MR3468737
Nakao HAYASHI and Pavel I. NAUMKIN
Nakao HAYASHI and Pavel I. NAUMKIN
Scattering Problem for the Supercritical Nonlinear Schrödinger Equation in 1d
Funkcialaj Ekvacioj. Serio Internacia
58
2015
451--470
http://fe.math.kobe-u.ac.jp/FE/FullPapers/58-3/58_451.pdf
http://www.ams.org/mathscinet-getitem?mr=MR3468737
We consider the one dimensional nonlinear Schrödinger equation $iu_{t}+u_{xx}/2=f(u)$, $x\in\mathbf{R}$, $t>0$, $u(0,x)=u_0(x)$, $x\in\mathbf{R}$, with a super critical nonlinearity $f(u)=\sum_{j\neq 0}f_j(u)$, and $f_j(u)$ are such that $f_j(u)=\lambda_j\vert u\vert^{\sigma_j-j}u^j$, where $\lambda_j\in\mathbf{C}$, $\sigma_j>3$. We prove the existence of the scattering operator in the weighted Sobolev spaces.
Scattering operator, Asymptotic behavior in time, Nonlinear Schrödinger, Power nonlinearity.
35Q55, 35B40.
58-451
2015
Scattering Problem for the Supercritical Nonlinear Schrödinger Equation in 1d
Nakao HAYASHI and Pavel I. NAUMKIN
Nakao HAYASHI and Pavel I. NAUMKIN
1
Bergh, J.; Löfström, J.
Interpolation spaces. An introduction
Grundlehren der Mathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York
1976
MR0482275
2
Cazenave, Th.
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
3
Cohn, S.
Resonance and long time existence for the quadratic semilinear Schrödinger equation
Comm. Pure Appl. Math.
45
1992
973-1001
MR1168116
4
Colin, M.; Colin, T.
On a quasilinear Zakharov system describing laser-plasma interactions
Differental Integral Equations
17
2004
297-330
MR2037980
5
Georgiev, V.; Lucente, S.
Decay for nonlinear Klein-Gordon equations
NoDEA Nonlinear Differential Equations Appl.
11
2004
529-555
MR2211299
6
Hayashi, N.; Naumkin, P. I.
Large time behavior of solutions for derivative cubic nonlinear Schrödinger equations without a self-conjugate property
Funkcial. Ekvac.
42
1999
311-324
MR1718755
1718755
7
Hayashi, N.; Naumkin, P. I.
Domain and range of the modified wave operator for Schrödinger equations with a critical nonlinearity
Comm. Math. Phys.
267
2006
477-492
MR2249776
8
Hayashi, N.; Naumkin, P. I.
Scattering Operator for Nonlinear Klein-Gordon Equations
Commun. Contemp. Math.
11
2009
771-781
MR2561936
9
Hayashi, N.; Naumkin, P. I.
Global existence for the cubic nonlinear Schrödinger equation in lower order Sobolev spaces
Differential Integral Equations
24
2011
801-828
MR2850366
10
Hayashi, N.; Li, C.; Naumkin, P. I.
On a system of nonlinear Schrödinger equations in 2d
Differental Integral Equations
24
2011
417-434
MR2809614
11
Hayashi, N.; Li, C.; Ozawa, T.
Small data scattering for a system of nonlinear Schrödinger equations
Differ. Equ. Appl.
3
2011
415-426
MR2856418
12
Hayashi, N.; Ozawa, T.
Scattering theory in the weighted $\mathbf{L}^2({\mathbf{R}}^n)$ spaces for some Schrödinger equations
Ann. Inst. H. Poincaré Phys. Théor.
48
1988
17-37
MR0947158
13
Klainerman, S.
Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions
Comm. Pure Appl. Math.
38
1985
631-641
MR0803252
14
Kosecki, R.
The unit condition and global existence for a class of nonlinear Klein-Gordon equations
J. Differential Equations
100
1992
257-268
MR1194810
15
Shatah, J.
Normal forms and quadratic nonlinear Klein-Gordon equations
Comm. Pure Appl. Math.
38
1985
685-696
MR0803256
16
Stein, E. M.
Singular Integrals and Differentiability Properties of Functions
Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ
1970
MR0290095
17
Strauss, W. A.
Nonlinear scattering theory
Scattering Theory in Mathematical Physics, Reidel, Dordrect
1979
pp. 53-79
18
Strauss, W. A.
Nonlinear scattering theory at low energy
J. Funct. Anal.
41
1981
110-133
MR0614228
19
Sunagawa, H.
On global small amplitude solutions to systems of cubic nonlinear Klein-Gordon equations with different mass in one space dimension
J. Differential Equations
192
2003
308-325
MR1990843