August  2011, 30(3): 807-834. doi: 10.3934/dcds.2011.30.807

On the critical nongauge invariant nonlinear Schrödinger equation

1. 

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

Received  January 2010 Revised  January 2011 Published  March 2011

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.

Citation: Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807
References:
[1]

H. Bateman and A. Erdelyi, "Tables of Integral Transforms,", McGraw-Hill Book Co., (1954).   Google Scholar

[2]

Th. Cazenave, "Semilinear Schrödinger Equations,", Courant Institute of Mathematical Sciences, (2003).   Google Scholar

[3]

N. Hayashi and E. I. Kaikina, Local existence of solutions to the Cauchy problem for nonlinear Schrödinger equations,, SUT J. Math., 34 (1998), 111.   Google Scholar

[4]

N. Hayashi and E. I. Kaikina, "Nonlinear Theory of Pseudodifferential Equations on a Half-line,", North-Holland Mathematics Studies, (2004).   Google Scholar

[5]

N. Hayashi and P. I. Naumkin, Large time behavior of solutions for derivative cubic nonlinear Schrödinger equations without a self-conjugate property,, Funkcialaj Ekvacioj, 42 (1999), 311.   Google Scholar

[6]

N. Hayashi and P. I. Naumkin, Asymptotics of small solutions to nonlinear Schrödinger equation with cubic nonlinearities,, International Journal of Pure and Applied Mathematics, 3 (2002), 255.   Google Scholar

[7]

N. Hayashi and P. I. Naumkin, Large time behavior for the cubic nonlinear Schrödinger equation,, Canadian Journal of Mathematics, 54 (2002), 1065.  doi: 10.4153/CJM-2002-039-3.  Google Scholar

[8]

N. Hayashi and P. I. Naumkin, On the asymptotics for cubic nonlinear Schrödinger equations,, Complex Var. Theory Appl., 49 (2004), 339.   Google Scholar

[9]

N. Hayashi and P. I. Naumkin, Nongauge invariant cubic nonlinear Schrödinger equations,, Pac. J. Appl. Math., 1 (2008), 1.   Google Scholar

[10]

N. Hayashi, P. I. Naumkin, A. Shimomura and S. Tonegawa, Modified Wave Operators for Nonlinear Schrödinger Equations in 1d or 2d,, Electronic Journal of Differential Equations, (2004), 1.   Google Scholar

[11]

N. Hayashi and T. Ozawa, Scattering theory in the weighted $\mathbfL^{2}(R^n)$spaces for some Schrödinger equations,, Ann. I.H.P. (Phys. Théor.), 48 (1988), 17.   Google Scholar

[12]

N. Hayashi and T. Ozawa, Modified wave operators for the derivative nonlinear Schrödinger equation,, Math. Ann., 298 (1994), 557.  doi: 10.1007/BF01459751.  Google Scholar

[13]

T. Ozawa, Long range scattering for nonlinear Schrödinger equations in one space dimension,, Commun. Math. Phys., 139 (1991), 479.  doi: 10.1007/BF02101876.  Google Scholar

[14]

J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations,, Commun. Pure Appl. Math., 38 (1985), 685.  doi: 10.1002/cpa.3160380516.  Google Scholar

[15]

S. Tonegawa, Global existence for a class of cubic nonlinear Schrödinger equations in one space dimension,, Hokkaido Math. J., 30 (2001), 451.   Google Scholar

show all references

References:
[1]

H. Bateman and A. Erdelyi, "Tables of Integral Transforms,", McGraw-Hill Book Co., (1954).   Google Scholar

[2]

Th. Cazenave, "Semilinear Schrödinger Equations,", Courant Institute of Mathematical Sciences, (2003).   Google Scholar

[3]

N. Hayashi and E. I. Kaikina, Local existence of solutions to the Cauchy problem for nonlinear Schrödinger equations,, SUT J. Math., 34 (1998), 111.   Google Scholar

[4]

N. Hayashi and E. I. Kaikina, "Nonlinear Theory of Pseudodifferential Equations on a Half-line,", North-Holland Mathematics Studies, (2004).   Google Scholar

[5]

N. Hayashi and P. I. Naumkin, Large time behavior of solutions for derivative cubic nonlinear Schrödinger equations without a self-conjugate property,, Funkcialaj Ekvacioj, 42 (1999), 311.   Google Scholar

[6]

N. Hayashi and P. I. Naumkin, Asymptotics of small solutions to nonlinear Schrödinger equation with cubic nonlinearities,, International Journal of Pure and Applied Mathematics, 3 (2002), 255.   Google Scholar

[7]

N. Hayashi and P. I. Naumkin, Large time behavior for the cubic nonlinear Schrödinger equation,, Canadian Journal of Mathematics, 54 (2002), 1065.  doi: 10.4153/CJM-2002-039-3.  Google Scholar

[8]

N. Hayashi and P. I. Naumkin, On the asymptotics for cubic nonlinear Schrödinger equations,, Complex Var. Theory Appl., 49 (2004), 339.   Google Scholar

[9]

N. Hayashi and P. I. Naumkin, Nongauge invariant cubic nonlinear Schrödinger equations,, Pac. J. Appl. Math., 1 (2008), 1.   Google Scholar

[10]

N. Hayashi, P. I. Naumkin, A. Shimomura and S. Tonegawa, Modified Wave Operators for Nonlinear Schrödinger Equations in 1d or 2d,, Electronic Journal of Differential Equations, (2004), 1.   Google Scholar

[11]

N. Hayashi and T. Ozawa, Scattering theory in the weighted $\mathbfL^{2}(R^n)$spaces for some Schrödinger equations,, Ann. I.H.P. (Phys. Théor.), 48 (1988), 17.   Google Scholar

[12]

N. Hayashi and T. Ozawa, Modified wave operators for the derivative nonlinear Schrödinger equation,, Math. Ann., 298 (1994), 557.  doi: 10.1007/BF01459751.  Google Scholar

[13]

T. Ozawa, Long range scattering for nonlinear Schrödinger equations in one space dimension,, Commun. Math. Phys., 139 (1991), 479.  doi: 10.1007/BF02101876.  Google Scholar

[14]

J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations,, Commun. Pure Appl. Math., 38 (1985), 685.  doi: 10.1002/cpa.3160380516.  Google Scholar

[15]

S. Tonegawa, Global existence for a class of cubic nonlinear Schrödinger equations in one space dimension,, Hokkaido Math. J., 30 (2001), 451.   Google Scholar

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