    July  1997, 3(3): 383-400. doi: 10.3934/dcds.1997.3.383

## Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited

 1 Department of Applied Mathematics, Science University of Tokyo, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162 2 Instituto de Física y Matemáticas, Universidad Michoacana, AP 2-82, CP 58040, Morelia, Michoacana

Received  September 1996 Published  April 1997

We continue to study the asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation

$i u_t + u_{x x} + ia(|u|^2u)_x = 0, \quad (t,x) \in \mathbf{R}\times \mathbf{R},$

$u(0,x) = u_0 (x), \quad x\in \mathbf{R},\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$(DNLS)

where $a \in \mathbf{R}$. We prove that if $||u_0||_{ H^{1,\gamma}} + ||u_0||_{ H^{1+\gamma,0}}$ is sufficiently small with $\gamma > 1/2$, then the solution of (DNLS) satisfies the time decay estimate

$||u(t)||_{L^\infty} + ||u_x(t)||_{L^\infty}\le C(1+|t|)^{-1/2},$

where $H^{m,s}= \{f\in \mathcal{S}'; ||f||_{m,s}= ||(1+|x|^2)^{s/2}(1-\partial_x^2)^{m/2}f||_{L^2} < \infty\}$, $m,s\in \mathbf{R}$. In the previous paper [4,Theorem 1.1] we showed the same result under the condition that $\gamma \ge 2$. Furthermore we show the asymptotic behavior in time of solutions involving the previous result [4,Theorem 1.2].

Citation: Nakao Hayashi, Pavel I. Naumkin. Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 383-400. doi: 10.3934/dcds.1997.3.383
  Thierry Cazenave, Zheng Han. Asymptotic behavior for a Schrödinger equation with nonlinear subcritical dissipation. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4801-4819. doi: 10.3934/dcds.2020202  Jean-Claude Saut, Jun-Ichi Segata. Asymptotic behavior in time of solution to the nonlinear Schrödinger equation with higher order anisotropic dispersion. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 219-239. doi: 10.3934/dcds.2019009  Alexander Komech, Elena Kopylova, David Stuart. On asymptotic stability of solitons in a nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1063-1079. doi: 10.3934/cpaa.2012.11.1063  Haoyue Cui, Dongyi Liu, Genqi Xu. Asymptotic behavior of a Schrödinger equation under a constrained boundary feedback. Mathematical Control and Related Fields, 2018, 8 (2) : 383-395. doi: 10.3934/mcrf.2018015  Nakao Hayashi, Elena I. Kaikina, Pavel I. Naumkin. Large time behavior of solutions to the generalized derivative nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 93-106. doi: 10.3934/dcds.1999.5.93  Kazuhiro Kurata, Tatsuya Watanabe. A remark on asymptotic profiles of radial solutions with a vortex to a nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2006, 5 (3) : 597-610. doi: 10.3934/cpaa.2006.5.597  Tetsu Mizumachi, Dmitry Pelinovsky. On the asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (5) : 971-987. doi: 10.3934/dcdss.2012.5.971  Xiaoyan Lin, Yubo He, Xianhua Tang. Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1547-1565. doi: 10.3934/cpaa.2019074  D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563  Ahmed Y. Abdallah. Asymptotic behavior of the Klein-Gordon-Schrödinger lattice dynamical systems. Communications on Pure and Applied Analysis, 2006, 5 (1) : 55-69. doi: 10.3934/cpaa.2006.5.55  Yongqin Liu. Asymptotic behavior of solutions to a nonlinear plate equation with memory. Communications on Pure and Applied Analysis, 2017, 16 (2) : 533-556. doi: 10.3934/cpaa.2017027  Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807  Tarek Saanouni. Remarks on the damped nonlinear Schrödinger equation. Evolution Equations and Control Theory, 2020, 9 (3) : 721-732. doi: 10.3934/eect.2020030  Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003  Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete and Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109  Hongwei Wang, Amin Esfahani. On the Cauchy problem for a nonlocal nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022039  Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450  Xiaoyu Zeng. Asymptotic properties of standing waves for mass subcritical nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1749-1762. doi: 10.3934/dcds.2017073  Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316  Mohamad Darwich. On the $L^2$-critical nonlinear Schrödinger Equation with a nonlinear damping. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2377-2394. doi: 10.3934/cpaa.2014.13.2377

2021 Impact Factor: 1.588