MR2829546
Masaya MAEDA and Jun-ichi SEGATA
Masaya MAEDA and Jun-ichi SEGATA
Existence and Stability of Standing Waves of Fourth Order Nonlinear Schrödinger Type Equation Related to Vortex Filament
Funkcialaj Ekvacioj. Serio Internacia
54
2011
1--14
http://fe.math.kobe-u.ac.jp/FE/FullPapers/54-1/54_1.pdf
http://www.ams.org/mathscinet-getitem?mr=MR2829546
In this paper, we study the fourth order nonlinear Schrödinger type equation (4NLS) which is a generalization of the Fukumoto-Moffatt [5] model that arising in the context of the motion of a vortex filament. Firstly, we mention the existence of standing wave solution and the conserved quantities. We next investigate the case that the equation is completely integrable and show that the standing wave obtained in [20] is orbitally stable in Sobolev spaces $H^m$ with $m\in \mathbb{N}$. Further, we show that the completely integrable equation is ill-posed in $H^s$ with $s\in(-1/2,1/2)$ by following Kenig-Ponce-Vega [13].
The fourth order nonlinear Schrödinger type equation, Standing wave, Ill-posedness.
35Q55, 35B35, 35Q51.
54-1
2011
Existence and Stability of Standing Waves of Fourth Order Nonlinear Schrödinger Type Equation Related to Vortex Filament
Masaya MAEDA and Jun-ichi SEGATA
Masaya MAEDA and Jun-ichi SEGATA
1
Cazenave, T.
Semilinear Schrödinger equations
Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI
2003
MR2002047
2
Cazenave, T.; Lions, P.-L.
Orbital stability of standing waves for some nonlinear Schrödinger equations
Comm. Math. Phys.
85
1982
549-561
MR0677997
3
Da Rios, L. S.
On the motion of an unbounded fluid with a vortex filament of any shape
Rend. Circ. Mat. Palermo
22
1906
117-135 (in Italian)
MR
4
Fukumoto, Y.
Asymptotic expansions for the motion of a curved vortex filament and the localized induction hierarchy
Sūrikaisekikenkyūsho Kōkyūroku No. 1225
2001
144-175
MR1875041
5
Fukumoto, Y.; Moffatt, H. K.
Motion and expansion of a viscous vortex ring. Part I. A higher-order asymptotic formula for the velocity
J. Fluid. Mech.
417
2000
1-45
MR1781877
6
Grillakis, M.; Shatah, J.; Strauss, W.
Stability theory of solitary waves in the presence of symmetry, I
J. Funct. Anal.
74
1987
160-197
MR0901236
7
Gidas, B.; Ni, Wei Ming; Nirenberg, L.
Symmetry of positive solutions of nonlinear elliptic equations in ${\bf R}\sp n$
Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London
1981
pp. 369-402
MR0634248
8
Hasimoto, H.
A soliton on a vortex filament
J. Fluid Mech.
51
1972
477-485
9
Hoseini, S. M.; Marchant, T. R.
Solitary wave interaction for a higher-order nonlinear Schrödinger equation
IMA J. Appl. Math.
72
2007
206-222
MR2316466
10
Huo, Z.; Jia, Y.
The Cauchy problem for the fourth-order nonlinear Schrödinger equation related to the vortex filament
J. Differential Equations
214
2005
1-35
MR2143510
11
Huo, Z.; Jia, Y.
A refined well-posedness for the fourth-order nonlinear Schrödinger equation related to the vortex filament
Comm. Partial Differential Equations
32
2007
1493-1510
MR2372477
12
Kavitha, L.; Daniel, M.
Integrability and soliton in a classical one dimensional site dependent biquadratic Heisenberg spin chain and the effect of nonlinear inhomogeneity
J. Phys. A
36
2003
10471-10492
MR2024907
13
Kenig, C. E.; Ponce, G.; Vega, L.
On the ill-posedness of some canonical dispersive equations
Duke Math. J.
106
2001
617-633
MR1813239
14
Kwong, M. K.
Uniqueness of positive solutions of $\Delta u-u+u\sp p=0$ in ${\bf R}\sp n$
Arch. Rational Mech. Anal.
105
1989
243-266
MR0969899
15
Langer, J.; Perline, R.
Poisson geometry of the filament equation
J. Nonlinear Sci.
1
1991
71-93
MR1102831
16
Lions, P.-L.
The concentration-compactness principle in the calculus of variations. The locally compact case, part 1
Ann. Inst. H. Poincaré Anal. Non Linéaire
1
1984
109-145
MR0778970
17
Lions, P.-L.
The concentration-compactness principle in the calculus of variations. The locally compact case, part 2
Ann. Inst. H. Poincaré Anal. Non Linéaire
1
1984
223-283
MR0778974
18
Segata, J.
Well-posedness for the fourth order nonlinear Schrödinger type equation related to the vortex filament
Differential Integral Equations
16
2003
841-864
MR0778974
19
Segata, J.
Remark on well-posedness for the fourth order nonlinear Schrödinger type equation
Proc. Amer. Math. Soc.
132
2004
3559-3568
MR2084077
20
Segata, J., Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation, preprint 2008
MR2629576
21
Shatah, J.; Strauss, W. A.
Instability of nonlinear bound states
Comm. Math. Phys.
100
1985
173-190
MR0804458
22
Zhidkov, P. E.
On an infinite sequence of invariant measures for the cubic nonlinear Schrödinger equation
Int. J. Math. Math. Sci.
28
2001
375-394
MR1893151