Large Time Behavior for the Cubic Nonlinear Schrödinger Equation

Hayashi, Nakao ; Naumkin, Pavel I.

Canadian journal of mathematics, Tome 54 (2002) no. 5, pp. 1065-1085

Résumé

We consider the Cauchy problem for the cubic nonlinear Schrödinger equation in one space dimension 1 $$\left\{ \begin{align}& i{{u}_{t}}\,+\,\frac{1}{2}{{u}_{xx}}\,+\,{{{\bar{u}}}^{3}}\,=\,0,\,\,\,\,\,t\,\in \,\mathbf{R},\,x\,\in \,\mathbf{R}, \\& u(0,\,x)\,=\,{{u}_{0}}(x),\,\,\,\,\,\,\,\,\,\,\,x\,\in \,\mathbf{R}. \\ \end{align} \right.$$ Cubic type nonlinearities in one space dimension heuristically appear to be critical for large time. We study the global existence and large time asymptotic behavior of solutions to the Cauchy problem (1). We prove that if the initial data ${{u}_{0}}\,\in \,{{\mathbf{H}}^{1,\,0}}\,\cap \,{{\mathbf{H}}^{0,\,1}}$ are small and such that ${{\sup }_{|\xi |\le 1}}\left| \arg \mathcal{F}{{u}_{0}}(\text{ }\xi \text{ )}\text{0}\frac{\pi n}{2} \right|<\frac{\pi }{8}$ for some n ∈ Z, and ${{\inf }_{\left| \xi\right|\le 1}}\,\left| \mathcal{F}{{u}_{0}}(\xi ) \right|\,>\,0$ , then the solution has an additional logarithmic timedecay in the short range region $\left| x \right|\,\le \,\sqrt{t}$ . In the far region $\left| x \right|\,>\,\sqrt{t}$ the asymptotics have a quasilinear character.

Détail

DOI : 10.4153/CJM-2002-039-3

Mots-clés : 35Q55

Bibliographie

[1] [1] Ginibre, J. and Velo, G., On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case; II Scattering theory, general case. J. Funct. Anal. 32 (1979), 1–71. Google Scholar

[2] [2] Ginibre, J. and Ozawa, T., Long range scattering for nonlinear Schrödinger and Hartree equations in space dimension n ≤ 2. Comm. Math. Phys. 151 (1993), 619–645. Google Scholar

[3] [3] Ginibre, J., Ozawa, T. and Velo, G., On the existence of wave operators for a class of nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Phys. Théor. 60 (1994), 211–239. Google Scholar

[4] [4] Hayashi, N., The initial value problem for the derivative nonlinear Schrödinger equation in the energy space. Nonlinear Anal. 20 (1993), 823–833. Google Scholar

[5] [5] Hayashi, N., Kaikina, E. I. and Naumkin, P. I., Large time behavior of solutions to the dissipative nonlinear Schrödinger equation. Proc. Royal Soc. Edinburgh 130 (2000), 1029–1043. Google Scholar

[6] [6] Hayashi, N. and Naumkin, P. I., Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited. Discrete Contin. Dynam. Systems 3 (1997), 383–400. Google Scholar

[7] [7] Hayashi, N. and Naumkin, P. I., Asymptotics in large time of solutions to nonlinear Schrödinger and Hartree equations. Amer. J. Math. 120 (1998), 369–389. Google Scholar

[8] [8] Hayashi, N. and Naumkin, P. I., Remarks on scattering theory and large time asymptotics of solutions to Hartree type equations with a long range potential. SUT J. Math. 34 (1998), 13–24. Google Scholar

[9] [9] Hayashi, N. and Naumkin, P. I., Large time behavior of solutions for derivative cubic nonlinear Schrödinger equations without a self-conjugate property. Funkcial. Ekvac. 42 (1999), 311–324. Google Scholar

[10] [10] Hayashi, N. and Naumkin, P. I., Asymptotics of small solutions to nonlinear Schrödinger equations with cubic nonlinearities. Internat. J. Pure Appl. Math., to appear. Google Scholar

[11] [11] Hayashi, N., Naumkin, P. I. and Uchida, H., Large time behavior of solutions for derivative cubic nonlinear Schrödinger equations. Publ. Res. Inst.Math. Sci. 35 (1999), 501–513. Google Scholar

[12] [12] Hayashi, N. and Ozawa, T., Modified wave operators for the derivative nonlinear Schrödinger equation. Math. Ann. 298 (1994), 557–576. Google Scholar

[13] [13] Hayashi, N. and Ozawa, T., Remarks on nonlinear Schrödinger equations in one space dimension. Differential Integral Equations 7 (1994), 453–461. Google Scholar

[14] [14] Katayama, S. and Tsutsumi, Y., Global existence of solutions for nonlinear Schrödinger equations in one space dimension. Comm. Partial Differential Equations 19 (1994), 1971–1997. Google Scholar

[15] [15] Kaup, D. J. and Newell, A. C., An exact solution for a derivative nonlinear Schrödinger equation. J. Math. Phys. 19 (1978), 789–801. Google Scholar

[16] [16] Mio, W., Ogino, T., Minami, K. and Takeda, S., Modified nonlinear Schrödinger equation for Alfven waves propagating along the magnetic field in cold plasmas. J. Phys. Soc. Japan 41 (1976), 265–271. Google Scholar

[17] [17] Mjølhus, E., On the modulation instability of hydromagnetic waves parallel to the magnetic field parallel to the magnetic field. J. Plasma Phys. 16 (1976), 321–334. Google Scholar

[18] [18] Naumkin, P. I., Cubic derivative nonlinear Schrödinger equations. SUT J. Math. 36 (2000), 9–42. Google Scholar

[19] [19] Ozawa, T., On the nonlinear Schrödinger equations of derivative type. Indiana Univ. Math. J. 45 (1996), 137–163. Google Scholar

[20] [20] Ozawa, T., Long range scattering for nonlinear Schrödinger equations in one space dimension. Comm. Math. Phys. 139 (1991), 479–493. Google Scholar

[21] [21] Shatah, J., Normal forms and quadratic nonlinear Klein-Gordon equations. Comm. Pure Appl. Math. 38 (1985), 685–696. Google Scholar

[22] [22] Tonegawa, S., Global existence for a class of cubic nonlinear Schrödinger equations in one space dimension. Hokkaido Math. J. 30 (2001), 451–473. Google Scholar

[23] [23] Tsutsumi, Y., The null gauge condition and the one dimensional nonlinear Schrödinger equation with cubic nonlinearity. Indiana Univ.Math. J. 44 (1994), 241–254. Google Scholar