-
Previous Article
Model distortions in Bayesian MAP reconstruction
- IPI Home
- This Issue
-
Next Article
Approximation errors and truncation of computational domains with application to geophysical tomography
Modified wave operator for Schrodinger type equations with subcritical dissipative nonlinearities
| 1. | Department of Mathematics, Graduate School of Science, Osaka University, Osaka Toyonaka 560-0043 |
| 2. | Instituto de Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Michoacán, Mexico |
$u_{t}-\frac{i}{\beta }\| partial _{x} |^{\beta }u=i\lambda \ |u| ^{\rho -1}u,$
for $\( t,x ) \in \mathbf{R}\times \mathbf{R.}$ We find the solutions in the neighborhood of suitable approximate solutions provided that $\beta \geq 2$, $\Im \lambda >0$ and $\rho <3$ is sufficiently close to $3.$ Also we prove the time decay estimate of solutions
$\ ||u ( t )| |\ _{\mathbf{L}^{2}}\leq Ct^{\frac{1}{2} -\frac{1}{\rho -1}}.$
When we prove the existence of a modified scattering operator, then a natural inverse problem arises to reconstruct the parameters $\beta ,\lambda $ and $\rho $ from the modified scattering operator.
| [1] |
Türker Özsarı. Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities. Communications on Pure and Applied Analysis, 2019, 18 (1) : 539-558. doi: 10.3934/cpaa.2019027 |
| [2] |
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 |
| [3] |
Juan Belmonte-Beitia, Víctor M. Pérez-García, Vadym Vekslerchik, Pedro J. Torres. Lie symmetries, qualitative analysis and exact solutions of nonlinear Schrödinger equations with inhomogeneous nonlinearities. Discrete and Continuous Dynamical Systems - B, 2008, 9 (2) : 221-233. doi: 10.3934/dcdsb.2008.9.221 |
| [4] |
Guan Huang. An averaging theorem for nonlinear Schrödinger equations with small nonlinearities. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3555-3574. doi: 10.3934/dcds.2014.34.3555 |
| [5] |
Filippo Dell'Oro. Global attractors for strongly damped wave equations with subcritical-critical nonlinearities. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1015-1027. doi: 10.3934/cpaa.2013.12.1015 |
| [6] |
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 |
| [7] |
Akihiro Shimomura. Modified wave operators for the coupled wave-Schrödinger equations in three space dimensions. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1571-1586. doi: 10.3934/dcds.2003.9.1571 |
| [8] |
Jaeyoung Byeon, Ohsang Kwon, Yoshihito Oshita. Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2015, 14 (3) : 825-842. doi: 10.3934/cpaa.2015.14.825 |
| [9] |
Binhua Feng. On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1785-1804. doi: 10.3934/cpaa.2018085 |
| [10] |
Bartosz Bieganowski, Jaros law Mederski. Nonlinear SchrÖdinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities. Communications on Pure and Applied Analysis, 2018, 17 (1) : 143-161. doi: 10.3934/cpaa.2018009 |
| [11] |
Tadahiro Oh, Mamoru Okamoto, Oana Pocovnicu. On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3479-3520. doi: 10.3934/dcds.2019144 |
| [12] |
Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1447-1478. doi: 10.3934/cpaa.2021028 |
| [13] |
Shuangjie Peng, Huirong Pi. Spike vector solutions for some coupled nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2205-2227. doi: 10.3934/dcds.2016.36.2205 |
| [14] |
Liping Wang, Chunyi Zhao. Infinitely many solutions for nonlinear Schrödinger equations with slow decaying of potential. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1707-1731. doi: 10.3934/dcds.2017071 |
| [15] |
Juncheng Wei, Wei Yao. Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1003-1011. doi: 10.3934/cpaa.2012.11.1003 |
| [16] |
Chuangye Liu, Rushun Tian. Normalized solutions for 3-coupled nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5115-5130. doi: 10.3934/cpaa.2020229 |
| [17] |
Seunghyeok Kim. On vector solutions for coupled nonlinear Schrödinger equations with critical exponents. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1259-1277. doi: 10.3934/cpaa.2013.12.1259 |
| [18] |
Xing Cheng, Ze Li, Lifeng Zhao. Scattering of solutions to the nonlinear Schrödinger equations with regular potentials. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 2999-3023. doi: 10.3934/dcds.2017129 |
| [19] |
Laurent Di Menza, Olivier Goubet. Stabilizing blow up solutions to nonlinear schrÖdinger equations. Communications on Pure and Applied Analysis, 2017, 16 (3) : 1059-1082. doi: 10.3934/cpaa.2017051 |
| [20] |
Zuji Guo. Nodal solutions for nonlinear Schrödinger equations with decaying potential. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1125-1138. doi: 10.3934/cpaa.2016.15.1125 |
2020 Impact Factor: 1.639
Tools
Metrics
Other articles
by authors
[Back to Top]