# American Institute of Mathematical Sciences

January  2002, 8(1): 237-255. doi: 10.3934/dcds.2002.8.237

## On the reduction of the modified Benjamin-Ono equation to the cubic derivative nonlinear Schrödinger equation

 1 Department of Mathematics, Graduate School of Science, Osaka University, Osaka Toyonaka 560-0043, Japan 2 Instituto de Física y Matemáticas, Universidad Michoacana, AP 2-82, Morelia, CP 58040, Michoacán, Mexico

Received  February 2001 Revised  July 2001 Published  October 2001

We study the asymptotic behavior for large time of small solutions to theCauchy problem for the modified Benjamin-Ono equation: $u_t + (u^3)_x + \mathcal H u_{x x} = 0$,where $\mathcal H$ is the Hilbert transformation, $x, t\in \mathbf R$. We investigate the reduction ofthe modified Benjamin-Ono equation to the cubic derivative nonlinear Shrödingerequation and then apply techniques developed in [11] - [14] to the resulting cubicnonlocal nonlinear Schrödinger equation. Our method is simpler than that usedin [10] because we can use the factorization of the free Schrödinger group. Ourpurpose in this paper is to show that solutions have the same $L^infty$ time decay rateas in the corresponding linear Benjamin-Ono equation and to prove the existence ofmodified scattering states, when the initial data are sufficiently small in the weightedSobolev spaces $\mathbf H^{2,0} \cap \mathbf H^{1,1}$, where $\mathbf H^{m, s} = \{ \phi\in S' : ||\phi||_{m,s} =||(1 + x^2)^{s/2}(1-\partial^2_x)^{m/2}\phi||_{\mathbf L^2}<\infty \}, m, s\in \mathbf R$. This is an improvement of the previous result [10],where we considered small initial data from the space $\mathbf H^{3,0}\cap \mathbf H^{1,2}$. Our method isbased on a certain gauge transformation and an appropriate phase function.
Citation: Nakao Hayashi, Pavel Naumkin. On the reduction of the modified Benjamin-Ono equation to the cubic derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 237-255. doi: 10.3934/dcds.2002.8.237
 [1] Xiaoyu Zeng, Yimin Zhang. Asymptotic behaviors of ground states for a modified Gross-Pitaevskii equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5263-5273. doi: 10.3934/dcds.2019214 [2] Satoshi Masaki, Jun-ichi Segata. Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1595-1611. doi: 10.3934/cpaa.2018076 [3] Nakao Hayashi, Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the modified witham equation. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1407-1448. doi: 10.3934/cpaa.2018069 [4] Bo-Qing Dong, Juan Song. Global regularity and asymptotic behavior of modified Navier-Stokes equations with fractional dissipation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 57-79. doi: 10.3934/dcds.2012.32.57 [5] G. Deugoué, T. Tachim Medjo. The Stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2593-2621. doi: 10.3934/cpaa.2018123 [6] Stephen Anco, Daniel Kraus. Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4449-4465. doi: 10.3934/dcds.2018194 [7] Jerry L. Bona, Stéphane Vento, Fred B. Weissler. Singularity formation and blowup of complex-valued solutions of the modified KdV equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4811-4840. doi: 10.3934/dcds.2013.33.4811 [8] Yong-Yong Li, Yan-Fang Xue, Chun-Lei Tang. Ground state solutions for asymptotically periodic modified Schr$\ddot{\mbox{o}}$dinger-Poisson system involving critical exponent. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2299-2324. doi: 10.3934/cpaa.2019104 [9] Per Christian Moan, Jitse Niesen. On an asymptotic method for computing the modified energy for symplectic methods. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1105-1120. doi: 10.3934/dcds.2014.34.1105 [10] Minkyu Kwak, Kyong Yu. The asymptotic behavior of solutions of a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 1996, 2 (4) : 483-496. doi: 10.3934/dcds.1996.2.483 [11] Yongqin Liu. Asymptotic behavior of solutions to a nonlinear plate equation with memory. Communications on Pure & Applied Analysis, 2017, 16 (2) : 533-556. doi: 10.3934/cpaa.2017027 [12] Shota Sato, Eiji Yanagida. Asymptotic behavior of singular solutions for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 4027-4043. doi: 10.3934/dcds.2012.32.4027 [13] Darya V. Verveyko, Andrey Yu. Verisokin. Application of He's method to the modified Rayleigh equation. Conference Publications, 2011, 2011 (Special) : 1423-1431. doi: 10.3934/proc.2011.2011.1423 [14] Yongsheng Mi, Chunlai Mu, Pan Zheng. On the Cauchy problem of the modified Hunter-Saxton equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2047-2072. doi: 10.3934/dcdss.2016084 [15] Yu Gao, Jian-Guo Liu. The modified Camassa-Holm equation in Lagrangian coordinates. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2545-2592. doi: 10.3934/dcdsb.2018067 [16] Min Zhu, Shuanghu Zhang. Blow-up of solutions to the periodic modified Camassa-Holm equation with varying linear dispersion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7235-7256. doi: 10.3934/dcds.2016115 [17] Min Zhu, Ying Wang. Blow-up of solutions to the periodic generalized modified Camassa-Holm equation with varying linear dispersion. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 645-661. doi: 10.3934/dcds.2017027 [18] Min Zhu, Shuanghu Zhang. On the blow-up of solutions to the periodic modified integrable Camassa--Holm equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2347-2364. doi: 10.3934/dcds.2016.36.2347 [19] Wen Tan, Bo-Qing Dong, Zhi-Min Chen. Large-time regular solutions to the modified quasi-geostrophic equation in Besov spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3749-3765. doi: 10.3934/dcds.2019152 [20] Zengji Du, Zhaosheng Feng. Existence and asymptotic behaviors of traveling waves of a modified vector-disease model. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1899-1920. doi: 10.3934/cpaa.2018090

2018 Impact Factor: 1.143