UMN, 49:4 (1994), 117

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

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

[34] Komech A. I., “On transitions to stationary states in Hamiltonian nonlinear wave equations”, Phys. Lett. A, 241 (1998), 311–323 | DOI | MR

[35] Komech A. I., Kunze M., Spohn H., “Effective dynamics for a mechanical particle coupled to a wave field”, Comm. Math. Phys., 203 (1999), 1–19 | DOI | MR | Zbl

[36] Komech A. I., Spohn H., “Soliton-like asymptotics for a classical particle interacting with a scalar wave field”, Nonlinear Anal., 33 (1998), 13–24 | DOI | MR | Zbl

[37] Komech A. I., Spohn H., “Long-time asymptotics for the coupled Maxwell–Lorentz equations”, Comm. Partial Differential Equations (to appear) | Zbl

[38] Komech A. I., Spohn H., Kunze M., “Long-time asymptotics for a classical particle interacting with a scalar wave field”, Comm. Partial Differential Equations, 22:1/2 (1997), 307–335 | MR | Zbl

[39] Komech A. I., Vainberg B. R., “On asymptotic stability of stationary solutions to nonlinear wave and Klein–Gordon equations”, Arch. Rational Mech. Anal., 134 (1996), 227–248 | DOI | MR | Zbl

[40] Ladyzhenskaya O. A., “Ob attraktorakh nelineinykh evolyutsionnykh dissipativnykh zadach”, Zap. nauch. semin. LOMI, 18, 1986, 72–85

[41] Lamb H., “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

[42] Lax P. D., Morawetz C. S., Phillips R. S., “Exponential decay of solutions of the wave equation in the exterior of a star-shaped obstacle”, Comm. Pure Appl. Math., 16 (1963), 477–486 | DOI | MR | Zbl

[43] Laks P., Fillips R., Teoriya rasseyaniya, Mir, M., 1971

[44] Morawetz C. S., “The decay of solutions to exterior initial-boundary value problem for the wave equation”, Comm. Pure Appl. Math., 14 (1961), 561–568 | DOI | MR | Zbl

[45] Morawetz C. S., Strauss W. A., “Decay and scattering of solutions of a nonlinear relativistic wave equation”, Comm. Pure Appl. Math., 25 (1972), 1–31 | DOI | MR | Zbl

[46] Ne'emann Y., “Derivation of strong interactions from a gauge invariance”, Nuclear Phys., 26 (1961), 222–229 | DOI | MR

[47] Novokshenov V. Yu., “Asimptotika pri $t\to\infty$ resheniya zadachi Koshi dlya nelineinogo uravneniya Shrëdingera”, Dokl. AN SSSR, 251:4 (1980), 799–801 | MR

[48] Pillet C.-A., Wayne C. E., “Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations”, J. Differential Equations, 141:2 (1997), 310–326 | DOI | MR | Zbl

[49] Reed M., Abstract non-linear wave equations, Lecture Notes in Math., 507, Springer-Verlag, Berlin, 1976 | Zbl

[50] Roffman E. H., “Localized solutions of nonlinear wave equations”, Bull. Amer. Math. Soc., 76 (1970), 70–71 | DOI | MR | Zbl

[51] Sazonov L. I., Yudovich V. I., “Ustoichivost statsionarnykh reshenii parabolicheskikh uravnenii i uravneniya Nave–Stoksa vo vsem prostranstve”, Sib. matem. zhurn., 29:1 (1988), 151–158 | MR

[52] Shrëdinger E., Chto takoe zhizn?, IL, M., 1947

[53] Segal I., “Dispersion for nonlinear relativistic wave equations”, Ann. Sci. Ecole Norm. Sup., 1 (1968), 459–497 | MR | Zbl

[54] Shabat A. B., “Ob uravneniyakh Kortevega–de Frisa”, Dokl. AN SSSR, 211:6 (1973), 1310–1313 | MR | Zbl

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

[56] Soffer A., Weinstein M. I., “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

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

[58] Strauss W. A., Nonlinear Invariant Wave Equations, Lecture Notes in Phys., 73, Springer-Verlag, Berlin, 1978 | Zbl

[59] Sukhanov V. V., “Asimptoticheskoe povedenie reshenii zadachi Koshi dlya sistemy tipa KdV pri bolshikh vremenakh”, Dokl. AN SSSR, 269:5 (1983), 1091–1094 | MR | Zbl

[60] Temam R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci., 68, Springer-Verlag, Berlin, 1988 | Zbl

[61] Vainberg B. R., “Ob analiticheskikh svoistvakh rezolventy dlya nekotorogo klassa puchkov operatorov”, Matem. sb., 77 (1968), 259–296 | MR | Zbl

[62] Vainberg B. R., “Povedenie reshenii zadachi Koshi dlya giperbolicheskikh uravnenii pri $t\to\infty$”, Matem. sb., 78:4 (1969), 542–578 | MR | Zbl

[63] Vainberg B. R., “Asimptoticheskoe povedenie pri $t\to\infty$ reshenii vneshnikh smeshannykh zadach dlya giperbolicheskikh uravnenii i kvaziklassika”, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 34, VINITI, M., 1991, 57–92

[64] Vainberg B. R., Asimptoticheskie metody v uravneniyakh matematicheskoi fiziki, Izd-vo MGU, M., 1982 | Zbl

[65] Weinberg S., “Conservation of isotopic spin and isotopic gauge invariant”, Rev. Modern Phys., 52 (1980), 515 | DOI | MR

[66] Wheeler J. A., Feynman R. P., “Interaction with the absorber as the mechanism of radiation”, Rev. Modern Phys., 17 (1945), 157–181 | DOI

[67] Yang C. N., Mills R. L., Phys. Rev., 96:1 (1954), 191–195 | DOI