Stability Solution of the Nonlinear Schrödinger Equation

ABSTRACT

In this paper we discuss stability theory of the mass critical, mass-supercritical and energy-subcritical of solution to the nonlinear Schrodinger equation. In general, we take care in developing a stability theory for nonlinear Schrodinger equation. By stability, we discuss the property: the approximate solution to nonlinear Schrodinger equation obeying with *e* small in a suitable space and small in and then there exists a veritable solution *u* to nonlinear Schrodinger equation which remains very close to in critical norms.

Cite this paper

M. M-Ali, "Stability Solution of the Nonlinear Schrödinger Equation,"*International Journal of Modern Nonlinear Theory and Application*, Vol. 2 No. 2, 2013, pp. 122-129. doi: 10.4236/ijmnta.2013.22015.

M. M-Ali, "Stability Solution of the Nonlinear Schrödinger Equation,"

References

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[8] T. Tao, M. Visan and X. Zhang, “The Nonlinear Schrodinger Equation with Combined Power-Type Nonlinearities,” Communications in Partial Differential Equations, Vol. 32, No. 7-9, 2007, pp. 1281-1343.

[9] X. Zhang, “On the Cauchy Problem of 3-D Energy-Critical Schrodinger Equations with Sub-Critical Perturbations,” Journal of Differential Equations, Vol. 230, No. 2, 2006, pp. 422-445.

[10] J. Colliander, M. Keel, G. Stafflani, H. Takaoka and T. Tao, “Global Well-Posedness and Scattering in the Energy Space for the Critical Nonlinear Schrodinger Equation in R3,” Analysis of PDEs.

[11] C. Kenig and F. Merle, “Global Well-Posedness, Scattering, and Blowup for the Energy-Critical, Focusing, Non-Linear Schrodinger Equation in the Radial Case,” preprint.

[12] E. Ryckman and M. Visan, “Global Well-Posedness and Scattering for the Defocusing Energy-Critical Nonlinear Schrodinger Equation in R1 + 4,” American Journal of Mathematics.

[13] T. Tao and M. Visan, “Stability of Energy-Critical Nonlinear Schrodinger Equations in High Dimensions,” Electronic Journal of Differential Equations, Vol. 2005, No. 118, 2005, pp. 1-28.

[14] M. Visan, “The Defocusing Energy-Critical Nonlinear Schrodinger Equation in Higher Dimensions,” Duke Mathematical Journal.

[15] T. Tao, M. Visan and X. Zhang, “Minimal-Mass Blowup Solutions of the Mass-Critical NLS,” Forum Mathematicum, Vol. 20, No. 5, 2008, pp. 881-919. doi:10.1515/FORUM.2008.042

[16] D. Foschi, “Inhomogeneous Strichartz Estimates,” Advances in Difference Equations, Vol. 2, No. 1, 2005, pp. 1- 24.

[1] S. Keraani, “On the Blow-Up Phenomenon of the Critical Nonlinear Schrodinger Equation,” Journal of Functional Analysis, Vol. 235, No. 1, 2006, pp. 171-192. doi:10.1016/j.jfa.2005.10.005

[2] R. Killip, M. Visan and X. Zhang, “Energy-Critical NLS with Quadratic Potentials,” Communications in Partial Differential Equations, Vol. 34, No. 12, 2009, pp. 1531-1565.

[3] T. Tao, M. Visan and X. Zhang, “Global Well-Posedness and Scattering for the Mass-Critical Nonlinear Schrodinger Equation for Radial Data in High Dimensions,” Duke Mathematical Journal, Vol. 140, No. 1, 2007, pp. 165-202.

[4] T. Cazenave, “Semilinear Schrodinger Equations,” Courant Lecture Notes in Mathematics, New York University, 2003.

[5] T. Cazenave and F. B. Weissler, “Critical Nonlinear Schrodinger Equation,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 14, No. 10, 1990, pp. 807-836. doi:10.1016/0362-546X(90)90023-A

[6] Y. Tsutsumi, “L2-Solutions for Nonlinear Schrodinger Equations and Nonlinear Groups,” Funkcial Ekvac, Vol. 30, No. 1, 1987, pp. 115-125.

[7] R. Killip and M. Visan, “Nonlinear Schrodinger Equations at Critical Regularity,” Clay Mathematics Proceedings, Vol. 10, 2009.

[8] T. Tao, M. Visan and X. Zhang, “The Nonlinear Schrodinger Equation with Combined Power-Type Nonlinearities,” Communications in Partial Differential Equations, Vol. 32, No. 7-9, 2007, pp. 1281-1343.

[9] X. Zhang, “On the Cauchy Problem of 3-D Energy-Critical Schrodinger Equations with Sub-Critical Perturbations,” Journal of Differential Equations, Vol. 230, No. 2, 2006, pp. 422-445.

[10] J. Colliander, M. Keel, G. Stafflani, H. Takaoka and T. Tao, “Global Well-Posedness and Scattering in the Energy Space for the Critical Nonlinear Schrodinger Equation in R3,” Analysis of PDEs.

[11] C. Kenig and F. Merle, “Global Well-Posedness, Scattering, and Blowup for the Energy-Critical, Focusing, Non-Linear Schrodinger Equation in the Radial Case,” preprint.

[12] E. Ryckman and M. Visan, “Global Well-Posedness and Scattering for the Defocusing Energy-Critical Nonlinear Schrodinger Equation in R1 + 4,” American Journal of Mathematics.

[13] T. Tao and M. Visan, “Stability of Energy-Critical Nonlinear Schrodinger Equations in High Dimensions,” Electronic Journal of Differential Equations, Vol. 2005, No. 118, 2005, pp. 1-28.

[14] M. Visan, “The Defocusing Energy-Critical Nonlinear Schrodinger Equation in Higher Dimensions,” Duke Mathematical Journal.

[15] T. Tao, M. Visan and X. Zhang, “Minimal-Mass Blowup Solutions of the Mass-Critical NLS,” Forum Mathematicum, Vol. 20, No. 5, 2008, pp. 881-919. doi:10.1515/FORUM.2008.042

[16] D. Foschi, “Inhomogeneous Strichartz Estimates,” Advances in Difference Equations, Vol. 2, No. 1, 2005, pp. 1- 24.