Stability and instability of standing waves for 1-dimensional nonlinear Schrödinger equation with multiple-power nonlinearity
Stability and instability of standing waves for 1-dimensional nonlinear Schrödinger equation with multiple-power nonlinearity
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DOI：
10.2996/kmj/1214442798
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发表时间：
2008-06
影响因子：
0.6
通讯作者：
Masaya Maeda
中科院分区：
文献类型：
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作者：
Masaya Maeda
1⁄4 0；Tb 0；x A R；Fin 1？式(1⁄1⁄4 pm J1 Jjuj Pju with aj A R and 1<；p1<；<；pm<；y.等式(1)出现在数学物理的不同领域。(1)的唯一局部存在性是众所周知的。也就是说，对任意u0 A H？R？，存在一个正常数T和唯一的局部解u A C？1⁄20；T？；H？R？？V C？1⁄20；T？；H？R？of(1)，其中u？1⁄4 u0.此外，Uüt？满足两个守恒定律ku？t？KL2 1⁄4 ku0kL2和Eüu？t？1⁄4 E？u0？，其中E？v？：1⁄4 12 kvxkL2-
iut þ uxx þ f ðuÞ 1⁄4 0; tb 0; x A R; ð1Þ where f ðuÞ 1⁄4 Pm j1⁄41 ajjuj pj u with aj A R and 1 < p1 < < pm < y. Equation (1) arises in various regions of mathematical physics. The unique local existence of (1) is well known. That is for any u0 A H ðRÞ, there exists a positive constant T and a unique local solution u A Cð1⁄20;TÞ;H ðRÞÞV Cð1⁄20;TÞ;H ðRÞÞ of (1) with uð0Þ 1⁄4 u0. Furthermore, uðtÞ satisfies the two conservation laws kuðtÞkL2 1⁄4 ku0kL2 and EðuðtÞÞ 1⁄4 Eðu0Þ, where EðvÞ :1⁄4 12 kvxkL2 Ð