# American Institute of Mathematical Sciences

May  2015, 14(3): 843-859. doi: 10.3934/cpaa.2015.14.843

## Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves

 1 Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan

Received  April 2014 Revised  December 2014 Published  March 2015

We consider the fourth order nonlinear Schrödinger type equation (4NLS) which arises in context of the motion of vortex filament. The purposes of this paper are twofold. Firstly, we consider the initial value problem for (4NLS) under the periodic boundary condition. By refining the modified energy method used in our previous paper [23], we prove the unique existence of the global solution for (4NLS) in the energy space $H_{p e r}^2(0,2L)$ with $L>0$. Secondly, we study the stability property of periodic standing waves for (4NLS). Using the spectrum properties of the Schrödinger operators associated to the periodic standing wave developed by Angulo [1], we prove that standing wave of dnoidal type is orbitally stable under the time evolution by (4NLS).
Citation: Jun-ichi Segata. Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves. Communications on Pure & Applied Analysis, 2015, 14 (3) : 843-859. doi: 10.3934/cpaa.2015.14.843
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##### References:
 [1] Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247 [2] Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020284 [3] Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450 [4] Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316 [5] Jason Murphy, Kenji Nakanishi. Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1507-1517. doi: 10.3934/dcds.2020328 [6] Xiaorui Wang, Genqi Xu, Hao Chen. Uniform stabilization of 1-D Schrödinger equation with internal difference-type control. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021022 [7] José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376 [8] Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276 [9] Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259 [10] Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264 [11] Riadh Chteoui, Abdulrahman F. Aljohani, Anouar Ben Mabrouk. Classification and simulation of chaotic behaviour of the solutions of a mixed nonlinear Schrödinger system. Electronic Research Archive, , () : -. doi: 10.3934/era.2021002 [12] Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $L^2$-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298 [13] Jonathan J. Wylie, Robert M. Miura, Huaxiong Huang. Systems of coupled diffusion equations with degenerate nonlinear source terms: Linear stability and traveling waves. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 561-569. doi: 10.3934/dcds.2009.23.561 [14] Jose Anderson Cardoso, Patricio Cerda, Denilson Pereira, Pedro Ubilla. Schrödinger Equations with vanishing potentials involving Brezis-Kamin type problems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020392 [15] Li Cai, Fubao Zhang. The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020125 [16] Norman Noguera, Ademir Pastor. Scattering of radial solutions for quadratic-type Schrödinger systems in dimension five. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021018 [17] Patrick Martinez, Judith Vancostenoble. Lipschitz stability for the growth rate coefficients in a nonlinear Fisher-KPP equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 695-721. doi: 10.3934/dcdss.2020362 [18] Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436 [19] Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456 [20] Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260

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