# American Institute of Mathematical Sciences

## Galerkin method of weakly damped cubic nonlinear Schrödinger with Dirac impurity, and artificial boundary condition in a half-line

 Department of Mathematics, Faculty of Science and Technology, Cadi Ayyad University, B.P. 549, Av. Abdelkarim Elkhattabi, Guéliz, Marrakesh, 40000, Morocco

* Corresponding author: Abderrazak Chrifi (abderrazak.chrifi@gmail.com)

Received  August 2020 Revised  January 2021 Early access  March 2021

We consider a weakly damped cubic nonlinear Schrödinger equation with Dirac interaction defect in a half line of $\mathbb{R}$. Endowed with artificial boundary condition at the point $x = 0$, we discuss the global existence and uniqueness of solution of this equation by using Faedo–Galerkin method.

Citation: Abderrazak Chrifi, Mostafa Abounouh, Hassan Al Moatassime. Galerkin method of weakly damped cubic nonlinear Schrödinger with Dirac impurity, and artificial boundary condition in a half-line. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021030
##### References:

show all references

##### References:
 [1] Vyacheslav A. Trofimov, Evgeny M. Trykin. A new way for decreasing of amplitude of wave reflected from artificial boundary condition for 1D nonlinear Schrödinger equation. Conference Publications, 2015, 2015 (special) : 1070-1078. doi: 10.3934/proc.2015.1070 [2] Minoru Murai, Kunimochi Sakamoto, Shoji Yotsutani. Representation formula for traveling waves to a derivative nonlinear Schrödinger equation with the periodic boundary condition. Conference Publications, 2015, 2015 (special) : 878-900. doi: 10.3934/proc.2015.0878 [3] Phan Van Tin. On the Cauchy problem for a derivative nonlinear Schrödinger equation with nonvanishing boundary conditions. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021028 [4] Razvan Mosincat, Haewon Yoon. Unconditional uniqueness for the derivative nonlinear Schrödinger equation on the real line. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 47-80. doi: 10.3934/dcds.2020003 [5] Meina Gao, Jianjun Liu. Quasi-periodic solutions for derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2101-2123. doi: 10.3934/dcds.2012.32.2101 [6] Nakao Hayashi, Elena I. Kaikina, Pavel I. Naumkin. Large time behavior of solutions to the generalized derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 93-106. doi: 10.3934/dcds.1999.5.93 [7] Hiroyuki Hirayama, Mamoru Okamoto. Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6943-6974. doi: 10.3934/dcds.2016102 [8] Kazumasa Fujiwara, Tohru Ozawa. On the lifespan of strong solutions to the periodic derivative nonlinear Schrödinger equation. Evolution Equations & Control Theory, 2018, 7 (2) : 275-280. doi: 10.3934/eect.2018013 [9] Nakao Hayashi, Pavel I. Naumkin. Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited. Discrete & Continuous Dynamical Systems, 1997, 3 (3) : 383-400. doi: 10.3934/dcds.1997.3.383 [10] Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1447-1478. doi: 10.3934/cpaa.2021028 [11] Masoumeh Hosseininia, Mohammad Hossein Heydari, Carlo Cattani. A wavelet method for nonlinear variable-order time fractional 2D Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2273-2295. doi: 10.3934/dcdss.2020295 [12] Christina A. Hollon, Jeffrey T. Neugebauer. Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition. Conference Publications, 2015, 2015 (special) : 615-620. doi: 10.3934/proc.2015.0615 [13] Nakao Hayashi, Pavel Naumkin. On the reduction of the modified Benjamin-Ono equation to the cubic derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 237-255. doi: 10.3934/dcds.2002.8.237 [14] Wulong Liu, Guowei Dai. Multiple solutions for a fractional nonlinear Schrödinger equation with local potential. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2105-2123. doi: 10.3934/cpaa.2017104 [15] Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107 [16] Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395 [17] Van Duong Dinh, Binhua Feng. On fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4565-4612. doi: 10.3934/dcds.2019188 [18] Songbai Peng, Aliang Xia. Normalized solutions of supercritical nonlinear fractional Schrödinger equation with potential. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021128 [19] Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3589-3610. doi: 10.3934/dcdss.2021021 [20] Chao Ji. Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6071-6089. doi: 10.3934/dcdsb.2019131

2020 Impact Factor: 2.425