On the critical nongauge invariant nonlinear Schrödinger equation

Pavel I. Naumkin; Isahi Sánchez-Suárez

doi:10.3934/dcds.2011.30.807

Article Contents

2011, Volume 30, Issue 3: 807-834. Doi: 10.3934/dcds.2011.30.807

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On the critical nongauge invariant nonlinear Schrödinger equation

Pavel I. Naumkin^1, and
Isahi Sánchez-Suárez^1,

1.
Instituto de Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Michoacán

Received: January 2010

Revised: January 2011

Published: September 2011

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Abstract

We consider the Cauchy problem for the critical nongauge invariant nonlinear Schrödinger equations

$iu_{t}+\frac{1}{2}$u_xx$=i\mu\overline{u}^{\alpha}u^{\beta},\text{ } x\in\mathbf{R},\text{ }t>0,$
$\ \ \ \ \ \ \ \ u(0,x) =u_{0}(x) ,\text{ }x\in\mathbf{R,} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$

where $\beta>\alpha\geq0,$ $\alpha+\beta\geq2,$ $\mu=-i^{\frac{\omega}{2} }t^{\frac{\theta}{2}-1},$ $\omega=\beta-\alpha-1,$ $\theta=\alpha+\beta-1.$ We prove that there exists a unique solution $u\in\mathbf{C}( [ 0,\infty) ;\mathbf{H}^{1}\cap\mathbf{H}^{0,1}) $ of the Cauchy problem (1). Also we find the large time asymptotics of solutions.

Keywords:

Asymptotics of solutions,
Nonlinear Schrödinger equations.

Mathematics Subject Classification: Primary: 35B40; 35Q55; Secondary: 35Q35.

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