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Stability of standing waves for a class of quasilinear Schrödinger equations

Published online by Cambridge University Press:  24 May 2012

JIANQING CHEN ,
YONGQING LI  and
ZHI-QIANG WANG
JIANQING CHEN
Affiliation:
School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, P. R. China email: jqchen@fjnu.edu.cn
YONGQING LI
Affiliation:
School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, P. R. China email: jqchen@fjnu.edu.cn
ZHI-QIANG WANG
Affiliation:
School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, P. R. China email: jqchen@fjnu.edu.cn Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA
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Abstract

This paper is concerned with the stability and instability of standing waves for the quasilinear Schrödinger equation of the form which has been derived in many models from mathematical physics. We find the exact threshold depending upon the interplay of quasilinear and nonlinear terms that separates stability and instability. More precisely, we prove that for α ∈ and odd p ∈ , when , the standing wave is stable, and when (where for N ≥ 3 and 2 α ċ 2* = +∞ for N = 2), the standing wave is strongly unstable. Our results show that the quasilinear term 2 α(△|φ|2α)|φ|2α−2φ makes the standing waves more stable, which is consistent with the physical phenomena.

Keywords

Quasilinear Schrödinger equationsStanding wavesStabilityInstability
Type
Papers
Information
European Journal of Applied Mathematics , Volume 23 , Issue 5 , October 2012 , pp. 611 - 633
Copyright
Copyright © Cambridge University Press 2012

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