Abstract
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 H2per (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). Fourth order nonlinear Schrödinger type equation, stability of standing waves.
| Original language | English |
|---|---|
| Pages (from-to) | 843-859 |
| Number of pages | 17 |
| Journal | Communications on Pure and Applied Analysis |
| Volume | 14 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - May 1 2015 |
| Externally published | Yes |
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All Science Journal Classification (ASJC) codes
- Analysis
- Applied Mathematics
Cite this
Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves. / Segata, Junichi.
In: Communications on Pure and Applied Analysis, Vol. 14, No. 3, 01.05.2015, p. 843-859.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves
AU - Segata, Junichi
PY - 2015/5/1
Y1 - 2015/5/1
N2 - 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 H2per (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). Fourth order nonlinear Schrödinger type equation, stability of standing waves.
AB - 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 H2per (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). Fourth order nonlinear Schrödinger type equation, stability of standing waves.
UR - http://www.scopus.com/inward/record.url?scp=84924013185&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84924013185&partnerID=8YFLogxK
U2 - 10.3934/cpaa.2015.14.843
DO - 10.3934/cpaa.2015.14.843
M3 - Article
AN - SCOPUS:84924013185
VL - 14
SP - 843
EP - 859
JO - Communications on Pure and Applied Analysis
JF - Communications on Pure and Applied Analysis
SN - 1534-0392
IS - 3
ER -