# American Institute of Mathematical Sciences

June  2015, 4(2): 233-240. doi: 10.3934/eect.2015.4.233

## Relating systems properties of the wave and the Schrödinger equation

 1 Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, Netherlands 2 FEMTO-ST AS2M, Université de Franche Comté/CNRS/ENSMM/UTBM, 24 rue Alain Savary, 25000 Besançon, France, France

Received  May 2014 Revised  September 2014 Published  May 2015

In this article we show that systems properties of the systems governed by the second order differential equation $\frac{d^{2}w}{dt^{2}}=-A_{0}w$ and the first order differential equation $\frac{dz}{dt}=iA_{0}z$ are related. This can be used to show that, for instance, exact observability of the $N$-dimensional wave equation implies the similar property for the $N$-dimensional Schrödinger equation.
Citation: Hans Zwart, Yann Le Gorrec, Bernhard Maschke. Relating systems properties of the wave and the Schrödinger equation. Evolution Equations & Control Theory, 2015, 4 (2) : 233-240. doi: 10.3934/eect.2015.4.233
##### References:

show all references

##### References:
 [1] Lassaad Aloui, Slim Tayachi. Local well-posedness for the inhomogeneous nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5409-5437. doi: 10.3934/dcds.2021082 [2] Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934/cpaa.2018027 [3] Takafumi Akahori. Low regularity global well-posedness for the nonlinear Schrödinger equation on closed manifolds. Communications on Pure & Applied Analysis, 2010, 9 (2) : 261-280. doi: 10.3934/cpaa.2010.9.261 [4] Seckin Demirbas. Local well-posedness for 2-D Schrödinger equation on irrational tori and bounds on Sobolev norms. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1517-1530. doi: 10.3934/cpaa.2017072 [5] Nobu Kishimoto. Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1123-1143. doi: 10.3934/cpaa.2008.7.1123 [6] Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for a periodic nonlinear Schrödinger equation in 1D and 2D. Discrete & Continuous Dynamical Systems, 2007, 19 (1) : 37-65. doi: 10.3934/dcds.2007.19.37 [7] Zihua Guo, Yifei Wu. Global well-posedness for the derivative nonlinear Schrödinger equation in $H^{\frac 12} (\mathbb{R} )$. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 257-264. doi: 10.3934/dcds.2017010 [8] Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for the $L^2$ critical nonlinear Schrödinger equation in higher dimensions. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1023-1041. doi: 10.3934/cpaa.2007.6.1023 [9] Jun-ichi Segata. Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 1093-1105. doi: 10.3934/dcds.2010.27.1093 [10] Xuan Liu, Ting Zhang. Local well-posedness and finite time blowup for fourth-order Schrödinger equation with complex coefficient. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021156 [11] Boling Guo, Jun Wu. Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021205 [12] Kelin Li, Huafei Di. On the well-posedness and stability for the fourth-order Schrödinger equation with nonlinear derivative term. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021122 [13] Massimo Cicognani, Michael Reissig. Well-posedness for degenerate Schrödinger equations. Evolution Equations & Control Theory, 2014, 3 (1) : 15-33. doi: 10.3934/eect.2014.3.15 [14] Benjamin Dodson. Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when $n = 3$ via a linear-nonlinear decomposition. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1905-1926. doi: 10.3934/dcds.2013.33.1905 [15] Shaoming Guo, Xianfeng Ren, Baoxiang Wang. Local well-posedness for the derivative nonlinear Schrödinger equation with $L^2$-subcritical data. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4207-4253. doi: 10.3934/dcds.2021034 [16] Abdelmouhcene Sengouga. Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints. Evolution Equations & Control Theory, 2020, 9 (1) : 1-25. doi: 10.3934/eect.2020014 [17] Tarek Saanouni. Global well-posedness of some high-order semilinear wave and Schrödinger type equations with exponential nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (1) : 273-291. doi: 10.3934/cpaa.2014.13.273 [18] Ademir Pastor. On three-wave interaction Schrödinger systems with quadratic nonlinearities: Global well-posedness and standing waves. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2217-2242. doi: 10.3934/cpaa.2019100 [19] Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007 [20] Lin Shen, Shu Wang, Yongxin Wang. The well-posedness and regularity of a rotating blades equation. Electronic Research Archive, 2020, 28 (2) : 691-719. doi: 10.3934/era.2020036

2020 Impact Factor: 1.081