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On the critical nongauge invariant nonlinear Schrödinger equation

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

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

    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.

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

    Citation:

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