Long range scattering for nonlinear Schrödinger equations in one space dimension. (English) Zbl 0742.35043

Summary: We consider the scattering problem for the nonlinear Schrödinger equation in \(1+1\) dimensions: \[ i\partial_ tu+(1/2)\partial^ 2u=\lambda| u|^ 2u+\mu| u| ^{p- 1}u,\quad(t,x)\in\mathbb{R}\times\mathbb{R}, (*) \] where \(\partial=\partial/\partial x\), \(\lambda\in\mathbb{R}\backslash\{0\}\), \(\mu\in\mathbb{R}\), \(p>3\). We show that modified wave operators for (*) exist on a dense set of a neighborhood of zero in the Lebesgue space \(L^ 2(\mathbb{R})\) or in the Sobolev space \(H^ 1(\mathbb{R})\). The modified wave operators are introduced in order to control the long range nonlinearity \(\lambda| u|^ 2u\).


35P25 Scattering theory for PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
Full Text: DOI


[1] Barab, J. E.: Nonexistence of asymptotically free solutions for a nonlinear Schrödinger equation. J. Math. Phys.25, 3270–3273 (1984) · Zbl 0554.35123 · doi:10.1063/1.526074
[2] Cazenave, T.: An introduction to nonlinear Schrödinger equations. Textos de Métodos Matemáticas vol.22, Instituto de Matemática, Rio de Janeiro (1989) · Zbl 0694.35170
[3] Cazenave, T., Weissler, F. B.: The Cauchy problem for the critical nonlinear Schrödinger equation inH s , Nonlinear Analysis. Theory Math. Appl.14, 807–836 (1990) · Zbl 0706.35127
[4] Dong, G. C., Li, S. J.: On the initial value problem for a nonlinear Schrödinger equation. J. Differ. Eqs.42, 353–365 (1981) · Zbl 0475.35036 · doi:10.1016/0022-0396(81)90109-1
[5] Flato, M., Simon, J., Taflin, E.: On global solutions of the Maxwell-Dirac equations. Commun. Math. Phys.112, 21–49 (1987) · Zbl 0641.35064 · doi:10.1007/BF01217678
[6] Friedman, A.: Partial Differential Equations, New York: Holt-Rinehart and Winston 1969 · Zbl 0224.35002
[7] Ginibre, J.: Unpublished, untitled note
[8] Ginibre, J., Velo, G.: On a class of non-linear Schrödinger equations III. Special theories in dimensions 1,2 and 3, Ann. Inst. Henri Poincaré, Physique théorique28, 287–316 (1978) · Zbl 0397.35012
[9] Ginibre, J., Velo, G.: Scattering theory in the energy space for a class of nonlinear Schrödinger equations. J. Math. Pures Appl.64, 363–401 (1985) · Zbl 0535.35069
[10] Glassey, R. T.: On the blowing-up solutions to the Cauchy problem for nonlinear Schrödinger equations. J. Math. Phys.18, 1794–1797 (1977) · Zbl 0372.35009 · doi:10.1063/1.523491
[11] Hayashi, N., Nakamitsu, K., Tsutsumi, M.: On solutions of the initial value problem for the nonlinear Schrödinger equations in one space dimension. Math. Z.192, 637–650 (1986) · Zbl 0617.35025 · doi:10.1007/BF01162710
[12] Hayashi, N., Ozawa, T.: Scattering theory in the weightedL 2(R n ) spaces for some Schrödinger equations. Ann. Inst. Henri Poincaré, Physique théorique48, 17–37 (1988)
[13] Hayashi, N., Tsutsumi, M.:L R n )-decay of classical solutions for nonlinear Schrödinger equations. Proceedings of the Royal Society of Edinburgh104, 309–327 (1986) · Zbl 0651.35014
[14] Hayashi, N., Tsutsumi, Y.: Sacattering theory for Hartree type equations. Ann. Inst.Henri Poincaré, Physique thérique46, 187–213 (1987)
[15] Kato, T.: On nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré, Physique théorique46, 113–129 (1987)
[16] Kato, T.: Nonlinear Schrödinger equations. In: Schrödinger Operators. Holden, H., Jensen, A. (eds.). Lecture Notes in Physics vol.345 Berlin, Heidelberg, New York: Springer 1989 · Zbl 0698.35131
[17] Makhankov, V. G.: Dynamics of classical solitons (in non-integrable systems). Phys. Reps.35, 1–128 (1978) · doi:10.1016/0370-1573(78)90074-1
[18] Reed, M., Simon, B.: Methods of Modern Mathematical Physics, III, Scattering theory. New York: Academic Press 1979 · Zbl 0405.47007
[19] Strauss, W. A.: Dispersion of low-energy waves for two conservative equations. Arch. Rat. Mech. Anal.55, 86–92 (1974) · Zbl 0289.35048 · doi:10.1007/BF00282435
[20] Strauss, W. A.: Nonlinear scattering theory at low energy. J. Funct. Anal.41, 110–133 (1981) · Zbl 0466.47006 · doi:10.1016/0022-1236(81)90063-X
[21] Thornhill, S. G., ter Haar, D.: Langmuir turbulence and modulational instability. Phys. Rep.43, 43–99 (1978) · doi:10.1016/0370-1573(78)90142-4
[22] Tsutsumi, Y.: Scattering problem for nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré, Physique théorique43, 321–347 (1985) · Zbl 0612.35104
[23] Tsutsumi, Y.: Global Existence and Asymptotic Behavior of Solutions for Nonlinear Schrödinger Equations. Doctoral Thesis, University of Tokyo (1985) · Zbl 0612.35104
[24] Tsutsumi, Y.L 2-solutions for nonlinear Schrödinger equations and nonlinear groups. Funkcialaj Ekvacioj30, 115–125 (1987) · Zbl 0638.35021
[25] Tsutsumi, Y., Yajima, K.: The asymptotic behavior of nonlinear Schrödinger equations. Bull. Am. Math. Soc.11, 186–188 (1984) · Zbl 0555.35028 · doi:10.1090/S0273-0979-1984-15263-7
[26] Weinstein, M. I.: Nonlinear Schrödinger equations and sharp interpolation estimates. Commun. Math. Phys.87, 567–576 (1983) · Zbl 0527.35023 · doi:10.1007/BF01208265
[27] Yajima, K.: Existence of solutions for Schrödinger evolution equations. Commun. Math. Phys.110, 415–426 (1987) · Zbl 0638.35036 · doi:10.1007/BF01212420
[28] Zakharov, V. E.: Collapse of Langmuir waves. Sov. Phys. JETP35, 908–914 (1972)
[29] Zakharov, V. E., Shabat, A. B.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP37, 62–69 (1972)
[30] Zakharov, V. E., Shabat, A. B.: Interaction between solutons in a stable medium. Sov. Phys. JETP37, 823–828 (1973)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.